Two or more atoms that bond together form a(n)

In ionic bonding, electrons are completely transferred from one atom to another. In the process of either losing or gaining negatively charged electrons, the reacting atoms form ions. The oppositely charged ions are attracted to each other by electrostatic forces, which are the basis of the ionic bond.

For example, during the reaction of sodium with chlorine:

sodium (on the left) loses its one valence electron to chlorine (on the right),
resulting in	↓
a positively charged sodium ion (left) and a negatively charged chlorine ion (right).

Interactive Animation: The reaction of sodium with chlorine

Notice that when sodium loses its one valence electron it gets smaller in size, while chlorine grows larger when it gains an additional valence electron. This is typical of the relative sizes of ions to atoms. Positive ions tend to be smaller than their parent atoms while negative ions tend to be larger than their parent. After the reaction takes place, the charged Na+ and Cl- ions are held together by electrostatic forces, thus forming an ionic bond. Ionic compounds share many features in common:

• Ionic bonds form between metals and nonmetals.
• In naming simple ionic compounds, the metal is always first, the nonmetal second (e.g., sodium chloride).
• Ionic compounds dissolve easily in water and other polar solvents.
• In solution, ionic compounds easily conduct electricity.
• Ionic compounds tend to form crystalline solids with high melting temperatures.

This last feature, the fact that ionic compounds are solids, results from the intermolecular forces (forces between molecules) in ionic solids. If we consider a solid crystal of sodium chloride, the solid is made up of many positively charged sodium ions (pictured below as small gray spheres) and an equal number of negatively charged chlorine ions (green spheres). Due to the interaction of the charged ions, the sodium and chlorine ions are arranged in an alternating fashion as demonstrated in the schematic. Each sodium ion is attracted equally to all of its neighboring chlorine ions, and likewise for the chlorine to sodium attraction. The concept of a single molecule does not apply to ionic crystals because the solid exists as one continuous system. Ionic solids form crystals with high melting points because of the strong forces between neighboring ions.

Sodium Chloride Crystal	NaCl Crystal Schematic
Cl-1	Na+1	Cl-1	Na+1	Cl-1
Na+1	Cl-1	Na+1	Cl-1	Na+1
Cl-1	Na+1	Cl-1	Na+1	Cl-1
Na+1	Cl-1	Na+1	Cl-1	Na+1

The second major type of atomic bonding occurs when atoms share electrons. As opposed to ionic bonding in which a complete transfer of electrons occurs, covalent bonding occurs when two (or more) elements share electrons. Covalent bonding occurs because the atoms in the compound have a similar tendency for electrons (generally to gain electrons). This most commonly occurs when two nonmetals bond together. Because both of the nonmetals will want to gain electrons, the elements involved will share electrons in an effort to fill their valence shells. A good example of a covalent bond is that which occurs between two hydrogen atoms. Atoms of hydrogen (H) have one valence electron in their first electron shell. Since the capacity of this shell is two electrons, each hydrogen atom will "want" to pick up a second electron. In an effort to pick up a second electron, hydrogen atoms will react with nearby hydrogen (H) atoms to form the compound H2. Because the hydrogen compound is a combination of equally matched atoms, the atoms will share each other's single electron, forming one covalent bond. In this way, both atoms share the stability of a full valence shell.

Interactive Animation: Covalent bonding between hydrogen atoms

Unlike ionic compounds, covalent molecules exist as true molecules. Because electrons are shared in covalent molecules, no full ionic charges are formed. Thus covalent molecules are not strongly attracted to one another. As a result, covalent molecules move about freely and tend to exist as liquids or gases at room temperature.

Multiple Bonds: For every pair of electrons shared between two atoms, a single covalent bond is formed.  Some atoms can share multiple pairs of electrons, forming multiple covalent bonds. For example, oxygen (which has six valence electrons) needs two electrons to complete its valence shell. When two oxygen atoms form the compound O2, they share two pairs of electrons, forming two covalent bonds.

Lewis Dot Structures: Lewis dot structures are a shorthand to represent the valence electrons of an atom. The structures are written as the element symbol surrounded by dots that represent the valence electrons. The Lewis structures for the elements in the first two periods of the periodic table are shown below.

	Lewis Dot Structures

Lewis structures can also be used to show bonding between atoms. The bonding electrons are placed between the atoms and can be represented by a pair of dots or a dash (each dash represents one pair of electrons, or one bond). Lewis structures for H2 and O2 are shown below.

H2	H:H	or	H-H
O2

There are, in fact, two subtypes of covalent bonds. The H2 molecule is a good example of the first type of covalent bond, the nonpolar bond. Because both atoms in the H2 molecule have an equal attraction (or affinity) for electrons, the bonding electrons are equally shared by the two atoms, and a nonpolar covalent bond is formed. Whenever two atoms of the same element bond together, a nonpolar bond is formed.

A polar bond is formed when electrons are unequally shared between two atoms. Polar covalent bonding occurs because one atom has a stronger affinity for electrons than the other (yet not enough to pull the electrons away completely and form an ion). In a polar covalent bond, the bonding electrons will spend a greater amount of time around the atom that has the stronger affinity for electrons. A good example of a polar covalent bond is the hydrogen-oxygen bond in the water molecule.

Water molecules contain two hydrogen atoms (pictured in blue) bonded to one oxygen atom (red). Oxygen, with six valence electrons, needs two additional electrons to complete its valence shell. Each hydrogen contains one electron. Thus oxygen shares the electrons from two hydrogen atoms to complete its own valence shell, and in return shares two of its own electrons with each hydrogen, completing the H valence shells.

Interactive Animation:Polar covalent bonding simulated in water

The primary difference between the H-O bond in water and the H-H bond is the degree of electron sharing. The large oxygen atom has a stronger affinity for electrons than the small hydrogen atoms. Because oxygen has a stronger pull on the bonding electrons, it preoccupies their time, and this leads to unequal sharing and the formation of a polar covalent bond.

Because the valence electrons in the water molecule spend more time around the oxygen atom than the hydrogen atoms, the oxygen end of the molecule develops a partial negative charge (because of the negative charge on the electrons). For the same reason, the hydrogen end of the molecule develops a partial positive charge. Ions are not formed; however, the molecule develops a partial electrical charge across it called a dipole. The water dipole is represented by the arrow in the pop-up animation (above) in which the head of the arrow points toward the electron dense (negative) end of the dipole and the cross resides near the electron poor (positive) end of the molecule.

Chemical bonding between atoms results in compounds that can be very different from the parent atoms. This module, the second in a series on chemical reactions, describes how atoms gain, lose, or share electrons to form ionic or covalent bonds. The module lists features of ionic and covalent compounds. Lewis dot structures and dipoles are introduced.

When discussing the history of chemistry it’s always dangerous to point to the specific origin of an idea, since by its very definition, the scientific process relies upon the gradual refinement of ideas that came before. However, as is the case with a number of such ideas, one can point to certain seminal moments, and in the case of chemical bonding, a famous early 18th century publication provides one such moment.

In his 1704 publication Opticks, Sir Isaac Newton makes mention of a force that points to the modern idea of the chemical bond. In Query 31 of the book, Newton describes ‘forces’ – other than those of magnetism and gravity – that allow ‘particles’ to interact.

In 1718, while translating Opticks into his native language, French chemist Étienne François Geoffroy created an Affinity Table. In this fascinating first look at the likelihood of certain interactions, Geoffroy tabulated the relative affinity that various substances had for other substances, and therefore described the strength of the interactions between those substances.

While Newton and Geoffroy’s work predated our modern understanding of elements and compounds, their work provided insight into the nature of chemical interactions. However, it was over 100 years before the concept of the combining power of elements was understood in a more modern sense. In a paper in the journal Philosophical Transactions entitled “On a new series of organic bodies containing metals” (Frankland, 1852), Edward Frankland describes the “"combining power of elements,” a concept now known as valency in chemistry. Frankland summarized his thoughts by proposing what he described as a ‘law’:

A tendency or law prevails (here), and that, no matter what the characters of the uniting atoms may be, the combining power of the attracting element, if I may be allowed the term, is always satisfied by the same number of these atoms.

Frankland’s work suggested that each element combined with only a limited number of atoms of another element, thus alluding to the concept of bonding. But it was two other scientists who performed the most important contemporary research on the concept of bonding.

In 1916, the American scientist Gilbert N. Lewis published a now famous paper on bonding entitled “The atom and the molecule” (Lewis, 1916). In that paper he outlined a number of important concepts regarding bonding that are still used today as working models of electron arrangement at the atomic level. Most significantly, Lewis developed a theory about bonding based on the number of outer shell, or valence, electrons in an atom. He suggested that a chemical bond was formed when two atoms shared a pair of electrons (later renamed a covalent bond by Irving Langmuir). His "Lewis dot diagrams" used a pair of dots to represent each shared pair of electrons that made up a covalent bond (Figure 2).

Lewis also championed the idea of ‘octets’ (groups of eight), that a filled valence shell was crucial in understanding electronic configuration as well as the way atoms bond together. The octet had been discussed previously by chemists such as John Newland, who felt it was important, but Lewis advanced the theory.

Comprehension Checkpoint

Lewis based his theory of bonding on

While still in college, a young chemist by the name of Linus Pauling familiarized himself with Lewis’s work and began to consider how it might be interpreted within the context of the newly developed field of quantum mechanics. The theory of quantum mechanics, developed in the first half of the 20th century, had redefined our modern understanding of the atom and so any theory of bonding would be incomplete if it were not consistent with this new theory (see our modules Atomic Theory II: Bohr and the Beginnings of Quantum Theory and Atomic Theory III: Wave-Particle Duality and the Electron for more information).

Pauling’s greatest contribution to the field was his book The Nature of the Chemical Bond (Pauling, 1939). In it, he linked the physics of quantum mechanics with the chemical nature of the electron interactions that occur when chemical bonds are made. Pauling’s work concentrated on establishing that true ionic bonds and covalent bonds sit at extreme ends of a bonding spectrum, and that most chemical bonds are classified somewhere between those extremes. Pauling further developed a sliding scale of bond type governed by the electronegativity of the atoms participating in the bond.

Pauling’s immense contributions to our modern understanding of the chemical bond led to his being awarded the 1954 Nobel Prize for "research into the nature of the chemical bond and its application to the elucidation of the structure of complex substances."

Chemical bonding and interactions between atoms can be classified into a number of different types. For our purposes we will concentrate on two common types of chemical bonds, namely covalent and ionic bonding.

Molecular bonds are formed when constituent atoms come close enough together such that the outer (valence) electrons of one atom are attracted to the positive nuclear charge of its neighbor. As the independent atoms approach one another, there are both repulsive forces (between the electrons in each atom and between the nuclei of each atom), and attractive forces (between the positive nuclei and the negative valence electrons). Some constituents require the addition of energy, called the activation energy, to overcome the initial repulsive forces. But at various distances, the atoms experience different attractive and repulsive forces, ultimately finding the ideal separation distance where the electrostatic forces are reduced to a minimum. This minimum represents the most stable position, and the distance between the atoms at this point is known as the bond length.

As the name suggests, covalent bonding involves the sharing (co, meaning joint) of valence (outer shell) electrons. As described previously, the atoms involved in covalent bonding arrange themselves in order to achieve the greatest energetic stability. And the valence electrons are shared – sometimes equally, and sometimes unequally – between neighboring atoms. The simplest example of covalent bonding occurs when two hydrogen atoms come together to ultimately form a hydrogen molecule, H2 (Figure 3).

The covalent bond in the hydrogen molecule is defined by the pair of valence electrons (one from each hydrogen atom) that are shared between the atoms, thus giving each hydrogen atom a filled valence shell. Since one shared pair of electrons represents one covalent bond, the hydrogen atoms in a hydrogen molecule are held together with what is known as a single covalent bond, and that can be represented with a single line, thus H-H.

There are many instances where more than one pair of valence electrons are shared between atoms, and in these cases multiple covalent bonds are formed. For example, when four electrons are shared (two pairs), the bond is called a double covalent bond; in the case of six electrons being shared (three pairs) the bond is called a triple covalent bond.

Common examples of such multiple bonds are those formed between atoms in oxygen and nitrogen gas. In oxygen gas (O2), two atoms share a double bond resulting in the structure O=O. In nitrogen gas (N2), a triple bond exists between two nitrogen atoms, N≡N (Figure 4).

Double covalent bonds are shorter and stronger than comparable single covalent bonds, and in turn, triple bonds are shorter and stronger than double bonds – nitrogen gas, for example, does not react readily because it is a strongly bonded stable compound.

Comprehension Checkpoint

When four electrons are shared between atoms, _____ bonds are formed.

Ionic bonding occurs when valence electrons are shared so unequally that they spend more time in the vicinity of their new neighbor than their original nuclei. This type of bond is classically described as occurring when atoms interact with one another to either lose or gain electrons. Those atoms that have lost electrons acquire a net positive charge and are called cations, and those that have gained electrons acquire a net negative charge and are referred to as anions. The number of electrons gained or lost by a constituent atom commonly conforms with Lewis’s valence octets, or filled valence shell principle.

In reality even the most classic examples of ionic bonding, such as the sodium chloride bond, contain characteristics of covalent bonding, or sharing of electrons of outer shell electrons. A common misconception is the idea that elements tend to bond with other elements in order to achieve these octets because they are 'stable' or, even worse, 'happy', and that’s what elements 'want'. Elements have no such feelings; rather, the actual reason for bond formation should be considered in terms of the energetic stability arising from the electrostatic interaction of positively charged nuclei with negatively charged electrons.

Substances that are held together by ionic bonds (like sodium chloride) can commonly separate into true charged ions when acted upon by an external force, such as when they dissolve in water. Further, in solid form, the individual atoms are not cleanly attracted to one individual neighbor, but rather they form giant networks that are attracted to one another by the electrostatic interactions between each atom’s nucleus and neighboring valence electrons. The force of attraction between neighboring atoms gives ionic solids an extremely ordered structure known as an ionic lattice, where the oppositely charged particles line up with one another to create a rigid, strongly bonded structure (Figure 5).

The lattice structure of ionic solids conveys certain properties common to ionic substances. These include:

• High melting and boiling points (due to the strong nature of the ionic bonds throughout the lattice).
• An inability to conduct electricity in solid form when the ions are held rigidly in fixed positions within the lattice structure. Ionic solids are insulators. However, ionic compounds are often capable of conducting electricity when molten or in solution when the ions are free to move.
• An ability to dissolve in polar solvents such as water, whose partially charged nature leads to an attraction to the oppositely charged ions in the lattice.

The special properties of ionic solids are discussed in further detail in the module Properties of Solids.

Comprehension Checkpoint

Atoms that lose electrons and acquire a net positive charge and are called

Lewis used dots to represent valence electrons. Lewis dot diagrams (see Figure 1) are a quick and easy way to show the valence electron configuration of individual atoms where no bonds have yet been made.

The dot diagrams can also be used to represent the molecules that are formed when different species bond with one another. In the case of molecules, dots are placed between two atoms to depict covalent bonds, where two dots (a shared pair of electrons) denote a single covalent bond. In the case of the hydrogen molecule discussed above, the two dots in the Lewis diagram represent a single pair of shared electrons and thus a single bond (Figure 6).

If ionic bonding and covalent bonding sit at the extreme ends of a bonding spectrum, how do we know where any particular compound sits on that spectrum? Pauling’s theory relies upon the concept of electronegativity, and it is the differences in electronegativity between the atoms that is crucial in determining where any bond might be placed on the sliding scale of bond type.

Pauling’s scale of electronegativity assigns numbers between 0 and 4 to each chemical element. The larger the number, the higher the electronegativity and the greater the attraction that element has for electrons. The difference in electronegativity between two species helps identify the bond type. Ionic bonds are those in which a large difference in electronegativity exists between two bonding species. Large differences in electronegativity usually occur when metals bond to non-metals, so bonds between them tend to be considered ionic.

When the difference in electronegativity between the atoms that make up the chemical bond is less, then sharing is considered to be the predominant interaction, and the bond is considered to be covalent. While it is by no means absolute, some consider the boundary between ionic and covalent bonding to exist when the difference in electronegativity is around 1.7 – less of a difference tends toward covalent, and a larger difference tends towards ionic. Smaller differences in electronegativity usually occur between elements that are both considered non-metals, so most compounds that are made up from two non-metal atoms are considered to be covalent.

Comprehension Checkpoint

If there is a big difference in electronegativity between two different elements, the bond between them will be

Once differences in electronegativity have been considered, and a bond has been determined as being covalent, the story is not quite over. Not all covalent bonds are created equally. The only true, perfectly covalent bond will be one where the difference in electronegativity between the two atoms within the bond is equal to zero. When this occurs, each atom has exactly the same attraction for the electrons that make up the covalent bond, and therefore the electrons are perfectly shared. This typically occurs in diatomic (two-atom) molecules such as H2, N2, O2, and those of the halogen compounds when the atoms in the bond are identical.

However, most covalent bonds occur between elements where even though the electronegativity difference is lower than 1.7, it is not zero. In these cases, the electrons are still considered shared, that is, the bond is still considered covalent, but the sharing is not perfect.

Most covalent bonds are formed between atoms of differing electronegativity, meaning that the shared electrons are attracted to one atom within the bond more than the other. As a result, the electrons tend to spend more time at one end of the bond than the other. This sets up what is known as a dipole, literally meaning ‘two poles’. One end of the bond is relatively positive (less attraction for electrons), and one end of the bond is relatively negative (more attraction for electrons). If this difference in electron affinity exists across the molecule, then the molecule is said to be polar – meaning that it will have two different, and opposite, partial charges at either end. Water (H2O) is an excellent example of a polar molecule. Electrons are not shared evenly since hydrogen and oxygen have different electronegativities. This creates dipoles in each H-O bond, and these dipoles do not cancel each other out, leaving the water molecule polar overall (Figure 7). (Read more about these bonds in our module Properties of Liquids.)

When the electrons in a bond are perfectly shared, there is no dipole, and neither end of the bond carries any partial charge. When no such overall charge exists, the molecule is said to be non-polar. An example of such a non-polar molecule is hydrogen, H2. In larger molecules with multiple covalent bonds, each bond will have either no dipole or a dipole with varying degrees of partial charge. When all of these dipoles are taken into consideration in three dimensions, the uneven distribution of charge caused by the dipoles may cancel out, making the molecule non-polar.

Alternatively, there may be a partial electrical charge across the molecule, making it a polar molecule. An example of a multiple atom non-polar molecule is carbon dioxide. Electrons are not shared evenly across the C=O bonds since carbon and oxygen have different electronegativities. This creates dipoles in each C=O bond, but because these are aligned oppositely across a linear molecule, with the oxygen atoms on either side of the carbon atom, they cancel via symmetry to leave the carbon dioxide molecule non-polar (Figure 8).

We have limited our discussion to ionic and covalent bonding and the sliding scale of bond type that exists between them. However, many other types of interactions and bonds between atoms exist, notably metallic bonding (the attractions that hold metal atoms together in metallic elements), and intermolecular forces (the interactions that exist between, rather than within, covalently bonded molecules). These each involve similar electrostatic interactions to the ones described in ionic and covalent bonds, but even those extensions are far from the end of the bonding story.

In 2014, researchers found the first experimental evidence for a new type of interaction between atoms that had been predicted in the 1980s (Fleming et al., 2014). Named a "vibrational bond," the theory describes a lightweight element (in this case, an isotope of hydrogen) oscillating or "bouncing" between two much heavier atoms (in this case, bromine) and effectively holding the larger atoms together. Donald Fleming, a chemist based at the University of British Columbia in Canada, described the new bond as being "like a Ping Pong ball bouncing between two bowling balls." As research continues, we can expect to understand interactions at the molecular level with increasing sophistication, and with it, a greater understanding of what we call chemical bonding.

The millions of different chemical compounds that make up everything on Earth are composed of 118 elements that bond together in different ways. This module explores two common types of chemical bonds: covalent and ionic. The module presents chemical bonding on a sliding scale from pure covalent to pure ionic, depending on differences in the electronegativity of the bonding atoms. Highlights from three centuries of scientific inquiry into chemical bonding include Isaac Newton’s ‘forces’, Gilbert Lewis’s dot structures, and Linus Pauling’s application of the principles of quantum mechanics.

Key Concepts

• When a force holds atoms together long enough to create a stable, independent entity, that force can be described as a chemical bond.

• The 118 known chemical elements interact with one another via chemical bonds, to create brand new, unique compounds that have entirely different chemical and physical properties than the elements that make them up.

• It is helpful to think of chemical bonding as being on a sliding scale, where at one extreme there is pure covalent bonding, and at the other there is pure ionic bonding. Most chemical bonds lie somewhere between those two extremes.

• When a chemical bond is formed between two elements, the differences in the electronegativity of the atoms determine where on the sliding scale the bond falls. Large differences in electronegativity favor ionic bonds, no difference creates non-polar covalent bonds, and relatively small differences cause the formation of polar-covalent bonds.

Beginning in the early Middle Ages, European and Persian philosophers became fascinated with the way that some substances seemed to “transmute” (or transform) into others. Simple stones, such as those that contained sulfur, seemed to magically burn; and otherwise unimpressive minerals were transformed, like the ore cinnabar becoming an enchanting silvery liquid metal mercury when heated. Alchemists based their approach on Aristotle’s ideas that everything in the world was composed of four fundamental substances - air, earth, fire, and water (Figure 2).

As such, they proposed, and spent generations trying to prove, that less expensive metals like copper and mercury could be turned into gold. Despite their misguided approach, many early alchemists performed foundational chemical experiments - transforming one substance into another, and so it is difficult to point to a specific date or event as the birth of the idea of an ordered, quantifiable chemical reaction. However, there are some important moments in history that have helped to make sense of it.

Antoine Lavoisier was a French nobleman in the 1700s who began to experiment with different chemical reactions. At the time, chemistry still couldn’t be described as being a true, quantitative science. Most of the theories that existed to explain the way that substances changed relied upon Greek philosophy, and there was precious little experimental detail attached to the alchemist’s tinkering.

However, during the second half of the 18th century, Lavoisier performed many quantitative experiments and observed that while substances changed form during a chemical reaction, the mass of the system – or a measure of the total amount of “stuff” present – did not change. In doing so, Lavoisier championed the idea of conservation of mass during transformations (Figure 3). In other words, unlike the alchemists before him who thought that they were creating matter out of nothing, Lavoisier proposed that substances are neither created nor destroyed, but rather change form during reactions. Lavoisier’s ideas were published in the seminal work Traité élémentaire de Chimie in 1789 (Lavoisier, 1789), which is widely hailed as the birth of modern chemistry as a quantitative science.

Joseph Proust was a French actor who followed in Lavoisier’s footsteps. Proust performed dozens of chemical reactions, starting with different amounts of various materials. Over time he observed that no matter how he started a certain chemical reaction, the ratio in which the reactants were consumed was always constant. For example, he worked extensively with copper carbonate and no matter how he changed the ratio of starting reactants, the copper, carbon, and oxygen all reacted together in a constant ratio (Proust, 1804). As a result, in the last few years of the 18th century, Proust formulated the law of constant composition (also referred to as the law of definite proportions, Figure 4).

He realized that any given chemical substance (that we now define as a compound) always consisted of the same ratio by mass of its elemental parts regardless of the method of preparation. This was a huge step forward in modern chemistry since it had been previously believed that the substances formed during chemical reactions were random and disordered.

The English chemist John Dalton helped make sense of the laws of conservation of mass and definite proportions in 1803 by proposing that matter was made of atoms of unique substances that could not be created or destroyed (see our module Early Ideas about Matter for more information).

Dalton extended Proust’s ideas by recognizing that it was possible for two elements to form more than one compound, but that whatever the compound was, it would always contain elements combined in whole number ratios (Dalton, 1808). This observation is known as the law of multiple proportions (Figure 5) and with his atomic theory, helped to cement Lavoisier’s observations.

These advancements, taken together, laid the groundwork for our modern understanding of chemical reactions, chemical equations, and chemical stoichiometry, or the process of expressing the relative quantities of reactants and products in a chemical reaction.

Comprehension Checkpoint

____ first theorized that while substances changed form during a chemical reaction, the mass of the system did not change.

There is a staggering array of chemical reactions. Chemical reactions occur constantly within our bodies, within plants and animals, in the air that circulates around us, in the lakes and oceans that we swim in, and even in the soil where we grow crops and build our homes. In fact, there are so many chemical reactions that occur that it would be difficult, if not impossible, to understand them all. However, one method that helps us to understand them is to categorize chemical reactions into a few, general types. While not a perfect system, placing reactions together according to their similarities helps us to identify patterns, which in turn allows predictions to be made about as yet unstudied reactions. In this module, we will consider and provide some context for a few categories of reactions, specifically: synthesis, decomposition, single replacement, double replacement, REDOX (including combustion), and acid-base reactions.

No matter the type of reaction, one universal truth applies to all chemical reactions. For a process to be classified as a chemical reaction, i.e., one where a chemical change takes place, a new substance must be produced. The formation of a new substance is nearly always accompanied by an energy change, and often with some kind of physical or observable change. The physical change can be of different types, such as the formation of bubbles of a gas, a solid precipitate, or a color change. These changes are clues to the existence of a chemical reaction and are important triggers for further research by chemists.

Prior to Lavoisier’s work, it was poorly understood that there were different gases made up of different elements. Instead, various gases were commonly mischaracterized as types of "air" or air missing parts – for example, terms commonly used were "inflammable air," or "dephlogisticated air." Lavoisier thought differently and was convinced that these were different substances. He conducted experiments where he mixed inflammable air with dephlogisticated air and a spark, and he found that the substances combined to produce water. In response, he renamed inflammable air "hydrogen" from the Greek hydro for "water" and genes for "creator." In so doing, Lavoisier was identifying a synthesis reaction. In general, a synthesis reaction is one in which simpler substances combine to form another more complex one. Hydrogen and oxygen (which Lavoisier also renamed dephlogisticated air) combine in the presence of a spark to form water, summarized by the chemical equation shown below (for more on chemical equations see the section called Anatomy of a chemical equation), it represents a simple synthesis reaction.

Equation 1

2H2(g) + O2(g) → 2H2O(l)

In 1774, the scientist Joseph Priestley turned his curiosity to a mineral called cinnabar – a brick red mineral. When he placed the mineral under sunlight amplified by a powerful magnifying glass, he found that a gas was produced which he described as having an “exalted nature” because a candle burned in the gas brightly (Priestley, 1775). Without realizing it, Priestley had discovered oxygen as a result of a decomposition reaction. Decomposition reactions are often thought of as the opposite of synthesis reactions since they involve a compound being broken down into simpler compounds or even elements. In the case of Priestley’s oxygen, he had broken down mercury (II) oxide (cinnabar) with heat into its individual elements. The reaction can be summarized in the following equation.

Equation 2

2HgO(s) → 2Hg(l) + O2(g)

The British chemist and meteorologist John Daniell, invented one of the very first practical batteries in 1836 (Figure 6). In his cell, Daniell utilized a very common single replacement reaction. His early cells were complicated affairs, with ungainly parts and complicated constructs, but by contrast, the chemistry behind them was really quite simple.

In certain chemical reactions, a single constituent can substitute for another one already joined in a chemical compound. The Daniell cell works because zinc can substitute for copper in a solution of copper sulfate, and in so doing exchange electrons that are used in the battery cell. The reaction can be summarized as follows:

Equation 3

Zn(s) + CuSO4(aq) → ZnSO4(aq) + Cu(s)

This particular single displacement is called a metal displacement since it involves one metal replacing another metal, and many types of batteries are based on metal replacement reactions. However, several other types of single replacement reactions exist, such as when a metal can replace hydrogen from an acid or from water, or a halogen can replace another halogen in certain salt compounds.

The controlled use of fire was a crucial development for early civilization. While it’s difficult to pin down the exact time that humans first tamed the combustion reactions that produce fire, recent research suggests it may have occurred at least a million years ago in a South African cave (Berna et al. 2012).

Chemically, combustion is no more than the reaction of a fuel (wood, oil, gasoline, etc.) with oxygen. For combustion to take place there must be a fuel and oxygen gas. However, these reactions often require activation energy (discussed in more detail in the module Chemical Bonding: The Nature of the Chemical Bond), which can be provided by a ‘spark’ or source of energy for ignition. Fuel, oxygen, and energy are the three things make up what is known as the fire triangle (Figure 7), and any one of them being absent means that combustion will not take place.

In the modern world, many of the fuels that are typically burned for energy, are hydrocarbons – substances that contain both hydrogen and carbon (as discussed in more detail in our Carbon Chemistry module). Plants produce hydrocarbons when they grow, and thus make an excellent fuel source, and other hydrocarbons are produced when plants or animals decay over time (such as natural gas, oil, and other substances). When these fuels combust, the hydrogen and carbon within them combine with oxygen to produce two very familiar compounds, water, and carbon dioxide. One simple example is the combustion of natural gas, or methane, CH4:

Equation 4

CH4(g) + 2O2(g) → CO2(g) + 2H2O(l)

As with the combustion of all fuels, heat and light are products, too, and it is these products that are used to cook our food or to heat our homes.

Each of the four types of reaction above are sub-categories of a single type of chemical reaction known as redox reactions. A redox reaction is one where reduction and oxidation take place together, hence the name. The individual processes of oxidation and reduction can be defined in more than one way, but whatever the definition, the two processes are symbiotic, i.e., they must take place together.

In one definition, oxidation is described as the process in which a species loses electrons, and reduction is a process where a species gains electrons. In this way, we can see how the pair must take place together. If a chemical substance is to lose electrons (and therefore be oxidized), then it must have another, interdependent chemical substance that it can give those electrons to. In the process, the second substance (the one gaining electrons) is said to be reduced. Without such an electron acceptor, the original species can never lose the electrons and no oxidation can take place. When the electron acceptor is present, it gets reduced and the redox combination process is complete. Redox reactions of this type can be summarized by a pair of equations – one to show the loss of electrons (the oxidation), and the other to show the gain of electrons (the reduction). Using the example of the Daniell cell above,

Equation 5

Oxidation: Zn → Zn2+ + 2e-

Reduction: Cu2+ + 2e- → Cu

The electrons shown being lost by zinc in the first reaction, are the same electrons being accepted by the copper ions in the second. Together, the reactions can be combined to cancel out the electrons on either side of the reactions, into the overall redox reaction:

Equation 6

Zn + Cu2+ → Zn2+ + Cu

Other definitions of oxidation and reduction also exist, but in every case, the two halves of the redox reaction remain symbiotic – one loses and the other gains. The loss from one species cannot happen without the other species gaining.

When soap won’t easily produce a lather in water, the water is said to be ‘hard’. Hard water causes all kinds of problems that go beyond just making it difficult to form a lather. The buildup of compounds in water pipes (known as ‘scale’), can block the flow of water and can cause problems in industrial processes. Textile manufacturing and the beverage industry rely heavily on water. In those situations, the quality of the water can make a difference to the end product, so controlling the water composition is crucial.

Hard water contains magnesium or calcium ions in the form of a dissolved salt such as magnesium chloride or calcium chloride. When soap (sodium stearate) comes into contact with either of those salts, it enters into a double displacement reaction that forms the insoluble precipitate known as ‘soap scum’.

A double displacement reaction (also known as a double replacement reaction) occurs when two ionic substances come together and both substances swap partners. In general:

Equation 7

AB + CD → AD + CB

Where A and C are cations (positively charged ions), and B and D are anions (negatively charged).

In the case of the reaction of soap with calcium chloride, the reaction is:

Equation 8

CaCl2(aq) + 2NA(C17H35COO)(aq) → 2NaCl(aq) + Ca(C17H35COO)2(s)

The solid calcium stearate is what we call soap scum, which is formed by the reaction of the soluble sodium stearate salt (the soap) in a double replacement reaction with calcium chloride.

Acid-base reactions happen around, and even inside of us, all the time. From the classic elementary school baking soda volcano to the process of digestion, we encounter acids and bases on a daily basis.

When a hydrogen atom loses its only electron, it forms a positive ion, H+. This hydrogen ion is the essential component of all acids, and indeed one definition of an acid is that of a hydrogen ion donor. Compounds such as the citric acid in lemon juice, the ethanoic acid in vinegar, or a typical laboratory acid like hydrochloric acid, all give their hydrogen ions away in chemical reactions known as acid-base reactions. The chemical opposites of acids are known as bases, and bases can be defined as hydrogen ion acceptors. Whenever an acid donates a hydrogen ion to a base, an acid-base reaction has taken place, for example, when hydrochloric acid donates a hydrogen ion to a base such as sodium hydroxide:

Equation 9a

HCl(aq) + NaOH(aq) → H2O(l) + NaCl(aq)

A closer look at this reaction reveals that in water the HCl gives away an H+ as shown below:

Equation 9b

HCl(aq) + H2O(l) → H3O+(aq) + Cl-(aq)

The resulting species, H3O+ (the hydronium ion), can, in turn, act as an acid when it comes into contact with any species that can accept a hydrogen ion, such as hydroxide ions from sodium hydroxide:

Equation 9c

H3O+(aq) + NaOH(aq) → 2H2O(l) + Na+(aq)

Combining equations #9a and #9b gives us equation #9c.

Equation #9c can be re-written to show the individual ions that are found in solution, thus:

Equation 9d

H+(aq) + Cl-(aq) + Na+(aq) + OH-(aq) → H2O(l) + Na+(aq) + Cl-(aq)

Removing the spectator ions from the equation above, we get the net ionic equation:

Equation 9e

H+(aq) + OH-(aq) → H2O(l)

Any chemical reaction that forms water from the reaction between an acid and base as in equation #9e is known as a neutralization reaction.

Comprehension Checkpoint

The type of chemical reaction where a single constituent can substitute for another one already joined in a chemical compound is:

Chemical equations are always linked to chemical reactions since they are the shorthand by which chemical reactions are described. That fact alone makes equations incredibly important, but equations also have a crucial role to play in describing the quantitative aspect of chemistry, something that we formally call stoichiometry.

All chemical reactions take on the same, basic format. The starting substances, or reactants, are listed using their chemical formula to the left-hand side of an arrow, with multiple reactants separated with plus signs. In the case of a reaction between carbon and oxygen:

To the right hand of the arrow one finds the chemical formulas of the new substance or substances (known as the products) that are produced by the chemical reaction. In this case, since carbon dioxide is the result of burning carbon in the presence of oxygen:

Equation 10b

[Reactants] C + O2 → CO2 [Products]

Since reactions can result in both physical as well as chemical changes, each substance is given a state symbol written as a subscript to the right of the formula, this describes the physical form of the reactants and products. Common state abbreviations are (s) for solids, (l) for liquids, (g) for gases and (aq) for any aqueous substances, i.e., those dissolved in water.

Equation 10c

C(s) + O2(g) → CO2(g)

Finally, in order to ensure that this representation abides by the law of conservation of mass, the equation may need to be balanced by the addition of numbers in front of each species that create equal numbers of atoms of each element on each side of the equation. In the case of the formation of carbon dioxide from carbon and oxygen, there is no need for the addition of such numbers (called the stoichiometric coefficients), since 1 carbon atom and 2 oxygen atoms appear on each side of the equation.

In nature, chemical reactions are often driven by exchanges in energy. In this respect, reactions are generally separated into two categories – those that release energy and those that absorb energy.

Exothermic reactions are those that release energy to the surroundings (Figure 8, right). Combustion reactions are an obvious example because the energy released by the reaction is converted into the light and heat seen in the immediate surroundings.

By contrast, endothermic reactions are those that absorb energy from the surroundings (Figure 8, left). In this situation, one may have to heat up the reaction or add some other form of energy to the system before seeing the reaction proceed.

In both cases it is important to note that energy is neither created nor destroyed, rather it is transferred from one type of energy to another, for example from chemical energy to that of heat or light. The energy that goes into the formation of chemical bonds is exchanged for other types of energy with the environment around that reaction. A classic example is the photosynthesis reaction, in which plants absorb light energy from the sun in order to create bonds between atoms that make up sugars, which are stored as chemical energy for later use by the plant. The process of respiration is essentially the reverse of photosynthesis, where the bonds in sugar molecules are broken and the released energy is then used by the plant.

Comprehension Checkpoint

_____ reactions are those that absorb energy from the surroundings.

Chemical reactions happen all around us every day. Whether it is a single replacement reaction in the battery of our flashlight, a synthesis reaction that occurs when iron rusts in the presence of water and oxygen, or an acid-base reaction that happens when we eat – we experience chemical reactions in almost everything we do. Understanding these reactions is not an abstract concept for a chemist in a far off laboratory, rather it is critical to understanding life and the world around us. To truly master chemical reactions, we need to understand the quantitative aspect of these reactions, something referred to as stoichiometry, and a concept we will discuss in another module.

This modules explores the variety of chemical reactions by grouping them into general types. We look at synthesis, decomposition, single replacement, double replacement, REDOX (including combustion), and acid-base reactions, with examples of each.

Key Concepts

• The steps from a qualitative science to quantitative one, were crucial in understanding chemistry and chemical reactions more completely.
• When a substance or substances (the reactants), undergo a change that results in the formation of a new substance or substances (the products), then a chemical reaction is said to have taken place.
• Mass and energy are conserved in chemical reactions. Matter is neither created or destroyed, rather it is conserved but rearranged to create new substances. No energy is created or destroyed, it is conserved but often converted to a different form.
• Chemical reactions can be classified into different types depending on their nature. Each type has its own defining characteristics in terms of reactants and products.
• Chemical reactions are often accompanied by observable changes such as energy changes, color changes, the release of gas or the formation of a solid.
• Energy plays a crucial role in chemical reactions. When energy is released into the surroundings the reaction is said to be exothermic; when energy is absorbed from the surroundings the reaction is said to be endothermic

That’s what chemical equations are for: they’re a type of shorthand used to precisely communicate exactly what’s happening in the reaction. In the most basic sense, a chemical equation describes the type and amount of each substance that reacts (the reactants) to form a given amount of specific substances produced (the products). Reactants will always react in proportions given in the equation. If the supply of one reactant runs out, the excess of the other will remain unreacted.

The shorthand that explains our rusty car part is this:

The large number in front of an atom or molecule, which we call the coefficient, tells us the relative amounts of each substance involved in or produced by the reaction. The numbers in subscript refer specifically to the element that’s in front of them.

In other words, 4 iron atoms in the muffler reacted with 3 oxygen molecules in the air. Each of these oxygen molecules contains 2 oxygen atoms. When the chemical bonds reassemble and these reactants combine, the result is 2 molecules of a specific iron oxide that contains 2 iron atoms and 3 oxygen atoms—a.k.a rust.

This equation conveys something much deeper than numbers of particles, though: it captures the centuries-long accumulation of knowledge about what our universe is made of and how matter interacts. Like an elegant poem (or like your rusty muffler drawing), a chemical equation conveys a world of complex concepts in just a few expressions.

Comprehension Checkpoint

If one reactant is completely consumed, the remaining reactant will

As our understanding of chemical processes deepened over time, chemical equations slowly became more sophisticated. Ultimately, such equations played a role in the recognition of chemistry as its own important science, separate from medicine, alchemy (which was popular in the 17th and 18th centuries), and physics.

The first known written chemical equation—which is really more of a diagram—appeared in what’s considered to be the first chemistry textbook, the Tyrocinium Chymicum (meaning “Begin Chemistry”). It was penned by French scientist and teacher Jean Beguin in Paris in 1615, and describes what Beguin observed when he heated antimony sulfide with a chloride of mercury. The mercury became vapor, leaving behind a residue of antimony oxychloride.

While Beguin’s diagram looks very different than modern chemical equations (and it isn’t entirely correct), it reflects an understanding of reactants and products in a chemical reaction.

Beguin revealed the beginnings of an understanding of what occurs in a chemical reaction, but not an explanation of why. More than a century passed before new diagrams revealed deeper insights into the phenomena that drive reactions.

These insights came from William Cullen, who was also a teacher, and founded the chemistry department at the University of Glasgow in 1747. His hand-written lecture notes contain diagrams using arrows and letters, indicating four different types of reactions:

The diagram on the lower left described what Cullen called a “single elective attraction,” or what we now call a single replacement reaction, one that involves a single element taking the place of another element within a compound. (For more on reaction types, see our Chemical Equations module. The upper left is his diagram for a double replacement reaction, or one that involves the exchange of a component from each of two different compounds. In both cases, the reactions produce precipitates, indicated by the squiggly lines.

The two diagrams on the right relate to the dissolution of salts. The prevailing wisdom during Cullen’s time was that “elective attractions,” which we now know as charge, caused a metal in one substance to be attracted to another substance. The metals swapped places, forming either a new solid precipitate or dissolving in solution (sometimes producing a solution that displayed new properties).

While he didn’t have all the details right, Cullen’s thinking was on track, and his diagrams represent an important step toward what is now a common generic chemical equation:

We may take this simple type of equation for granted now, but at the time, it demonstrated some very perceptive ideas about what is actually happening during a chemical reaction. Remember, atoms and molecules were still not understood in Cullen’s time. It took a trio of chemists working over the next 50 years to clarify the picture that gave us the modern chemical equation. So, to grasp the idea that substances combined and were exchanged in chemical reactions is really significant.

In 1774, French chemist Antoine Lavoisier made an important observation – he noted that while substances in a chemical reaction changed form in ways described by Cullen, the mass of the system did not change. In other words, the amount of each element present remained the same, meaning that matter and mass are conserved. This is a crucial concept that will be important when we discuss balancing chemical reactions – all elements have to be fully accounted for at the start and at the end of a chemical reaction.

Around the same time as Lavosier, Joseph Proust, another Frenchman, was working extensively with copper carbonate. He found that regardless of how he changed the ratio of starting reactants – adding more copper sometimes, or more carbon or oxygen at others, the copper, carbon, and oxygen all reacted together in a constant ratio. His insight brought us the law of definite proportions: in any given compound, the elements occur in fixed ratios, regardless of their source. (Visit our module on Chemical Reactions to learn more about the work of these scientists). Again, this seems obvious to us today as we know that elements only react together in certain ways, but Proust completed his work before it was widely recognized that atoms and compounds existed.

Finally, in 1803, English chemist John Dalton tied these threads together by proposing that matter is made of atoms of unique substances that could not be created or destroyed (see our module Early Ideas about Matter for more information). He showed that each element could combine with multiple others to form different compounds, and always in whole number ratios.

This knowledge, taken together, provided the foundation for the chemical notation we use today. Chemical equations aren’t like mathematical equations, which have been around much longer. While the quantity of each element must be equal on both sides of the equation, you will never see an “equals” sign in a chemical equation. That’s because a chemical equation describes a process of change.

Comprehension Checkpoint

Chemical equations are written basically the same way today as they were in the 17th century.

Again, iron and oxygen combine to form a specific iron oxide. The arrow indicates that this reaction proceeds to the right as it is written, meaning that iron oxide is formed. In some cases, the reaction can also go backward, and we use a double arrow showing that some reactions go in both directions. Unfortunately for the muffler, that’s not the case here.

Now look at the equation more closely: How many iron atoms are on the reactant side (left) versus the product side (right)? How many oxygen atoms? You will see that they’re not equal as the equation is currently written.

However, we know from the law of conservation of matter that atoms can’t be created or destroyed. In other words, we can’t just get rid of an iron atom or create an oxygen atom to make the equation work. Nor can we change the subscripts in the reaction, because doing so would suggest that we are starting with a different reactant or obtaining a different product.

What we can do is adjust the numbers of reactant and product components (iron atoms and/or oxygen molecules on the left side and iron oxide molecules on the right), because doing so doesn’t imply creating or destroying matter. It’s a way of bookkeeping – in order to produce a molecule that has two iron atoms, you need to find a source of those atoms.

Let’s see how this works with the reaction that creates rust.

We already know that the numbers of each type of atom aren’t equal on each side of the reaction. To address this, we can add coefficients in front of the reactants and products to adjust the number of particles and create a balanced equation. (If there is no coefficient, it means there is only one of that type of particle.)

Let’s look at the reaction atom by atom.

Since the rust molecule has two iron atoms, we must balance the Fe atoms by adding the coefficient “2” in front of the iron atom on the reactant side, since that is the only place where iron appears on the left side of the equation. Now we have two irons on each side of the equation.

Moving on to the oxygen atoms, we find two on the left side of the equation, but three on the right side. We could mathematically balance the equation by using one and a half oxygen molecules, since each molecule is made up of two oxygen atoms. To better understand this, picture one oxygen molecule as two atoms bonded together O-O, therefore 1.5 oxygen molecules (O-O and O-O) provides three oxygen atoms:

In the real world, though, oxygen doesn’t occur in half-molecules. We can solve this conundrum by simply multiplying all the coefficients by two:

Now we have 4 atoms of iron on the left side, and 4 (2 molecules, each containing 2 iron atoms) on the right side. For oxygen, there are 6 atoms on the left side (3 molecules of 2 atoms each) and 6 on the right side (2 molecules containing 3 oxygen atoms). Now we have a balanced equation.

Comprehension Checkpoint

To create a balanced equation, we can

Along with telling us exactly how much of a chemical compound is involved in a reaction, balanced chemical equations tell us is the proportions of “ingredients” required to make a particular product. It’s a bit like a recipe. Let’s say you’re making a batch of cookies. The recipe calls for:

• 2 cups of flour
• 1 cup of sugar

and promises you a batch of 12 cookies. You can follow the recipe and use 2 cups of flour and 1 cup of sugar and expect 12 cookies, or you can double the recipe and use 4 cups of flour and 2 cups of sugar and expect 24 cookies (or you can triple the recipe, or half it, or so on).

Similarly, to make a “batch” of 2 rust molecules, you need 4 iron atoms and 3 oxygen molecules:

So, for every four atoms of iron and three molecules of oxygen, we get two molecules of rust. Much as you would do with a cookie recipe, you could double this: start with eight iron atoms and six oxygen molecules and make four molecules of rust. No matter how many times you multiply it, the base proportion always remains constant.

In the real world of chemistry, though, we don’t deal with individual atoms and molecules; there are a lot more than two molecules of rust on a muffler that falls off a car. That’s where the concept of the mole comes in handy. A mole is a quantity of particles, specifically a mole is 6.022 x 1023 particles. In other words, one mole of iron atoms contains 6.022 x 1023 atoms. Four moles of iron atoms contain four times that amount, or 28.088 x 1023 iron atoms. (To brush up on your mole math, see our module, The Mole and Atomic Mass).

Note that when we were balancing the equation, we were thinking about the numbers of individual atoms on each side, attending to the law of conservation of matter that we can’t create or destroy atoms. When we’re thinking about proportions, however, we’re thinking about the coefficients in front of the particles, which represent the number of moles of each type of particle consumed or produced. These coefficients tell us how many moles of product can be produced with the number of moles of reactant present at the beginning of the reaction.

A chemical equation is an ingeniously compact way of communicating a lot of information in a short sequence of parts. Modern chemical equations reflect our understanding of matter being composed of atoms and of chemical reactions as a process of breaking bonds and rearranging atoms into new compounds. Mass is conserved in a chemical reaction, and the number of particles on each side of the equation must reflect this. A balanced equation also communicates the proportions of products and reactants involved in that reaction.

Chemical equations are an efficient way to describe chemical reactions. This module explains the shorthand notation used to express how atoms are rearranged to make new compounds during a chemical reaction. It shows how balanced chemical equations convey proportions of each reactant and product involved. The module traces the development of chemical equations over the past four centuries as our understanding of chemical processes grew. A look at chemical equations reveals that nothing is lost and nothing is gained in a typical chemical reaction–matter simply changes form.

The universe is in constant motion: from the orbiting of planets around the sun, to the movement of particles from one area to another. And while on a grand scale it may appear that there is a rationale to this movement – for example, the planets in our solar system have regular revolutions that can be predicted – in truth there is a great deal of motion that occurs randomly.

When we learn about diffusion, we often hear about the movement of particles from an area of high concentration to an area of low concentration, as if the particles themselves are somehow motivated to move in this direction. But this movement is in fact a by-product of what scientists refer to as the “random walk” of particles. Molecules do not move in straight paths from Point A to Point B. Instead, they interact with their environment, bumping into other molecules and barriers encountered along their way, as well as interacting with the medium through which they are moving.

The observation of the spontaneous, random movement of small particles was first recorded in the first century BCE. Lucretius, a Roman poet and philosopher, described the dust seen in sunbeams coming through a window (Figure 1):

You will see a multitude of tiny particles mingling in a multitude of ways... their dancing is an actual indication of underlying movements of matter that are hidden from our sight... It originates with the atoms which move of themselves [i.e., spontaneously]… So the movement mounts up from the atoms and gradually emerges to the level of our senses, so that those bodies are in motion that we see in sunbeams, moved by blows that remain invisible.

While Lucretius’s “dancing” particles were likely dust particles or pollen grains that are affected by air currents and other phenomenon, his description is a wonderfully accurate assessment of what goes on at the molecular level. Many scientists have explored this random molecular motion in a variety of contexts, most famously by the Scottish botanist Robert Brown in the 19th century.

In 1828, while observing pollen granules suspended in water under a microscope, Brown discovered that the motion of the granules were “neither from currents in the fluid, nor from its gradual evaporation, but belonged to the particle itself.” After suspending various organic and inorganic substances in water and seeing this same inherent, random movement, he concluded that this random walk of particles – later termed Brownian motion in his honor – was a general property of matter that is suspended in a liquid medium. However, it would take nearly a century for scientists to mathematically quantify Brownian motion and demonstrate that this random movement of molecules dictates diffusion.

Comprehension Checkpoint

When molecules move from an area of high concentration to an area of low concentration,

About the same time that Brown was making his observations, a group of scientists including the French engineer Sadi Carnot and German physicist Rudolph Clausius were establishing a whole new field of scientific study: the field of Thermodynamics (see our Thermodynamics I module for more information). Clausius’s work in particular led to the development of the kinetic theory of heat – the idea that atoms and molecules are in motion and the speed of that motion is related to a number of things, including the heat of the substance. The molecules of a solid are generally considered to be locked in place (though they vibrate); however, the molecules of a liquid or a gas are free to move around, and they do: bumping in to one another or the walls of their container like balls on a pool table.

As molecules in a liquid or gas move through space, they bump into one another and follow random paths – moving in a straight line until something blocks their way and then bouncing off of that thing. This random molecular movement is constantly occurring and can be measured, giving a molecule’s mean free path – or, the average distance a particle moves between impacts with other particles.

It is this spontaneous and random motion that leads to diffusion. For example, as the scent molecules from baking cookies move into the air, they interact with air molecules – crashing into them and changing direction. Over time, these random processes will cause the scent molecules to disperse throughout the room. Diffusion is presented as a process in which a substance moves down a concentration gradient – from an area of high concentration to an area of low concentration. However, it is important to recognize that there is no directional force at play – the scent molecules are not pushed to the edge of the room because the concentration is lower there. It is the random movement of these molecules within the roomful of moving air molecules that causes them to evenly spread out throughout the entire space – bouncing off walls, moving through doors, and eventually moving through the whole house. In this way, it appears to move along a concentration gradient – from the kitchen oven to the most distant rooms of the house.

It may sound like a paradox – the movement of molecules are random, yet at the same time appear to occur along a gradient – but in practice, it’s actually quite logical. A simple illustration of this process can be seen using a glass of water and food coloring. When a drop of food coloring enters the water, the food coloring molecules are highly concentrated at the location where the dye molecules meet the water molecules, giving the water in that area a very dark color (Figure 2). The bottom of the glass initially has few or no food coloring molecules and so remains clear. As the food coloring molecules begin to interact with the water molecules, molecular collisions cause them to move randomly around the glass. As collisions continue, the molecules spread out, or diffuse, over space.

Eventually, the molecules spread throughout the entire glass, becoming evenly distributed and filling the space. At this point, the molecules have reached a state of equilibrium in which no net diffusion is taking place and the concentration gradient no longer exists. In this state, the molecules are still moving haphazardly and colliding with each other; we just can’t see that motion because the water and color molecules are evenly dispersed throughout the space. Once equilibrium has been reached, the probability that a molecule will move from the top to the bottom is equal to the probability a molecule will move from the bottom to the top.

Comprehension Checkpoint

When equilibrium is achieved and molecules are equally distributed,

We know that diffusion involves the movement of particles from one place to another; thus, the speed at which those particles move affects diffusion. Since molecular motion can be measured by the heat of an object, it follows that the hotter a substance is the faster diffusion will take place in that substance. (Click the animation below to see how temperature affects diffusion.) If you were to repeat your food coloring and water experiment comparing a glass of cold to a glass of hot water, you would see that the color disperses much more quickly in the hot water. But what other factors influence the speed, or rate, at which diffusion takes place?

Interactive Animation: The Effect of Temperature on Diffusion

In 1829, the Scottish physical chemist Thomas Graham first quantified diffusion behavior before the idea of atoms and molecules was widely established. Basing his observations on real-life “substances,” Graham measured the diffusion rates of gases through plaster plugs, fine tubes, and small orifices that were meant to slow down the diffusion process so that he could quantify it. One of his experiments, detailed in Figure 3, used an apparatus with the open end of a tube sitting in a beaker of water and the other end sealed with a plaster stopper containing holes large enough for gases to enter and leave the tube. Graham filled the open end of the tube with various gases (as indicated by the red tube in Figure 3), and observed the rate at which the gases effused, or escaped through the plaster plug. If the gas effused from the tube faster than the air outside of the tube moved in, the water level in the tube would rise. On the other hand, if the outside air moved through the plaster faster than the gas in the tube escaped to the outside, the water level in the tube would go down. He used the rate of change in the water level to determine the relative rate at which the different gases diffused into air.

Graham experimented with many combinations of different gases and published his findings in an 1829 publication of the Quarterly Journal of Science, Literature, and Art titled “A Short Account of Experimental Researches on the Diffusion of Gases Through Each Other, and Their Separation by Mechanical Means.” He stated that when gases come into contact with each other, “indefinitely minute volumes” of the gases spontaneously intermix with each other until they reach equilibrium (Graham, 1829). However, he discovered that different types of gases did not mix at the same rate – rather, the rates at which two gases diffuse is inversely proportional to the square root of their densities, a relationship now known as Graham’s law. Although Graham’s original relationship used density, or mass per unit volume, the modern form of the equation uses molar mass, or the mass of one mole of a substance.

What Graham showed was that the molecular weight of a molecule directly affects the speed at which that molecule can move. Graham’s work actually helped lay the foundations of kinetic molecular theory because it recognized that at a given temperature, a heavy molecule would move more slowly than a light molecule. In other words, more kinetic energy is needed to move a large molecule at the same speed as a small molecule. You can think of it this way: A small push will get a tennis ball rolling quickly; however, it takes a much harder push to move a bowling ball at the same speed. At a given temperature, small molecules move faster, and will diffuse more quickly than large ones. View the animation below to see how atomic mass affects diffusion.

Interactive Animation: The Effect of Atomic Mass on Diffusion

Comprehension Checkpoint

Rate of diffusion is influenced by

Graham later studied the diffusion of salts into liquids and discovered that the diffusion rate in liquids is several thousand times slower than in gases. This seems relatively obvious to us today, as we know that the molecules of a gas move faster and are more spread out than molecules in a liquid. Therefore, the movement of one substance within a gas occurs more freely than in a liquid. Diffusion in liquids is proportional to temperature, as it is in gases, as well as to the viscosity of the specific liquid into which the material is diffusing. (View the animation below to compare diffusion in gases and liquids.) Diffusion, in fact, can even take place in solids. While this is a very slow process, Sir William Chandler Roberts-Austen, a British metallurgist, fused gold plates to the end of cylindrical rods made of lead. He analyzed the lead rods after a period of 31 days and actually found that gold atoms had “flowed” into the solid rods.

While we have talked extensively about diffusion and concentration gradients, it was not until the mid-1800s when a German-born physicist and physiologist named Adolf Fick built upon Graham’s work and introduced the notion of a diffusion coefficient, or diffusivity, to characterize how fast molecules diffuse.

In his 1855 publication “On Diffusion” in Annalen der Physik, Fick described an experimental setup in which he connected cylindrical and conical tubes with solid salt crystals at the bottom to an “infinitely large” reservoir filled with freshwater (Figure 4). The solid salt crystals dissolved into the water in the tubes and diffused toward the water reservoir. A stream of freshwater swept the saltwater out of the reservoir. This stream of water kept the salt concentration at the very top of the tubes (the point where the salt solution met the water reservoir) close to zero. The dissolving salt at the bottom of the tube maintained a high salt concentration in the water at that end of the tube. Because the tubes had a different shape (conical versus cylindrical), the concentration gradient in the tubes differed, setting up a system in which diffusion could be compared in relation to a concentration gradient.

Fick then calculated the diffusion rate of the salt by measuring the amount of salt that passed through the top of the respective tubes (just before they met the freshwater in the reservoir) within a given time period. He discovered that the movement rate of the salt solution into the water reservoir depended on the concentration difference between the solution at the bottom of the tube and the concentration of the solution leaving the tube and entering the reservoir. In other words – the higher the concentration of salt at the top of the tube, the faster it diffused into the water reservoir. You can see how concentration affects diffusion in the animation below.

Interactive Animation: The Effect of Concentration on Diffusion

After studying the phenomenon, Fick hypothesized that the relationship between the concentration gradient and the diffusion rate was similar to what Joseph Fourier, a French mathematician and physicist, found in his study of heat conduction in 1822. Fourier had described the rate of heat transfer through a substance as proportional to the difference in temperature between two regions. Heat moves from warmer to cooler objects, and the greater the temperature difference between the two objects, the faster the heat moves. (This is why your mug of hot coffee cools off much faster outside on a cold morning than when you leave it in your heated apartment). Using Fourier’s law of thermal conduction as a model, Fick created a mathematical framework for the movement of salt into the water, proposing that the diffusion rate of a substance is proportional to the difference in concentration between the two regions. What this means for diffusion of a substance is that if the concentration of a given substance is high in relation to the substance it is diffusing into (e.g., food coloring into water), it will diffuse faster than if the concentration difference is low (e.g., food coloring into food coloring). The application of a successful principle from one branch of science to another is not uncommon, and Fick was a classic example of this process. Fick knew of Fourier’s work because he had modeled his experimental apparatus on that of Fourier. Thus it was natural for him to apply Fourier’s law to diffusion. While he had no way to know that the underlying mechanism of heat conduction and diffusion were both based on atomic collisions (in fact, some researchers at the time still doubted the existence of atoms), he had a feeling. That feeling, and the existence of atoms themselves, would be mathematically proven some 50 years later when Albert Einstein published his seminal work, Investigations on the Theory of the Brownian Movement (Einstein, 1905).

The diffusion coefficient, or diffusivity D, defined by Fick is a proportionality constant between the diffusion rate and the concentration gradient. The diffusion coefficient is defined for a specific solute-solvent pair, and the higher the value for the coefficient, the faster two substances will diffuse into one another. For example, at 25°C the diffusivity of gaseous air into gaseous water is 0.282 cm2/sec (Cussler, 1997). At the same temperature, the diffusivity of dissolved air into liquid water is 2.00 x 10-5 cm2/sec, a much lower number than that for the two gases, representing the much slower diffusion rate in liquids compared to gases. And the diffusivity of dissolved helium into liquid water at 25°C is 6.28 x 10-5 cm2/sec – higher than that of dissolved air, representing the smaller size of helium atoms compared to the nitrogen and oxygen molecules in air.

Comprehension Checkpoint

When dissolved into water, helium has a higher diffusivity than air. This means that helium dissolves at a ______ rate than air.

Yet another factor that influences the rate at which diffusion occurs is the distance a molecule travels before bumping into something (referred to as a molecule’s mean free path). Imagine taking a container filled with a gas and putting it under pressure so that the molecules in the gas are squeezed together. This would slow the rate of diffusion through the gas because the molecules travel a shorter distance before colliding with something else and changing direction. (The animation below shows the effect of pressure on diffusion.)

Interactive Animation: The Effect of Pressure on Diffusion

This is an important factor affecting the difference in diffusion rates in gases versus liquids versus solids; because gas particles are the most spread out of the three, molecules travel the furthest between collisions and diffusion occurs most rapidly in this state (Figure 5).

To fully understand why we can smell the cookies baking in the kitchen from the bedroom we also have to consider another process at work here – advection. Advection involves the transfer of a material or heat due to the movement of a fluid. So, because people walk through the rooms of your house and because heat rises from your radiators, the air is constantly moving, and that movement carries and mixes the scent molecules in your house. In many situations (such as your house), the effects of advection exceed those of diffusion, but these processes work in tandem to bring you the cookie smell.

From the traveling smells of cookies to the dissolving of salt into water, diffusion is a process happening around (and within!) us every second of every day. It is a process that is critical to moving oxygen across the membranes of our lungs, moving nutrients through soil to be taken up by plants, dispersing pollutants that are released into the atmosphere, and a whole host of other events that are necessary for life to exist.

The process of diffusion is critical to life, as it is necessary when our lungs exchange gas during breathing and when our cells take in nutrients. This module explains diffusion and describes factors that influence the process. The module looks at historical developments in our understanding of diffusion, from observations of “dancing” particles in the first century BCE to the discovery of Brownian motion to more recent experiments. Topics include concentration gradients, the diffusion coefficient, and advection.

Key Concepts

• Diffusion is the process by which molecules move through a substance, seemingly down a concentration gradient, because of the random molecular motion and collision between particles.

• Many factors influence the rate at which diffusion takes place, including the medium through with a substance is diffusing, the size of molecules diffusing, the temperature of the materials, and the distance molecules travel between collisions.

• The diffusion coefficient, or diffusivity, provides a relative measure at specific conditions of the speed at which two substances will diffuse into one another.

Despite its scarcity across the universe, water is so abundant on Earth that we aren’t always aware of how special it is. For starters, water is the only substance that exists naturally on our planet as a solid (ice and snow), liquid (rivers, lakes, and oceans), and a gas (water in the atmosphere as humidity). As you might recall (or can read about in our module on States of Matter), water molecules are in a different energy state in each phase. The amount of energy required to go from solid to liquid and liquid to gas is related to how water molecules interact with each other. Those interactions are, in turn, related to how the atoms within a water molecule interact with each other.

Our Chemical Bonding: The Nature of the Chemical Bond module discussed how a dipole forms across a water molecule; in the bond between oxygen and hydrogen, the electrons are shared unequally, drawn a bit more to the oxygen. As a result, a partial negative charge (ð-) forms at the oxygen end of the molecule, and a partial positive charge (ð+) forms at each of the hydrogen atom ends (Figure 1).

Since the hydrogen and oxygen atoms in the molecule carry opposite (though partial) charges, nearby water molecules are attracted to each other like tiny little magnets. The electrostatic attraction between the ð+ hydrogen (ð stands for partial charge, a value less than the charge of an electron) and the ð- oxygen in adjacent molecules is called hydrogen bonding (Figure 2).

Hydrogen bonds make water molecules "stick" together. These bonds are relatively weak compared to other types of covalent or ionic bonds. In fact, they are often referred to as an attractive force as opposed to a true bond. Yet, they have a big effect on how water behaves. There are many other compounds that form hydrogen bonds, but the ones between water molecules are particularly strong. Figure 2 shows why. If you look at the central molecule in this figure you see that the oxygen end of the molecule forms hydrogen bonds with two other water molecules; in addition, each hydrogen on the central molecule is attracted to a separate water molecule. As the illustration shows, each water molecule forms attractions with four other water molecules, a network of connections that makes the hydrogen bonding in water particularly strong and lends the substance its many unique properties.

Now it’s time to make use of that glass of water. If you have some ice cubes, drop one in your glass. You’ll notice that it floats. Its ability to bob to the top of the water line means that the ice (water in its solid state) is less dense than liquid water. (To review density and buoyancy, see our Density module) This isn’t a common state of affairs; if you put a chunk of solid wax into a vat of molten wax, it will sink toward the bottom (and possibly melt before it gets there).

To understand what causes ice to float but solid wax to sink, let’s think first about what happens when a liquid turns to a solid (again, the States of Matter module can be a handy review here). In a liquid, the molecules have enough kinetic energy to keep moving around. As molecules come near to each other, they are drawn together by intermolecular forces. At the same time, molecules have enough kinetic energy to break free of those forces and be drawn to other nearby molecules. Thus the liquid flows because intermolecular attractions can be broken and reformed.

A liquid freezes when the kinetic energy is reduced (i.e. the temperature is reduced) enough that the attractive forces between molecules can no longer be broken, and the molecules become locked in a static lattice. For nearly all compounds, the lower energy and lack of movement between molecules means the molecules in a solid are packed together more tightly than the liquid state. This is the case with wax and so solid wax is denser than the liquid and sinks.

In the case of water, though, the shape of the molecule and the strength of the hydrogen bonds affect the arrangement of the molecules. In liquid water, hydrogen bonding pulls molecules closely together. As water freezes, the dipole ends with like charges repel each other, forcing the molecules into a fixed lattice in which they are farther from each other than they are in liquid water (Figure 3). More space between molecules makes the ice less dense than liquid water, and thus it floats.

Water is sometimes referred to as the “universal solvent,” because it dissolves more compounds than any other liquid known. The polarity of the water molecule allows it to readily dissolve other polar molecules, as well as ions. (See our Solutions, solubility, and colligative properties module for a deeper discussion of dissolution.)

This ability to dissolve substances is one of the properties that makes water vital for life. Most biological molecules, such as DNA, proteins, and vitamins are polar, and important ions such as sodium and potassium are also charged. In order for any of these compounds to carry out functions in the body, they have to be able to circulate in the blood and the fluid within and between cells, all of which are mostly water. Because of its polarity, water is able to dissolve these and other substances, allowing their free movement around the body. A few biomolecules, such as fats and cholesterol, aren’t polar, and don’t dissolve in water – however, the body has developed unique ways to circulate and store these substances.

Water is also able to dissolve gasses such as oxygen, allowing fish, plants, and other aquatic life to access this dissolved oxygen (Figure 4). O2 isn’t a polar molecule; it dissolves because the polar charges in the water molecule induce a dipole in the oxygen, making it soluble and so available to aquatic life. (Learn more about induced-dipole interactions in our Properties of Liquids module.)

Let’s return to your water glass. Fill the glass just to the rim and stop. Then, slowly, add a little bit more. You’ll see that you can actually fill the glass a bit past its rim, and the edges of the water will round out against the glass, holding the water in.

Once again, hydrogen bonding is behind this act, resulting in cohesion. Cohesion occurs when molecules of the same kind are attracted to each other. In the case of water, the molecules form strong hydrogen bonds, which hold the substance together. As a result, water is highly cohesive, in fact, it is the most cohesive of all non-metallic liquids.

Cohesion occurs throughout your glass of water, but it’s especially strong at the surface. Molecules there have fewer neighbors (because they have none at the very surface), and so create stronger bonds with the molecules that are near them. The result is called surface tension, or the ability of a substance to resist disruption to its surface. Dip your finger into your water glass and then pull it out. The drop that forms at the end of your fingertip is held together by surface tension.

Surface tension was the misunderstood central player in a raucous debate between Galileo Galilei and his chief rival, Ludovico delle Colombe in 1611. Delle Colombe, a philosopher, was at odds with some of Galileo’s ideas, including his explanation that ice floats on water because it is less dense. So the philosopher challenged Galileo to a debate, which delle Colombe believed would prove his own intellectual superiority.

Delle Colombe championed the (incorrect) idea that ice floats not because of density, but because of its shape, which he saw as broad and flat, as is ice on a lake. To prove the “truth” of his theory, he used ebony wood, which is slightly denser than water, in a demonstration before an audience of curious spectators. He dropped a sphere of the wood into water, and it sank. He then placed a thin wafer of the wood flat on the water’s surface, and it floated. Delle Colombe pronounced himself the winner.

Galileo left frustrated. His observations of the world gave him evidence that his explanation, not delle Colombe’s, was right, but he couldn’t explain the outcome of delle Colombe’s experiment.

Had he known about molecules and dipoles and hydrogen bonds at the time, Galileo certainly would have offered this explanation: When delle Colombe floated the thin ebony disc, he was taking advantage of the cohesive nature of water and the surface tension that arises from it (Figure 5). As the ebony wafer appeared to float on the water, the force exerted by its mass was distributed throughout the surface of the water beneath it. In other words, a single pinpoint-sized area of surface water only had to support the pinpoint-sized piece of ebony just above it. The hydrogen bonds between the water molecules were strong enough to support the weight of the disc. When delle Colombe placed the sphere in the water, however, the pinpoint-sized area that first touched the water bore the weight of the entire sphere, which was more than the water’s surface tension could support. Had Galileo known this at the time, he could have disproved delle Colombe easily – had he simply pushed the wafer through the surface to break the surface tension, the wafer would have sunk.

This same surface tension is what allows leaves to stay at the surface of a lake and dewdrops to adhere to a spider’s web. Even some animals take advantage of this phenomenon – the Basilisk lizard (Figure 6), water striders, and a few other small animals and bugs appear to "walk" on water by taking advantage of the surface tension of water.

For your next observation, take another sip of water, and notice the side of the glass. Chances are you’ll see a few drops stuck to it. Gravity is pulling down on these drops, so something else must be keeping them stuck there. That something else is adhesion, the attraction of water to other kinds of molecules; in this case, the molecules that make up the glass. Because of the polarity of the molecule, water exhibits stronger adhesion to those surfaces that have some net electrical charge, and glass is one such surface. But place a drop of water on a non-polar surface, such as a piece of wax paper and you will see it take a different shape than one to which it adheres. On the wax paper, the water droplets take the shape of a true droplet because there is little adhesion and the cohesive forces pull the drop into a sphere. But on glass you will see the droplets flatten and deform a bit as the adhesive forces draw it more to the surface of the glass.

Both cohesion and adhesion (Figure 7) occur with many compounds besides water. Pressure sensitive tapes, for example, stick to surfaces because they are coated with a high viscosity fluid that adheres to the surface to which they are pressed. Generally, you can overcome this adhesive force by pulling, for example – you can easily lift a Post-it® Note from a page. But sometimes the adhesive forces are stronger than the forces holding the surface together – pull tape off of a piece of paper and you remove pieces of the paper with the tape.

Let’s return to our glass of water, and look inside to where the water surface meets the glass. The very edge of the water surface curves upward slightly on the glass. That’s also adhesion – the water is drawn up the surface by adhesion with the glass. If you have a clear plastic straw, you can put one end of it into the water and see that the liquid climbs up the straw a bit, above the surface of the remaining glass of water. It’s actually moving upward against gravity!

What’s happening in your straw is a phenomenon called capillary action (Figure 7). Capillary action occurs in small tubes, where the surface area of the water is small, and the force of adhesion—water’s attraction to the polar glass or other material—overcomes the force of cohesion between those surface molecules.

Another way to see the effects of adhesion and cohesion is to compare the behavior of polar and nonpolar liquids. When you put water in a test tube, adhesion makes the water along the edges move slightly upward and creates a concave meniscus. Liquid mercury, on the other hand, is not polar and therefore not attracted to glass. In a test tube, cohesion at the surface of the mercury is much stronger than adhesion to the glass. The surface tension in the mercury forms a convex meniscus, much the same as the way water forms a slight bulge over the top of your very full glass (Figure 8).

Adhesion and capillary action are among the forces at play that help plants take up water (and dissolved nutrients) in their roots. Capillary action also keeps your eyes from drying out, as saline water flows from tiny ducts in the outer corners of your eyes. With each blink, you spread the water away from the duct, and capillary action brings more fluid to the surface.

If you want to see capillary action at work, put a few drops of red food coloring in your glass of water, and then drop a stalk or two of leafy celery into it. After a day or two, your green celery will be streaked with red.

Water is a truly unusual and important substance. The unique chemical properties of water that give rise to surface tension, capillary action, and the low density of ice play vital roles in life as we know it. Floating ice protects aquatic organisms and keeps them from being frozen in the winter. Capillary action keeps plants alive. Surface tension allows lily pads to stay on the surface of a lake. In fact, water’s chemistry is so complex and important that scientists today are still striving to understand all the feats this simple substance can perform.

Key Concepts

• Water has a number of unique properties that make it important in both the chemical and biological worlds.

• The polarity of water molecules allows liquid water to act as a "universal solvent," able to dissolve many ionic and polar covalent compounds.

• The polarity of water molecules also results in strong hydrogen bonds that give rise to phenomena such as surface tension, adhesion, and cohesion.

The English scientist Michael Faraday can reasonably be considered one of the greatest minds ever in the fields of electrochemistry and electromagnetism. Somewhat paradoxically, all of Faraday’s pioneering work was carried out prior to the discovery of the fundamental particle that these electrical phenomena depend upon. However, one of Faraday’s earliest experimental observations was a crucial precursor to the discovery of the first subatomic particle, the electron.

As early as the mid-17th century, scientists had been experimenting with glass tubes filled with what was known then as rarefied air. Rarefied air referred to a system in which most of the gaseous atoms had been removed, but where the vacuum was not complete. In 1838, Faraday noted that when passing a current through such a tube, an arc of electricity was observed. The arc started at the negative plate (known as the cathode) and traveled through the tube to the oppositely charged anode (Faraday, 1838).

In his experiments, Faraday observed a luminescence that started part way down the tube, and traveled toward the anode. This left an area between the cathode and the start of the luminescence that was not illuminated, and subsequently became known as Faraday’s dark space (Figure 1). Faraday couldn’t fully explain his observations, and it took a number of further developments in terms of the technology of the tubes, before a greater understanding emerged.

Comprehension Checkpoint

Faraday is famous for discovering and naming the electron.

In 1857 the German glassblower Heinrich Geissler, while working for fellow countryman and physicist Julius Plücker at the University of Bonn, improved the quality of the vacuum that could be achieved in such tubes. However, Geissler’s tubes still contained enough gaseous atoms that when the electrical current travelled in the tube, there was an interaction between the two, causing the tubes to glow. In the mid-19th century these Geissler tubes were largely nothing more than a curiosity, but interestingly, a curiosity that proved to be a forerunner of neon lights.

Englishman William Crookes repeated experiments similar to those of Faraday and Geissler, but this time with the ‘new and improved’ vacuums. The number of gas atoms (and hence the pressure) was drastically reduced in Crookes’ tubes. This caused an interesting effect: Faraday’s dark space was observed even further down the tube, again extending away from cathode toward the anode.

In addition to the extension of the dark space, fluorescence was observed on the glass behind the anode at the positive end of the tube. When further experimentation revealed that shadows of objects placed in the tube were cast onto the glass behind the anode, the German physicist Johann Hittorf proposed that the shadows must have been created by something travelling in a straight line from the cathode to the anode. Yet another German physicist Eugen Goldstein, christened these invisible beams cathode rays.

As discussed in our module Early Ideas about Matter, Dalton’s atomic theory suggested that the atom was indivisible, i.e., that it was the smallest particle that made up matter, and that all matter was based upon that single unit. Experiments with cathode ray tubes dramatically changed that view when they led to the discovery of the first subatomic particle.

J.J. Thomson was an English physicist who worked with cathode ray tubes similar to those used by Crookes and others in the mid-19th century. Thomson’s experiments (Thomson, 1897) went further than those before him and provided evidence of the properties of the “something” hinted at by Hittorf. Thomson noted that cathode rays were deflected by magnetic fields and that the deflection was the same no matter what the source of the rays. This suggested that the rays were universal in their properties and that they had some magnetic charge. Thomson further demonstrated that cathode rays were charged because they could be deflected by an electric field. He found that the rays were deflected toward a positive plate and away from a negative plate, thus determining that they were made up of negatively charged particles of some sort.

Finally, Thomson applied both electrical and magnetic fields to the cathode ray at the same time. Knowing the strength of the fields applied and measuring the deflection of the particle stream in the tube, he was first able to measure the velocity of the particles in the stream. Then, by measuring the deflection of the ray while varying the two fields, Thomson was able to measure the mass-to-charge ratio of the particles in the stream and he found something astonishing. The negative particles had a mass-to-charge ratio that was over 1,000 times lower than that of a hydrogen atom, suggesting that the particles were incredibly tiny – much smaller than the smallest atom known. This fact allowed Thomson to definitively say that the atom was not the fundamental building block of matter and that smaller (subatomic) particles existed. Thomson originally called these particles corpuscles, but later they became known as electrons.

Comprehension Checkpoint

J.J. Thomson determined that cathode rays were made up of

Thomson’s discovery made sense of all of the previous observations made by Faraday, Geissler, and Crookes. Zipping through a tube filled with gas but partially under vacuum, electrons would eventually slam into those gas atoms, knocking off some of their electrons and making them fluoresce. The dark space that Faraday first noted was due to the distance needed for the electrons to accelerate to the speed necessary to ionize the tube’s gas atoms.

In the better vacuums achieved in the Crookes’ tubes, the electrons could travel further distances without interacting with gas molecules because of the lower density of molecules in the tube, thus extending the dark space.

Comprehension Checkpoint

The less gas there was in a cathode ray tube, the _____ dark space could be observed.

With the electron now discovered, Thomson went on to propose an entirely new model of the atom that was known as "The Plum Pudding Model." The model was so called since it mimicked the British desert of the same name that had dried fruit (primarily raisins not plums), dispersed in a body of suet and eggs that made a dough.

In his model Thomson proposed that the negatively charged electrons (analogous to the raisins) were randomly spread out among what he called "a sphere of uniform positive electrification" (analogous with the dough or body of the pudding) (see Figure 2).

Thompson’s model of the atom as a doughy clump of positive and negative particles persisted until 1911, when Ernest Rutherford, a former student of Thomson’s, advanced atomic theory yet another notch.

During the years 1908–1911, Ernest Marsden and Hans Geiger performed a series of experiments under the direction of Ernest Rutherford at the University of Manchester in England. In these experiments alpha particles (tiny, positively charged particles) were fired at a thin piece of gold foil (Figure 3). Under the Thomson Plum Pudding model of the atom, the "sphere of uniform positive electrification" was thought to be so diffuse that the tiny, fast moving alpha particles would pass straight through. Similarly, the electrons in the model were thought to be so tiny that any electrostatic interactions between them and the positive alpha particles would be minimal, so the path of the alpha particles would hardly be affected.

As predicted, Rutherford and his co-workers observed that most of the alpha particles passed straight through the gold foil, and some particles were deflected at small angles. However, contradictory to what the Plum Pudding model predicted, a few rebounded at very sharp angles, some even flying straight back toward the source! These particles were acting as if they were encountering a hard object, like a tennis ball bouncing off a brick wall (Figure 4).

The fact that most of the alpha particles passed straight through the gold foil suggested to Rutherford that atoms are made up of largely empty space. However, contrary to the Thomson Plum Pudding model, Rutherford’s work suggested that there was a dense, positively charged area in an atom that caused the observed repulsion and backscattering of alpha particles. Rutherford was astonished by these observations and famously said:

It was quite the most incredible event that has ever happened to me in my life. It was almost as incredible as if you fired a 15-inch shell at a piece of tissue paper and it came back and hit you. On consideration, I realized that this scattering backward must be the result of a single collision, and when I made calculations I saw that it was impossible to get anything of that order of magnitude unless you took a system in which the greater part of the mass of the atom was concentrated in a minute nucleus. It was then that I had the idea of an atom with a minute massive centre, carrying a charge.

Over a series of experiments and papers (Rutherford, 1911, 1913, 1914), Rutherford developed a model of the atom with a dense, positively charged area of the atom at the center, now known as the nucleus – and the nuclear model of the atom was born.

Comprehension Checkpoint

Rutherford and his colleagues were surprised that

Following the discovery of the electron, Nobel Prize-winning physicist Robert Millikan conducted an ingenious experiment that allowed for the specific value of the negative charge of the electron to be calculated. In his famous oil drop experiment, Millikan and co-workers sprayed tiny oil droplets from an atomizer into a sealed chamber (Millikan, 1913). The oil drops fell downward, under the influence of gravity, into a space between two electrical plates. There they became charged, by interacting with air that had been ionized by X-rays.

By adjusting the voltage between the two electrical plates, Millikan applied an electrical force upward that exactly matched the gravitational force downward, thus suspending the drops motionless. When suspended, the electrical force and the force of gravity were working in opposite directions but were equal in magnitude. Hence:

where q is the charge on the oil drop, E is the electric field, m is the mass of the oil drop, and g is the gravitational field. By measuring the mass of each oil drop and knowing both the gravitational and the electrical field, the charge on each drop could be determined.

Millikan found that there were differing charges on different oil drops. However, in each case the charges on the oil drops were found to be multiples of 1.60 x 10-19 coulombs. He concluded that the differing charges were due to different numbers of electrons, each having a negative charge of 1.60 x 10-19 coulombs, and hence the charge on the electron was found.

Comprehension Checkpoint

Millikan found that the different oil drops in his experiment

Thomson’s electron and Rutherford’s nuclear model were tremendous advancements. The Japanese scientist Hantaro Nagaoka had previously rejected Thomson’s Plum Pudding model on the grounds that opposing charges could not penetrate each other, and he counter-proposed a model of the atom that resembled the planet Saturn with rings of electrons revolving around a positive center. Upon hearing of Rutherford’s work, he wrote to him in 1911 saying, "Congratulations on the simpleness of the apparatus you employ and the brilliant results you obtained."

But, the planetary model was not perfect, and several inconsistent experimental observations meant much work was still to be done. At the time the electron was still thought of as a small particle, and it was thought to spin almost randomly around the nucleus of the atom. It would take the additional experiments and the genius of Neils Bohr, Max Planck, and others to make the paradigm shift from classical physics in which atoms consist of tiny particles and are governed by laws of motion, to quantum mechanics in which electrons behave like waves and exhibit strange and exotic behaviors. (See our interactive animation comparing orbital and quantum models of the first 12 elements.) To learn more about the strange behaviors of quantum physics, read the other entries in our Atomic Theory series: II: Bohr and the Beginnings of Quantum Theory, III: Wave-Particle Duality and the Electron, and IV: Quantum Numbers and Orbitals.

Interactive Animation: Atomic and ionic structure of the first 12 elements

The 19th and early 20th centuries saw great advances in our understanding of the atom. This module takes readers through experiments with cathode ray tubes that led to the discovery of the first subatomic particle: the electron. The module then describes Thomson’s plum pudding model of the atom along with Rutherford’s gold foil experiment that resulted in the nuclear model of the atom. Also explained is Millikan’s oil drop experiment, which allowed him to determine an electron’s charge. Readers will see how the work of many scientists was critical in this period of rapid development in atomic theory.

Key Concepts

• Atoms are not dense spheres but consist of smaller particles including the negatively charged electron.

• The research on passing electrical currents through vacuum tubes by Faraday, Geissler, Crookes, and others laid the groundwork for discovery of the first subatomic particle.

• J.J. Thomson’s observations of cathode rays provide the basis for the discovery of the electron.

• Rutherford, Geiger, and Marsden performed a series of gold foil experiments that indicated that atoms have small, dense, positively-charged centers – later named the nucleus.

• Millikan’s oil drop experiment determines the fundamental charge on the electron as 1.60 x 10-19 coulombs.

The story of atomic theory first encounters reproducible, scientific (evidence based) proof in the late 18th century. French chemists Antoine Lavoisier and Joseph Proust, with their Law of Conservation of Mass in 1789 and Law of Definite Proportions in 1799, respectively, each laid the groundwork for Englishman John Dalton’s work on the Law of Multiple Proportions (Dalton, 1803). Given that many centuries had elapsed between the earliest ideas of the atom and Dalton’s work, it would be fair to say that the evolution of atomic theory had been a gradual one, with progression in the field being steady rather than spectacular. But that was all about to change, and quite dramatically.

The most intense period of progress took place between the late 19th and early 20th century, and it hinged heavily on the work of a Danish physicist named Niels Bohr. Like so many before him, Bohr built upon the work of his predecessors, and for Bohr, part of that foundation had been built by Ernest Rutherford.

Based upon a series of experiments, Rutherford proposed the planetary model of the atom in which electrons swirled around a hard, dense nucleus (see Atomic Theory I: The Early Days). While Rutherford’s model explained many observations accurately, it was found to have flaws.

Rutherford’s planetary model of the atom was based upon classical physics – a system that deals with physical particles, force, and momentum. Unfortunately, this same system predicted that electrons orbiting in the manner that Rutherford described would lose energy, give off radiation, and ultimately crash into the nucleus and destroy the atom. However, for the most part, atoms are stable, lasting literally billions of years. Furthermore, the radiation predicted by the Rutherford model would have been a continuous spectrum of every color – in essence white light that when passed through a prism would display all of the colors of the rainbow (Figure 1).

But when pure gases of different elements are excited by electricity, as they would have been when placed in the newly discovered electric-discharge tube, they emit radiation at distinct frequencies. In other words, different elements do not emit white light, they emit light of different colors, and when that light is passed through a prism it does not produce a continuous rainbow of colors, but a pattern of colored lines, now referred to as line spectra (Figure 2). Clearly, Rutherford’s model did not fit with all of the observations, and Bohr made it his business to address these inconsistencies.

In 1911, Niels Bohr (Figure 3) had just completed his doctorate in physics at the University of Copenhagen and was invited to continue his work at the University of Manchester in England by Rutherford. Rutherford had already significantly advanced atomic theory with his groundbreaking gold foil experiment, but Bohr’s genius was in taking the Rutherford model and advancing it further.

It wasn’t Bohr who had come up with the original idea of a planetary model of the atom, but he was the one who took the fundamental concept and applied new ideas about quantum theory to it. This leap was necessary to explain the new evidence that challenged the old model, and to subsequently formulate a new, ‘better’ model.

Interactive Animation: Bohr’s Atom

Comprehension Checkpoint

The planetary model of the atom was based on

It’s easy to think of light, and other forms of energy, as continuous. Turn up the dimmer switch on your lamp, and the lamp gets gradually brighter. However, by the late 1800s, physicists were beginning to suspect that this was not in fact true. Classical physics models failed to accurately predict black-body radiation; in other words, classical physics did not accurately predict the energy given off by an object when it was heated.

The German physicist Max Planck solved this problem in 1903 by proposing that black-body radiation energy had to be quantized, i.e., that it could only be released or absorbed in specific ‘packets’ that were associated with specific frequencies. This solved the black-body problem and was consistent with the observed experimental data. Thus, quantum mechanics was born.

Despite the advances that he and others made using this idea, interestingly, Planck remained quite skeptical of quantized energy for many years. He insisted that the calculations that he had done, and the conclusions that he had reached, were somehow a sophisticated mathematical trick and that ultimately the old, classical model would prevail. After all, it had been around for approximately 200 years and had stood up to some pretty intense scrutiny.

In 1905, Albert Einstein published a series of papers proposing that light also exhibited quantum behavior (Einstein, 1905). Sometimes described as Einstein’s Annus mirabilis (miracle year), the papers taken together, and combined with Planck’s work, allowed Bohr to marry the nature of the atom with physics to usher in a new dawn of understanding in atomic theory.

In 1913, building on Planck’s and Einstein’s theories of quantization, Bohr proposed that the electron itself was quantized – that it could not exist just anywhere around an atom (as suggested by the Rutherford model) but instead could only be found in specific positions, with specific energies. The electron could transition to different positions, but only in discrete, defined steps. It could not spin in any location around the nucleus of an atom but instead was restricted to specific areas of space – much like the planets in our solar system are restricted to specific paths.

Being negatively charged, electrons are attracted to the positive protons in the nucleus of an atom, and will normally occupy the orbital, or path, within an atom that is closest to the nucleus if it is available. This state, which has low potential energy, is called the ground state. By exposing the electrons to an external source of energy such as an electric discharge, it is possible to promote the electrons from their ground state to other positions that have higher potential energies, called excited states. These ‘excited’ electrons quickly return to lower energy positions (in order to regain the stability associated with lower energies), and in doing so they release energy in specific frequencies that correspond to the energy differences between electron orbitals, or shells (see quantum behavior simulation). Bohr’s mathematical equations further predicted that electrons would not crash into the nucleus in a manner that classical physics – and Rutherford’s model – had predicted. This was another crucial realization that made a jump from one paradigm (classical physics) to a new one (quantum physics).

Interactive Animation: Atomic and ionic structure of the first 12 elements

Bohr’s discovery that Planck’s quantum theory could be applied to the classical Rutherford model of the atom, and could account for the observed shortcomings in the original model, is another beautiful example of how scientific theory uses prior evidence, coupled with new experimental observations, to adapt, develop, and change models and understanding over time. Science is usually advanced by contemporary scientists building on the work of predecessors and, as Isaac Newton put it in his 1676 letter to Robert Hooke (both prominent scientists of their time), by their “standing on the shoulders of giants.” Bohr’s work built on the theories of those before him, and extended them to explain the experimentally observed line spectra of atoms in a mathematical proof that made perfect sense.

Comprehension Checkpoint

In an atom, the ground state is the orbital with the ______ potential energy.

While Bohr’s work seemed to explain the curious phenomenon of line spectra, a number of lines had been observed in the spectra of hydrogen that did not fit Bohr’s theory. At first glance this appeared to poke holes in Bohr’s ideas, but Bohr was quick to offer an explanation. He suggested that the lines in the spectrum for hydrogen that could not be accounted for were actually caused not by hydrogen atoms, but rather by an entirely different species altogether. So, what were these different species, and how did they come to be?

Almost 30 years prior to Bohr's publishing his famous trilogy of papers in the Philosophical Magazine and Journal of Science in 1913, the idea of particles having the ability to carry some kind of charge had been established by the Swedish scientist Svante Arrhenius and the Englishman Michael Faraday. These charged particles had been christened ions.

Atoms are electrically neutral, meaning that the number of positive protons in any given atom equals the number of negative electrons in the same. Since the positive charge is exactly cancelled by the negative, the atom has no overall electrical charge. However, just as it is possible to excite an electron to a higher orbital, as Bohr’s work had shown, it is also possible to give an electron sufficient energy to overcome the attraction of the nucleus completely and to remove it from the atom entirely. This has the effect of unbalancing the electrical charge and results in the formation of a species with an overall positive charge – called a cation. For example, a sodium atom can lose an electron to form a positively charged sodium cation (Equation 1); the energy associated with the ejection of the first electron from any atom is called the first ionization energy.

Na(s) → Na+(s) + e-

Cations that are formed by the ejection of one electron can be further ionized by losing additional electrons, and in the process forming another ion, this time with a 2+ charge. The energy required for the ejection of the second electron is known as the second ionization energy. Although it rarely occurs with larger atoms (or indeed with the sodium example given above), it is theoretically possible to remove all of the electrons from any given atom, leading to third, fourth, fifth, etc. ionization energies for atoms with large numbers of electrons.

Once an electron (or electrons) has been ejected from an atom in this manner, it can be accepted by other atoms, and as such, electrons can be transferred from one atom to another. Just as releasing electrons unbalances the charge, accepting electrons causes atoms to become unbalanced in terms of their charge as well, and once again an ion is formed. This time, the ion has a negative charge (a greater number of electrons than protons), and the species is called an anion. (A hydrogen cation and anion are shown in Figure 4.) For example, a neutral chlorine atom, with equal numbers of protons and electrons, can accept an electron from an external source to form a negatively charged chloride anion (Equation 2).

Cl(g) + e- → Cl-(g)

Ions have wildly different properties when compared to their parent atoms, and this transfer of electrons from one atom to another is an example of how a small change in the structure of an atom can make a large difference in the behavior and nature of the particles. For example, sodium metal is unstable, reacting violently with water and corroding instantaneously in air. Thus, coming into contact with free sodium metal would be extremely dangerous. Similarly, chlorine exists as a gas under ambient conditions and it is highly poisonous, scarring the lungs of anyone who breathes it (in fact, free chlorine gas was used as a chemical weapon in World War I). However, when the two substances react with one another, sodium loses an electron forming a cation, and chlorine accepts the same electron to form an anion. The two resulting ions then bond together as a result of their charges, and together create a very common substance – table salt, which is neither reactive nor poisonous.

Comprehension Checkpoint

_____ have a positive charge, while _____ have a negative charge.

In Bohr’s work, the ionization of one particular element, helium, proved to be the key to unlock the explanation for the unexpected lines that he observed in hydrogen’s spectrum. When a helium atom, which has two electrons, loses an electron to form the helium ion, He+, its electronic structure mimics that of atomic hydrogen, since both species only have only one electron – the helium ion is said to be isoelectronic with the hydrogen atom. However, the helium ion possesses a nucleus with double the charge of a hydrogen atom (two protons as opposed to one proton). Bohr realized this, and suggested that the greater attraction between the electron and He nucleus accounted for the spectral lines that were previously unexplained – the charge of the nucleus affected the energy associated with transfers of electrons between the orbitals. Bohr’s theory was proven correct when spectra were generated using ionized helium that had been purged of hydrogen.

At the close of the 19th century, two different particles were known to exist in the atom, and both possessed an electrical charge – the very small and negatively charged electron and the much larger and positively charged proton. However, by the beginning of the 20th century, evidence began to mount that this was not a complete picture of the atom. Specifically, the mass of protons and electrons in an atom did not appear sufficient to justify the mass of the whole atom, and certain types of nuclear decay suggested that something else might be going on in the nucleus. In 1932, James Chadwick, a British physicist who had studied with, and was working for, Ernest Rutherford at the time, set out to solve the problem. Rutherford had proposed the idea of a neutral atomic particle that had mass as early as 1920, but he never managed to gain traction in the hunt for this mysterious particle.

In 1932 Chadwick further developed an experiment that had been first performed by Frederic Joliot-Curie and Irene Joliot-Curie. They found that by using polonium as a source of alpha particles, they could cause beryllium to emit radiation that, in turn, could be used to knock protons out of a piece of paraffin wax. The Joliot-Curies proposed that this radiation was gamma radiation, a packet of energy with no true mass. As an accomplished researcher of gamma rays and of the nucleus of the atom, Chadwick realized what others had not – that protons were too massive to be ejected from paraffin by mass-less gamma rays. By more carefully measuring the impact of the mystery particle on the paraffin wax and combining this with other measurements, Chadwick concluded that the particles being emitted were not gamma radiation, but a relatively heavy particle that had no charge – a particle named the neutron (Figure 5).

Chadwick wrote a paper about his discovery entitled “The Possible Existence of a Neutron,” and it was published in the journal Nature (Chadwick, 1932). In 1935, he was awarded the Nobel Prize in Physics for his discovery. The Joliot-Curies did not go without recognition either. Their work on radioactivity and radioactive isotopes won them the Nobel Prize for Chemistry, also in 1935.

Chadwick’s discovery marked the genesis of induced nuclear reactions, where the neutron is accelerated and crashed into the nuclei of other elements, generating massive amounts of energy (the neutron can easily do this since being neutral it is not repelled from the nucleus in the way that positively charged particles would be). These reactions had a massive impact upon the world as a whole since they spawned thoughts of the atomic bomb and of nuclear energy.

Comprehension Checkpoint

A _____ is a heavy particle that has no charge.

The neutron also explains the existence of atoms of the same element that have different atomic masses. Isotopes are atoms of the same element (i.e., they have the same numbers of protons) but that differ in the number of neutrons they possess. As a result, different isotopes have similar chemical properties, but their masses, and in some cases their physical behavior, differ. Isotopes are differentiated by their atomic mass, which can be indicated by writing the element’s symbol, followed by a dash and then the mass, or, more commonly, by writing the mass as a superscript before the element symbol. For example, carbon-12 (C-12, or 12C) and carbon-14 (C-14, or 14C) are both naturally occurring isotopes of carbon. Carbon-12 is a stable isotope that accounts for almost 99% of naturally occurring carbon. Carbon-14 is a radioactive isotope that accounts for only about 1 x 10-10 % of naturally occurring carbon, but because it decays to nitrogen with a half-life of approximately 5,730 years, it can be used to date some carbon-containing objects. (Three isotopes of carbon are depicted in Figure 6.)

There is often more than one naturally occurring isotope of any given, individual element. As a result, the atomic masses that are given on a modern periodic table are the weighted masses of all the known isotopes of each element. For example, naturally occurring chlorine has two principal isotopes, one with 18 neutrons (mass of 35), and one with 20 neutrons (mass of 37). The lighter isotope 35Cl, accounts for almost 76% of its natural abundance, while the 37Cl isotope accounts for only about 24%. Thus the weighted mean is the average mass of naturally occurring chlorine atoms weighted by their relative abundance, or 35.45.

Comprehension Checkpoint

Isotopes are atoms of the same element that have a different

Bohr’s work provided a bridge between several disparate ideas that may never have been linked together without his intervention. He changed the paradigm from applying classical (particle) physics to the model of the atom, to thinking about the application of quantum theory and waves – a truly crucial development in the grand scheme of atomic theory and one that laid the groundwork for future scientists to build upon.

Having traveled from the earliest work on atomic theory through the crucial early part of the 20th century, we still have some way to go in the story of the atom. The advancements described in this module were grounded in Rutherford’s work, modified by Planck’s insights, and pieced together by Bohr’s genius. Still, there would be at least another decade that had to pass before work by Pauli, Heisenberg, and ultimately Schrödinger led to the full development of modern quantum mechanics that completely describes the atom as we know it today.

This module is an updated version of our previous content, to see the older module please go to this link.

The 20th century brought a major shift in our understanding of the atom, from the planetary model that Ernest Rutherford proposed to Niels Bohr’s application of quantum theory and waves to the behavior of electrons. With a focus on Bohr’s work, the developments explored in this module were based on the advancements of many scientists over time and laid the groundwork for future scientists to build upon further. The module also describes James Chadwick’s discovery of the neutron. Among other topics are anions, cations, and isotopes.

Key Concepts

• Drawing on experimental and theoretical evidence, Niels Bohr changed the paradigm of modern atomic theory from one that was based on physical particles and classical physics, to one based in quantum principles.

• Under Bohr’s model of the atom, electrons cannot rotate freely around the atom, but are bound to certain atomic orbitals that both constrain and define an atom's electronic behavior.

• Atoms can gain or lose electrons to become electrically charged ions.

• James Chadwick completed the early picture of the atom with his discovery of the neutron, a neutral, nuclear particle that affects an atom’s mass and the different physical properties of atomic isotopes.

In the late 19th century, the father of the periodic table, Russian chemist Dmitri Mendeleev, had already determined that the elements could be grouped together in a manner that showed gradual changes in their observed properties. (This is discussed in more detail in our module The Periodic Table of Elements.) By the early 1920s, other periodic trends, such as atomic volume and ionization energy, were also well established.

The German physicist Wolfgang Pauli made a quantum leap by realizing that in order for there to be differences in ionization energies and atomic volumes among atoms with many electrons, there had to be a way that the electrons were not all placed in the lowest energy levels. If multi-electron atoms did have all of their electrons placed in the lowest energy levels, then very different periodic patterns would have resulted from what was actually observed. However, before we reach Pauli and his work, we need to establish a number of more fundamental ideas.

The development of early quantum theory leaned heavily on the concept of wave-particle duality. This simultaneously simple and complex idea is that light (as well as other particles) has properties that are consistent with both waves and particles. The idea had been first seriously hinted at in relation to light in the late 17th century. Two camps formed over the nature of light: one in favor of light as a particle and one in favor of light as a wave. (See our Light I: Particle or Wave? module for more details.) Although both groups presented effective arguments supported by data, it wasn’t until some two hundred years later that the debate was settled.

Interactive Animation: Atomic and ionic structure of the first 12 elements

At the end of the 19th century the wave-particle debate continued. James Clerk Maxwell, a Scottish physicist, developed a series of equations that accurately described the behavior of light as an electromagnetic wave, seemingly tipping the debate in favor of waves. However, at the beginning of the 20th century, both Max Planck and Albert Einstein conceived of experiments which demonstrated that light exhibited behavior that was consistent with it being a particle. In fact, they developed theories that suggested that light was a wave-particle – a hybrid of the two properties. By the time of Bohr’s watershed papers, the time was right for the expansion of this new idea of wave–particle duality in the context of quantum theory, and in stepped French physicist Louis de Broglie.

In 1924, de Broglie published his PhD thesis (de Broglie, 1924). He proposed the extension of the wave-particle duality of light to all matter, but in particular to electrons. The starting point for de Broglie was Einstein’s equation that described the dual nature of photons, and he used an analogy, backed up by mathematics, to derive an equation that came to be known as the “de Broglie wavelength” (see Figure 1 for a visual representation of the wavelength).

The de Broglie wavelength equation is, in the grand scheme of things, a profoundly simple one that relates two variables and a constant: momentum, wavelength, and Planck's constant. There was support for de Broglie’s idea since it made theoretical sense, but the very nature of science demands that good ideas be tested and ultimately demonstrated by experiment. Unfortunately, de Broglie did not have any experimental data, so his idea remained unconfirmed for a number of years.

It wasn’t until 1927 that de Broglie’s hypothesis was demonstrated via the Davisson-Germer experiment (Davisson, 1928). In their experiment, Clinton Davisson and Lester Germer fired electrons at a piece of nickel metal and collected data on the diffraction patterns observed (Figure 2). The diffraction pattern of the electrons was entirely consistent with the pattern already measured for X-rays and, since X-rays were known to be electromagnetic radiation (i.e., waves), the experiment confirmed that electrons had a wave component. This confirmation meant that de Broglie’s hypothesis was correct.

Interestingly, it was the (experimental) efforts of others (Davisson and Germer), that led to de Broglie winning the Nobel Prize in Physics in 1929 for his theoretical discovery of the wave-nature of electrons. Without the proof that the Davisson-Germer experiment provided, de Broglie’s 1924 hypothesis would have remained just that – a hypothesis. This sequence of events is a quintessential example of a theory being corroborated by experimental data.

Comprehension Checkpoint

Theories must be backed up by

In 1926, Erwin Schrödinger derived his now famous equation (Schrödinger, 1926). For approximately 200 years prior to Schrödinger’s work, the infinitely simpler F = ma (Newton’s second law) had been used to describe the motion of particles in classical mechanics. With the advent of quantum mechanics, a completely new equation was required to describe the properties of subatomic particles. Since these particles were no longer thought of as classical particles but as particle-waves, Schrödinger’s partial differential equation was the answer. In the simplest terms, just as Newton’s second law describes how the motion of physical objects changes with changing conditions, the Schrödinger equation describes how the wave function (Ψ) of a quantum system changes over time (Equation 1). The Schrödinger equation was found to be consistent with the description of the electron as a wave, and to correctly predict the parameters of the energy levels of the hydrogen atom that Bohr had proposed.

Schrödinger’s equation is perhaps most commonly used to define a three-dimensional area of space where a given electron is most likely to be found. Each area of space is known as an atomic orbital and is characterized by a set of three quantum numbers. These numbers represent values that describe the coordinates of the atomic orbital: including its size (n, the principal quantum number), shape (l, the angular or azimuthal quantum number), and orientation in space (m, the magnetic quantum number). There is also a fourth quantum number that is exclusive to a particular electron rather than a particular orbital (s, the spin quantum number; see below for more information).

Schrödinger’s equation allows the calculation of each of these three quantum numbers. This equation was a critical piece in the quantum mechanics puzzle, since it brought quantum theory into sharp focus via what amounted to a mathematical demonstration of Bohr’s fundamental quantum idea. The Schrödinger wave equation is important since it bridges the gap between classical Newtonian physics (which breaks down at the atomic level) and quantum mechanics.

The Schrödinger equation is rightfully considered to be a monumental contribution to the advancement and understanding of quantum theory, but there are three additional considerations, detailed below, that must also be understood. Without these, we would have an incomplete picture of our non-relativistic understanding of electrons in atoms.

German mathematician and physicist Max Born made a very specific and crucially important contribution to quantum mechanics relating to the Schrödinger equation. Born took the wave functions that Schrödinger produced, and said that the solutions to the equation could be interpreted as three-dimensional probability “maps” of where an electron may most likely be found around an atom (Born, 1926). These maps have come to be known as the s, p, d, and f orbitals (Figure 3).

In the year following the publication of Schrödinger’s work, the German physicist Werner Heisenberg published a paper that outlined his uncertainty principle (Heisenberg, 1927). He realized that there were limitations on the extent to which the momentum of an electron and its position could be described. The Heisenberg Uncertainty Principle places a limit on the accuracy of simultaneously knowing the position and momentum of a particle: As the certainty of one increases, then the uncertainty of other also increases.

The crucial thing about the uncertainty principle is that it fits with the quantum mechanical model in which electrons are not found in very specific, planetary-like orbits – the original Bohr model – and it also dovetails with Born’s probability maps. The two contributions (Born and Heisenberg’s) taken together with the solution to the Schrödinger equation, reveal that the position of the electron in an atom can only be accurately predicted in a statistical way. That is to say, we know where the electron is most likely to be found in the atom, but we can never be absolutely sure of its exact position.

Comprehension Checkpoint

The Heisenberg uncertainty principle concerning the position and momentum of a particle states that as the certainty of one increases, the _____ of the other increases.

In 1922 German physicists Otto Stern, an assistant of Born’s, and Walther Gerlach conducted an experiment in which they passed silver atoms through a magnetic field and observed the deflection pattern. In simple terms, the results yielded two distinct possibilities related to the single, 5s valence electron in each atom. This was an unexpected observation, and implied that a single electron could take on two, very distinct states. At the time, nobody could explain the phenomena that the experiment had demonstrated, and it took a number of scientists, working both independently and in unison with earlier experimental observations, to work it out over a period of several years.

In the early 1920s, Bohr’s quantum model and various spectra that had been produced could be adequately described by the use of only three quantum numbers. However, there were experimental observations that could not be explained via only three mathematical parameters. In particular, as far back as 1896, the Dutch physicist Pieter Zeeman noted that the single valence electron present in the sodium atom could yield two different spectral lines in the presence of a magnetic field. This same phenomenon was observed with other atoms with odd numbers of valence electrons. These observations were problematic since they failed to fit the working model.

In 1925, Dutch physicist George Uhlenbeck and his graduate student Samuel Goudsmit proposed that these odd observations could be explained if electrons possessed angular momentum, a concept that Wolfgang Pauli later called “spin.” As a result, the existence of a fourth quantum number was revealed, one that was independent of the orbital in which the electron resides, but unique to an individual electron.

By considering spin, the observations by Stern and Gerlach made sense. If an electron could be thought of as a rotating, electrically-charged body, it would create its own magnetic moment. If the electron had two different orientations (one right-handed and one left-handed), it would produce two different ‘spins,’ and these two different states would explain the anomalous behavior noted by Zeeman. This observation meant that there was a need for a fourth quantum number, ultimately known as the “spin quantum number,” to fully describe electrons. Later it was determined that the spin number was indeed needed, but for a different reason – either way, a fourth quantum number was required.

Comprehension Checkpoint

Some experimental observations could not be explained mathematically using three parameters because

In 1922, Niels Bohr visited his colleague Wolfgang Pauli at Göttingen where he was working. At the time, Bohr was still wrestling with the idea that there was something important about the number of electrons that were found in ‘closed shells’ (shells that had been filled).

In his own later account (1946), Pauli describes how building upon Bohr’s ideas and drawing inspiration from others’ work, he proposed the idea that only two electrons (with opposite spins) should be allowed in any one quantum state. He called this ‘two-valuedness’ – a somewhat inelegant translation of the German zweideutigkeit (Pauli, 1925). The consequence was that once a pair of electrons occupies a low energy quantum state (orbitals), any subsequent electrons would have to enter higher energy quantum states, also restricted to pairs at each level.

Using this idea, Bohr and Pauli were able to construct models of all of the electronic structures of the atoms from hydrogen to uranium, and they found that their predicted electronic structures matched the periodic trends that were known to exist from the periodic table – theory met experimental evidence once again.

Pauli ultimately formed what came to be known as the exclusion principle (1925), which used a fourth quantum number (introduced by others) to distinguish between the two electrons that make up the maximum number of electrons that could be in any given quantum level. In its simplest form, the Pauli exclusion principle states that no two electrons in an atom can have the same set of four quantum numbers. The first three quantum numbers for any two electrons can be the same (which places them in the same orbital), but the fourth number must be either +½ or -½, i.e., they must have different ‘spins’ (Figure 4). This is what Uhlenbeck and Goudsmit’s research suggested, following Pauli’s original publication of his theories.

The period described here was rich in the development of the quantum theory of atomic structure. Literally dozens of individuals, some mentioned throughout this module and others not, contributed to this process by providing theoretical insights or experimental results that helped shape our understanding of the atom. Many of the individuals worked in the same laboratories, collaborated together, or communicated with one another during the period, allowing the rapid transfer of ideas and refinements that would shape modern physics. All these contributions can certainly been seen as an incremental building process, where one idea leads to the next, each adding to the refinement of thinking and understanding, and advancing the science of the field.

The 20th century was a period rich in advancing our knowledge of quantum mechanics, shaping modern physics. Tracing developments during this time, this module covers ideas and refinements that built on Bohr’s groundbreaking work in quantum theory. Contributions by many scientists highlight how theoretical insights and experimental results revolutionized our understanding of the atom. Concepts include the Schrödinger equation, Born’s three-dimensional probability maps, the Heisenberg uncertainty principle, and electron spin.

Key Concepts

• Electrons, like light, have been shown to be wave-particles, exhibiting the behavior of both waves and particles.

• The Schrödinger equation describes how the wave function of a wave-particle changes with time in a similar fashion to the way Newton’s second law describes the motion of a classic particle. Using quantum numbers, one can write the wave function, and find a solution to the equation that helps to define the most likely position of an electron within an atom.

• Max Born’s interpretation of the Schrödinger equation allows for the construction of three-dimensional probability maps of where electrons may be found around an atom. These ‘maps’ have come to be known as the s, p, d, and f orbitals.

• The Heisenberg Uncertainty Principle establishes that an electron’s position and momentum cannot be precisely known together, instead we can only calculate statistical likelihood of an electron’s location.

• The discovery of electron spin defines a fourth quantum number independent of the electron orbital but unique to an electron. The Pauli exclusion principle states that no two electrons with the same spin can occupy the same orbital.

As we saw in earlier reading, the electron is not a true particle, but a wave-particle similar to the photon. Since we measure the position of objects with light, the small size of the electron introduces a challenge. If we shine a light beam on a moving tennis ball, the light has little effect on the tennis ball and we can measure both its position and momentum with a high degree of accuracy. However, the electron is so tiny that even a single photon will influence its trajectory – thus if we shine a beam of light on it to measure its position, the energy of the photon will affect its momentum, and vice versa. Werner Heisenberg developed a principle to describe this uncertainty, called appropriately the Heisenberg Uncertainty Principle (Heisenberg, 1927). It tells us that mathematically, the product of the uncertainty in position (Δx) and the uncertainty in momentum (Δp) of an electron cannot be less than the reduced Planck constant ℏ/2 (Equation 1).

This is a very small number that can usually be ignored, but when dealing with a particle as small as an electron, it is significant. Because of this, it becomes necessary to describe the position of an electron in terms of probability, rather than that of absolute certainty. We have to say that an electron is likely to be found within the atom in certain areas of high probability, but we cannot be 100% sure of its precise position.

Comprehension Checkpoint

The Heisenberg Uncertainty Principle tells us that

Because the electron is not a true particle, we cannot describe its movement or location in traditional terms. In other words, those that would normally be defined by simple x, y, and z coordinates. The challenges raised by the combination of wave particle duality and the Heisenberg Uncertainty Principle are simplified and expressed by Schrödinger’s equation that, when solved, produces wave functions denoted by Ψ. When Ψ is squared, the resulting solution gives the probability of finding an electron at a particular place in the atom. The Ψ2 term describes how electron density and electron probability are distributed in space around the nucleus of an atom.

The application of the Schrödinger equation is most easily understood in the case of the hydrogen atom because the one-electron atom allows us to avoid the complex interactions of multiple electrons. The time-independent form of the Schrödinger equation (Equation 2) can be solved to provide solutions that correspond to the energy levels in a hydrogen atom. In solving this equation, we find that there are multiple solutions for Ψ that we call Ψ1, Ψ2, Ψ3, etc.

Each of these solutions has a different energy that corresponds to what we think of as different energy levels, or electron shells, within the atom. The lowest energy electron distribution is that which is closest to the nucleus, and it is called the ground state. The ground state is the most stable state of the single electron within the hydrogen atom. Other, higher energy states exist and are called excited states. As the energy of each level increases above the ground state, we refer them to them as the second, third, fourth, etc. energy levels, respectively.

Comprehension Checkpoint

The energy of the electron shells ______ as you move away from the nucleus of the atom.

Acceptable solutions to the wave equation for an electron cannot be just any values, but rather they are restricted to those that obey certain parameters. Those parameters are described by a set of three quantum numbers that are named the principal quantum number, the azimuthal quantum number, and the magnetic quantum number. These quantum numbers are given the symbols n, l, and m, respectively.

The principal quantum number n is a positive integer starting with a value of 1, where 1 corresponds to the first (lowest energy) shell known as the ground state in hydrogen. The principal quantum number then increases with the increasing energy of the shells (the excited states in hydrogen) to values of n = 2, n = 3, n = 4, etc., as one moves farther away from the nucleus to higher energy levels.

The azimuthal quantum number l (also called the orbital angular momentum quantum number) determines the physical, three-dimensional shape of the orbitals in any subshell. The value of l is dependent on the principal quantum number n, and there are multiple values of l for each value of n. Values of l are positive integers or 0, and are determined by subtracting integers from the corresponding value of n. For example, when n = 3 we can subtract 3, 2, and 1 from n to yield values for l of 0, 1, and 2, respectively. Thus, the ground state shell in hydrogen (principal quantum number 1) has only one s subshell, shell 2 has s and p subshells, and so on. The subshells indicated by l are also given the letter designations s, p, d, and f where l = 0 corresponds to an s subshell, l = 1 a p subshell, l = 2 a d subshell, and l = 3 an f subshell. Each subshell has a unique shape in 3D-space.

The magnetic quantum number m, defines the orientation (i.e., the position) and the number of orbitals within any given subshell. Values of m depend upon values of l, the azimuthal quantum number, and m can take on values of +1, -1, and all integers (including 0) in between. So for example, when l = 1, m can be +1, 0, or -1, yielding three separate orbitals, on axes x, y, and z in space. So in the case of l = 1 (the p subshell), there are three orbitals oriented in different directions in space.

Thus each energy level is given a unique set of quantum numbers to fully describe it. Only certain values for quantum numbers in any one energy level are allowed, and those combinations are summarized in Table 1. Analysis of all of the allowed solutions to the wave equation shows that orbitals can be grouped together in sets according to their l values: s, p, d, and f sets.

Table 1: The allowed solutions to the wave equation shows that orbitals can be grouped together in sets according to their l values (i.e., the s, p, d, and f sets).

The first set of sub-orbitals are described by a type of wave function whose probability density depends only on the distance from the nucleus, and whose probability is the same in all directions. Thus they have a spherical shape and are called s orbitals. These wave functions are all found to have values of l = 0 and therefore values of m = 0, and every energy level has such a wave function starting with 1s, and moving to 2s, 3s, etc. (Figure 1).

The second type of wave function has a probability density that depends on both the distance from the nucleus and the orientation along either the x-, y-, or z-axis in space. This leads to three, separate, equivalent (or degenerate) wave functions that have a "figure eight" shape in 3D-space. These wave functions are all found to have values of l = 1 and can take on three, different m values. These are called p orbitals and exist for every energy level except the first, thus appearing as 2p, 3p, 4p, etc. (Figure 2).

The third type of wave function has a probability density that depends on both the distance from the nucleus and the orientation along two of the x-, y-, and z-axes in space. This leads to five separate, equivalent wave functions. When the wave functions have equivalent energy, they are given the label "degenerate." They have complex shapes in 3D-space. These wave functions are all found to have values of l = 2 and can take on five different m values. These are called d orbitals, and every energy level except the first and second has such a wave function (Figure 3).

The fourth type is a set of orbitals with even more complexity in terms of their wave function, positions, and shape in 3D-space. These wave functions are all found to have values of l = 3 and can take on seven different m values. They, too, are all degenerate. These are called f orbitals and every energy level except the first, second, and third has such a wave function, starting with 4f and moving to 5f (Figure 4).

To summarize, the orbitals available in the first four energy levels are as follows in Table 2:

Table 2: Orbitals associated with the first four energy levels.

Comprehension Checkpoint

The principal quantum number n

Since orbitals are defined by very specific solutions to the Schrödinger equation, electrons must absorb very specific quantities of energy to be promoted from one energy level to another. By exposing electrons to an external source of energy, such as light, it is possible to promote the electrons from their ground state to other positions that have higher potential energies, called excited states. These ‘excited’ electrons quickly return to lower energy positions (in order to regain the stability associated with lower energies), and in doing so they release energy in specific frequencies that correspond to the energy differences between electron orbitals, or shells. Light energy is related to frequency (f) and Planck’s constant (h) in the equation E = hf.

Electrons within a hydrogen atom will absorb specific frequencies of energy that correspond to the energy gaps (ΔE) between two energy levels, thus allowing the promotion of an electron from a relatively low energy level to a relatively high energy level. Examining the absorption spectrum produced when hydrogen is irradiated, we observe a pattern of lines that supports this (Figure 5).

Bohr proposed that electrons can transition to different positions, but only in discrete, defined steps. He thought that electrons were restricted to a specific area of space around the nucleus, much like the planets in our solar system are restricted to specific paths. The hydrogen absorption spectrum of discrete lines (rather than a continuous spectrum) shows that only very specific transitions can be made and is evidence for the quantum model.

In addition to the absorption spectrum shown above, another identical spectrum can be observed, this time with the energy being emitted rather than absorbed. In this case the electrons fall back from higher energies to lower energies rather than being promoted as in the absorption spectrum. This is called the emission spectrum of the atom and is once again made up of discrete, individual lines.

Comprehension Checkpoint

When energy is emitted, electrons

Each series of lines that appear in such spectra are named after their discoverers. The Lyman series are those lines in which the electrons are either promoted from, or to, the 1s orbital. In the Balmer, Paschen, Brackett, and Pfund series, electrons travel either from or to the orbitals where n = 2, 3, 4, and 5, respectively. The differing energy gaps (and therefore the differing frequencies) correspond to lines in different regions of the electromagnetic spectrum as a whole (Figure 6). Lyman lines are in the ultraviolet region, Balmer in the visible, and Paschen, Brackett, and Pfund in various parts (near and far) of the infrared.

In hydrogen atoms, and other single electron species such as He+ and Li2+, it is found that the energy of the orbitals is only dependent on their distance from the nucleus, with increasing energies as n increases. The lowest energy orbital is 1s, followed by 2s and 2p (that have the same energy), and 3s, 3p, and 3d (that have the same energy) etc. As such, the ground state (lowest energy state) for a hydrogen atom with only one electron is with the electron residing in the 1s orbital. The differences in energies (ΔE) required to promote electrons from one level to another can be determined by using one version of the Rydberg Formula, where RH is a constant for hydrogen and ni and nf are the initial and final energy levels respectively.

Since other single electron species such as He+ and Li2+ contain more protons than the hydrogen atom, the electron experiences a greater attraction from the nucleus. As such, differing, larger amounts of energy are required for promotion of electrons in these species and a modified Rydberg constant is required in the calculation of ΔE for single electron species other than the hydrogen atom.

Comprehension Checkpoint

In atoms that have only one electron, the energy of the orbitals is dependent on the

The hydrogen atom is the simplest case to study since its single electron is both uninfluenced by other electrons and does not influence any other electrons. In this case, the wave function is relatively easy to compute. However, in more complex atoms the calculations are not so easy. For example, when two electrons are present, as in helium, things become considerably more complex. Rather than there being just one simple potential energy attraction between the nucleus and the single electron in a hydrogen atom, in helium there are now three potential energy terms to consider: the attractive forces between the nucleus and each electron, and the repulsion between the two electrons. The addition of even more electrons leads to the equation and the calculations becoming increasingly complex.

The complexity of the Schrödinger equation under these many electron circumstances means that it has to be solved by a series of approximations rather than directly. One method is to effectively apply the single electron system over and over again to produce what amounts to an approximate answer. Although not entirely correct, such approximations prove to be very workable and produce reasonable answers to the multi-electron problem.

The approximation works well enough to once again produce Ψ2 wave functions that predict the electron density in space of multi-electron atoms. The orbitals in the single electron situation and those in the multi-electron situation are still defined by the same set of restricted quantum numbers, but one difference arises regarding the specific energies of those orbitals. Previously, the energy depended only on the distance from the nucleus (that is, the energy n in multi-electron systems is found to be dependent on the value of l). This complicates the idea that relates electron energy simply to distance from the nucleus (i.e., orbital energy follows the pattern 1s < (2s = 2p) < (3s = 3p = 3d), etc.). Instead, it produces the following sequence of relative energies for the orbitals.

1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p, etc.

Note the insertion of 4s and 5s earlier in the sequence than expected. This order is reflected on the periodic table. When reaching the end of period three on the periodic table (element 18, argon), we will have filled the 1s, 2s, 2p, 3s, and 3p orbitals giving a total (as expected) of 18 electrons. One would perhaps expect the next orbitals to be filled to be the 3d, since after all, they are also in the third energy level. However, the next elements on the periodic table, potassium and calcium (elements 19 and 20, respectively), fill their 4s orbitals before their 3d. Moving further along the fourth period to element 21, scandium, we find that the 3d subshell now begins to populate. When that subshell is full at element 30, zinc, we then fill the 4p subshell beginning with element 31, gallium. In short, the periodic table is ordered in such a way that it reflects the sequence of relative energies of orbitals given above.

As seen previously, the concept of electron spin introduced the need for the fourth and final quantum number, the spin quantum number, s. Its introduction ensures that each electron has a unique set of quantum numbers according to the Pauli Exclusion Principle. With there being only two values for s (+½ or -½), it follows that each orbital may only hold a maximum of two electrons.

The notation used to describe the electronic configuration of an atom involves the use of a superscript to denote the number of electrons in any given orbital. For example, in the case of beryllium that has a total of four electrons, 1s2 2s2 shows that both the 1s and the 2s orbitals are each filled with two electrons.

When identifying which orbitals the electrons in an atom occupy, we generally follow the principle referred to as the Aufbau principle, where lower energy orbitals are filled first. So, for example, 1s is filled before 2s, and 2s before 2p, etc. But what about degenerate orbitals, those that have the same energy, such as the three orbitals that exist in the 2p sub-level?

Hund’s rule states that when electrons are placed into any set of degenerate orbitals, one electron is placed into each orbital before any spin pairing takes place. This means, for example, if we consider three electrons being placed into the degenerate 2p orbitals, one would see an electronic configuration of 2px1 2py1 2pz1 (recall the p orbitals are oriented along x, y, and z axes, respectively) before we see any pairing. The fourth electron to enter the 2p sub-shell would create the need for an unavoidable pairing, hence 2px2 2py1 2pz1 (Figure 7).

Now that we have a sense of electron configuration, their energies, and their most probable positions within atoms, the next step is to describe the behavior of electrons when atoms interact with one another. Electron interaction forms the basis of a key chemical process called chemical bonding. And understanding the behavior of electrons provides us with a better understanding of the chemical behavior of the elements and their compounds.

Our Atomic Theory series continues, exploring the quantum model of the atom in greater detail. This module takes a closer look at the Schrödinger equation that defines the energies and probable positions of electrons within atoms. Using the hydrogen atom as an example, the module explains how orbitals can be described by type of wave function. Evidence for orbitals and the quantum model is provided by the absorption and emission spectra of hydrogen. Other concepts include multi-electron atoms, the Aufbau Principle, and Hund’s Rule.

Key Concepts

• The wave-particle nature of electrons means that their position and momentum cannot be described in simple physical terms but must be described by wave functions.

• The Schrödinger equation describes how the wave function of a wave-particle changes with time in a similar fashion to the way Newton’s second law describes the motion of a classical particle. The equation allows the calculation of each of the three quantum numbers related to individual atomic orbitals (principal, azimuthal, and magnetic).

• The Heisenberg uncertainty principle establishes that an electron’s position and momentum cannot be precisely known together; instead we can only calculate statistical likelihood of an electron’s location.

• The discovery of electron spin defines a fourth quantum number independent of the electron orbital but unique to an electron. The Pauli exclusion principle states that no two electrons with the same spin can occupy the same orbital.

• Quantum numbers, when taken as a set of four (principal, azimuthal, magnetic and spin) describe acceptable solutions to the Schrödinger equation, and as such, describe the most probable positions of electrons within atoms.

• Orbitals can be thought of as the three dimensional areas of space, defined by the quantum numbers, that describe the most probable position and energy of an electron within an atom.

In the 17th century, the Italian mathematician Evangelista Torricelli built the first mercury barometer by filling a glass tube sealed at one end with mercury and then inverting the open end into a tub full of the liquid metal. To the surprise of his contemporaries, the tube remained partially filled—almost as if something was pushing down on the mercury in the tub, and forcing the liquid metal up the tube (Figure 2). Most significantly, the level at which mercury rose in the tube changed from day to day, challenging scientists to explain how mercury was forced up the closed glass tube.

The British scientists Robert Boyle and Robert Hooke devised an experiment to figure out what was pressing down on the mercury. Working with a candy cane-shaped glass tube that had its short leg sealed off, Boyle poured in just enough mercury to fill the tube’s curve and trap air inside the short leg. When he poured in still more mercury, Boyle saw that the while the volume of trapped air shrank, the air somehow pushed against the mercury and forced it back up the long leg. Boyle reasoned that air must also weigh down on the mercury in the Torricelli’s tub and exert the force driving mercury partially up the tube (to learn more about Boyle’s experiment, see our module Properties of Gases).

But how could air—which many scientists had regarded as an indivisible element—weigh down on mercury? By the early 18th century, scientists realized that air was made up of tiny particles. However, these same scientists couldn’t imagine air particles just floating in space. They assumed that the particles vibrated and spun while being held in place by an invisible substance called ether.

The Swiss mathematician Daniel Bernoulli had a different idea about how particles could be suspended in air. In his 1738 book Hydrodynamica, Bernoulli sketched out a thought experiment illustrating how the linear motion of air particles could exert pressure. Bernoulli first asked his readers to imagine a cylinder fitted with a movable piston. Next, he directed the reader to picture the air particles as tiny spheres that zipped around in all directions, colliding with each other and the piston. These numerous, constant collisions would “kick” the piston aloft. Furthermore, Bernoulli suggested that if the air was heated, the particles would zoom faster, striking the piston more often and kicking the piston still higher in the cylinder.

With his thought experiment, Bernoulli was the first to develop several assumptions about molecules and heat that are integral to modern KMT. Like modern KMT, Bernoulli assumed that molecules behave like tiny spheres in constant linear motion. Working from this assumption, he reasoned that the molecules would constantly collide with each other and the walls of a container, thereby exerting pressure on these walls. Importantly, he also assumed that heat affects the movement of molecules.

Though largely correct, Bernoulli’s ideas were mostly dismissed by contemporary scientists. He didn’t offer any experimental data to support his ideas. Furthermore, accepting his ideas required a literal belief in atoms, which many scientists were skeptical existed up to the 19th century. Equating motion with temperature implied that there must be an absolute minimum temperature at which point all motion ceased.

And finally, Bernoulli’s kinetic theory competed with caloric theory, a prominent idea at the time. According to caloric theory, caloric was a “heat substance” that engulfed gas molecules, causing them to repel each other so forcefully they would shoot into the walls of a container. This competing idea about what caused air pressure was championed by influential chemists such as Antoine Lavoisier and John Dalton, while Bernoulli’s idea was largely ignored until the 19th century. Unfortunately, it is not uncommon for good ideas to take time to be accepted in science as previously accepted theories are disproven. However, over time, scientific progress assures that theories which better explain the data collected take hold, as Bernoulli’s did.

Comprehension Checkpoint

Like modern kinetic-molecular theory, Bernoulli theorized that

Unlike Lavoisier and Dalton, the 19th century German physicist Rudolf Clausius rejected caloric theory. Instead of regarding heat as a substance that surrounds molecules, Clausius proposed that heat is a form of energy that affects the temperature of matter by changing the motion of molecules in matter. This kinetic theory of heat enabled Clausius to study and predict the flow of heat—a field we now call thermodynamics (for more information, see our module Thermodynamics I).

In his 1857 paper, “On the nature of the motion which we call heat,” Clausius speculated on how heat energy, temperature, and molecule motion could explain gas behavior. In doing this, he proposed several ideas about the molecules of gases. These ideas have come to be accepted for ideal gases—theoretical gases that perfectly obey the ideal gas equation (for more information, see our Properties of Gases module). Clausius proposed that the space taken up by ideal gas molecules should be regarded as infinitesimal when compared to the space occupied by the whole gas – in other words, a gas consists mostly of empty space. Second, he suggested that the intermolecular forces between molecules should be treated as infinitesimal.

A key part of Clausius’s ideas was his work on the mathematical relationship between heat, temperature, molecule motion, and kinetic energy—the energy of motion. He proposed that the net kinetic energy of the molecules in an ideal gas is directly proportional to the gas’s absolute temperature, T. Its kinetic energy, Ek, is therefore determined by the number of gas molecules, n, which each have a molecule mass of m, and are moving with velocity u, as shown below:

With this equation and observational data on the weights and volumes of gases at specific temperatures, Clausius was able to calculate the average speeds of gas molecules such as oxygen (an astonishing 461 m/s!). However, the Dutch meteorologist Christoph Hendrik Diederik Buys-Ballot quickly pointed out a problem with these speed calculations. If gas molecules moved hundreds of meters per second, then an odorous gas (like in perfume) should spread across a room almost instantly. Instead, perfumes and other scents usually took several minutes to reach people across a room. This suggested that either Clausius’s mathematical relationship was wrong, or that something more complicated was happening with real gas molecules.

Comprehension Checkpoint

Clausius proposed that

Buys-Ballot’s objection forced Clausius to re-think his ideas about gas molecules. If a gas molecule could travel at 461 m/s, but still take minutes to cross a room, it must be encountering lots of obstacles—such as other gas molecules. Clausius realized then that one of his core ideas about ideal gas molecules had to change.

In 1859, Clausius published a paper proposing that, instead of being infinitesimally small, gas molecules had to be big enough that it could collide with another molecule, and there had to be so many fast-moving gas molecules present that it couldn’t travel far before doing so. The average distance that a molecule travels between collisions has come to be known as its mean free path. Clausius realized that while the mean free path must be very big compared to the actual size of the molecules, it would still have to be small enough that a fast-moving molecule would collide with other molecules many times each second (Figure 3).

Thus, gas molecules are constantly colliding and changing directions. While it’s tempting to picture molecules colliding every few seconds like bumper cars at an amusement park, the collisions are so much more frequent. For example, at room temperature, one oxygen molecule travels an average distance of 67 nm (almost 1,500 times narrower than the width of a human hair) before colliding with another molecule. And this single molecule collides with others 7.2 billion times per second! This astounding frequency of collisions explains how gas molecules can zip along at hundreds of meters per second, but still take minutes to cross a room.

Clausius’s idea about mean free path was vital to how Langmuir solved the blackening light bulb problem. Thanks to Clausius, Langmuir understood that he needed to decrease the mean free path for the tungsten atoms sublimating off the filament. In a vacuum, the mean free path was very long, and the tungsten atoms quickly make their way from the filament to the inside of the bulb. By adding an inert gas like argon, Langmuir increased the number of molecules in the bulb and the frequency of collisions—thereby decreasing the mean free path, and increasing the bulb’s life.

Clausius’s ideas about mean free path, molecule motion, and kinetic energy intrigued James Clerk Maxwell, a contemporary Scottish physicist. In 1860, Maxwell published his first kinetic theory paper expanding on Clausius’s work. Building on Clausius’s calculations for the average speed of oxygen and other molecules, Maxwell further developed the idea that the gas molecules in a sample of gas are moving at different speeds. Within this range of possible speeds, some molecules are slower than the average and some faster (Figure 4); furthermore, a molecule’s speed can change when it collides with another molecule. Only a tiny number of the gas molecules are actually moving at the slowest and fastest speeds possible—but we know now that this small number of speedy molecules are especially important, because they are the most likely molecules to undergo a chemical reaction.

Along with these ideas, Maxwell proposed that gas particles should be treated mathematically as spheres that undergo perfectly elastic collisions. This means that the net kinetic energy of the spheres is the same before and after they collide, even if their velocities change. Together, Clausius and Maxwell developed several key assumptions that are vital to modern KMT.

Comprehension Checkpoint

The average distance that a molecule travels between collisions is known as

A major use of modern KMT is as a framework for understanding gases and predicting their behavior. KMT links the microscopic behaviors of ideal gas molecules to the macroscopic properties of gases. In its current form, KMT makes five assumptions about ideal gas molecules:

1. Gases consist of many molecules in constant, random, linear motion.
2. The volume of all the molecules is negligible compared to the gas’s total volume.
3. Intermolecular forces are negligible.
4. The average kinetic energy of all molecules does not change, so long as the gas’s temperature is constant. In other words, collisions between molecules are perfectly elastic.
5. The average kinetic energy of all molecules is proportional to the absolute temperature of the gas. This means that, at any temperature, gas molecules in equilibrium have the same average kinetic energy (but NOT the same velocity and mass).

With KMT’s assumptions, scientists are able to describe on a molecular level the behaviors of gases. These behaviors are common to all gases because of the relationships between gas pressure, volume, temperature, and amount, which are described and predicted by the gas laws (for more on the gas laws, please see our Properties of Gases module). But KMT and the gas laws are useful for understanding more than abstract ideas about chemistry. With KMT and the gas laws, we can better understand the behaviors of real gases, such as the air we use to inflate tires, as we’ll explore more below.

Comprehension Checkpoint

Gases consist of many molecules in _____, random, linear motion.

Boyle’s law describes how, for a fixed amount of gas, its volume is inversely proportional to its pressure (Figure 5). This means that if you took all the air from a fully inflated bike tire and put the air inside a much larger, empty car tire, the air would not be able to exert enough pressure to inflate the car tire. While this example about the relationship between gas volume and pressure may seem intuitive, KMT can help us understand the relationship on a molecular level. According to KMT, air pressure depends on how often and how forcefully air molecules collide with tire walls. So when the volume of the container increases (like when we transfer air from the bike tire to the car tire), the air molecules have to travel farther before they can collide with the tire’s walls. This means that there are fewer collisions per unit of time, which results in lower pressure (and an underinflated car tire).

Charles’s law describes how a gas’s volume is directly proportional to its absolute temperature (Figure 6)—and also why your car tire pressure increases the longer you drive (and thus why you should always measure tire pressure after your car has been parked for a long period). Using KMT, we can understand that as the friction between the tires and road raises the air temperature inside the tires, the air molecules’ kinetic energy and speed are also increasing. Because the molecules are zipping around faster, they collide more frequently and more forcefully with the tire’s walls, thereby increasing the pressure. Both Charles’s law and KMT also explain why tire pressure decreases after you park. As the tires cool down, the air molecules move more slowly and collide less with the tire’s walls, thereby exerting less pressure.

While KMT is a useful tool for understanding the linked behaviors of molecules and matter, particularly gases, KMT does have limitations related to how its theoretical assumptions differ from the behavior of real matter. In particular, KMT’s assumptions that intermolecular forces are negligible, and the volume of molecules is negligible, aren’t always valid. Real gas molecules do experience intermolecular forces. As pressure on a real gas increases and forces its molecules closer together, the molecules can attract one another. This attraction slows down the molecules just a little bit before they slam into one another or the walls of a container, so that the pressure inside a container of real gas molecules is slightly lower than we would expect based on KMT. These intermolecular forces are particularly influential when gas molecules are moving more slowly, such as at low temperature.

While growing pressure on a real gas initially allows its intermolecular forces to have more influence, a different factor gains more influence as the pressure continues to grow. While KMT assumes that gas molecules have no volume, real gas molecules do have volume. This gives a real gas greater volume at high pressure than would be predicted from KMT. Furthermore, as a real gas is compressed, the mean free path of its molecules decreases and the molecules collide more often—thereby increasing the pressure exerted by a real gas compared to KMT’s prediction.

Ultimately, KMT is most useful and accurate when gases are under conditions that cause molecules to behave consistently with KMT’s assumptions. These conditions often happen at low pressure, where molecules have lots of empty space to move in, and the molecule volumes are very small compared to the total volume. And the conditions often occur at high temperature, when the molecules possess a high kinetic energy and fast speed, which lets them overcome the attractive forces between molecules.

Ultimately, KMT provides assumptions about molecule behavior that can be used both as the basis for other theories about molecules, and to solve real-world problems. Clausius’ concept of a molecule’s mean free path underlies our modern ideas of diffusion and Brownian motion, which can explain why scents from perfume or baking cookies take so long to cross a room. And while Langmuir used KMT’s assumptions to develop a longer-lasting light bulb, still other scientists are deploying their knowledge of KMT in more dramatic and controversial ways. By understanding how real gas molecules behave and move, scientists are able to separate gas molecules from each other based on tiny differences in mass—a key principle behind, for example, how uranium isotopes are enriched for use in nuclear weapons.

Over four hundred years, scientists including Rudolf Clausius and James Clerk Maxwell developed the kinetic-molecular theory (KMT) of gases, which describes how molecule properties relate to the macroscopic behaviors of an ideal gas—a theoretical gas that always obeys the ideal gas equation. KMT provides assumptions about molecule behavior that can be used both as the basis for other theories about molecules and to solve real-world problems.

Key Concepts

• Kinetic-molecular theory states that molecules have an energy of motion (kinetic energy) that depends on temperature.

• Rudolf Clausius developed the kinetic theory of heat, which relates energy in the form of heat to the kinetic energy of molecules.

• Over four hundred years, scientists have developed the kinetic-molecular theory of gases, which describes how molecule properties relate to the macroscopic behaviors of an ideal gas—a theoretical gas that always obeys the ideal gas equation.

• The kinetic-molecular theory of gases assumes that ideal gas molecules (1) are constantly moving; (2) have negligible volume; (3) have negligible intermolecular forces; (4) undergo perfectly elastic collisions; and (5) have an average kinetic energy proportional to the ideal gas’s absolute temperature.

In 1902, Frederick Soddy proposed the theory that "radioactivity is the result of a natural change of an isotope of one element into an isotope of a different element." Nuclear reactions involve changes in particles in an atom's nucleus and thus cause a change in the atom itself. All elements heavier than bismuth (Bi) (and some lighter) exhibit natural radioactivity and thus can "decay" into lighter elements. Unlike normal chemical reactions that form molecules, nuclear reactions result in the transmutation of one element into a different isotope or a different element altogether (remember that the number of protons in an atom defines the element, so a change in protons results in a change in the atom). There are three common types of radiation and nuclear changes:

Alpha Radiation (α) is the emission of an alpha particle from an atom's nucleus. An α particle contains two protons and two neutrons (and is similar to a He nucleus:

(Note: in nuclear chemistry, element symbols are traditionally preceded by their atomic weight [upper left] and atomic number [lower left].)

Beta Radiation (β) is the transmutation of a neutron into a proton and an electron (followed by the emission of the electron from the atom's nucleus:

Gamma Radiation (γ) involves the emission of electromagnetic energy (similar to light energy) from an atom's nucleus. No particles are emitted during gamma radiation, and thus gamma radiation does not itself cause the transmutation of atoms; however, γ radiation is often emitted during, and simultaneous to, α or β radioactive decay. X-rays, emitted during the beta decay of cobalt-60, are a common example of gamma radiation.

Comprehension Checkpoint

Radiation can result in an atom having a different atomic number.

Radioactive decay proceeds according to a principle called the half-life. The half-life (T½) is the amount of time necessary for one-half of the radioactive material to decay. For example, the radioactive element bismuth (210Bi) can undergo alpha decay to form the element thallium (206Tl) with a reaction half-life equal to five days. If we begin an experiment starting with 100 g of bismuth in a sealed lead container, after five days we will have 50 g of bismuth and 50 g of thallium in the jar. After another five days (ten from the starting point), one-half of the remaining bismuth will decay and we will be left with 25 g of bismuth and 75 g of thallium in the jar. As illustrated, the reaction proceeds in halves, with half of whatever is left of the radioactive element decaying every half-life period.

The fraction of parent material that remains after radioactive decay can be calculated using the equation:

The amount of a radioactive material that remains after a given number of half-lives is therefore:

The decay reaction and T½ of a substance are specific to the isotope of the element undergoing radioactive decay. For example, Bi210 can undergo a decay to Tl206 with a T½of five days. Bi215, by comparison, undergoes bdecay to Po215 with a T½ of 7.6 minutes, and Bi208 undergoes yet another mode of radioactive decay (called electron capture) with a T½ of 368,000 years!

Comprehension Checkpoint

All radioactive material decays at the same rate.

While many elements undergo radioactive decay naturally, nuclear reactions can also be stimulated artificially. Although these reactions also occur naturally, we are most familiar with them as stimulated reactions. There are two such types of nuclear reactions:

1) Nuclear fission: reactions in which an atom's nucleus splits into smaller parts, releasing a large amount of energy in the process. Most commonly this is done by "firing" a neutron at the nucleus of an atom. The energy of the neutron "bullet" causes the target element to split into two (or more) elements that are lighter than the parent atom.

Interactive Animation: The Fission of U235

During the fission of U235, three neutrons are released in addition to the two daughter products. If these released neutrons collide with nearby U235 nuclei, they can stimulate the fission of these atoms and start a self-sustaining nuclear chain reaction. This chain reaction is the basis of nuclear power. As uranium atoms continue to split, a significant amount of energy is released from the reaction. The heat released during this reaction is harvested and used to generate electrical energy.

Interactive Animation: Two Types of Nuclear Chain Reactions

Nuclear fusion: reactions in which two or more elements "fuse" together to form one larger element, releasing energy in the process. A good example is the fusion of two "heavy" isotopes of hydrogen (deuterium: H2 and tritium: H3) into the element helium.

Interactive Animation: Nuclear Fusion

Fusion reactions release tremendous amounts of energy and are commonly referred to as thermonuclear reactions. Although many people think of the sun as a large fireball, the sun (and all stars) are actually enormous fusion reactors. Stars are primarily gigantic balls of hydrogen gas under tremendous pressure due to gravitational forces. Hydrogen molecules are fused into helium and heavier elements inside of stars, releasing energy that we receive as light and heat.

Beginning with the work of Marie Curie and others, this module traces the development of nuclear chemistry. It describes different types of radiation: alpha, beta, and gamma. The module then applies the principle of half-life to radioactive decay and explains the difference between nuclear fission and nuclear fusion.