Fugacity

Type of physical science: Fugacity, Thermodynamics, Gases; behavior of, Fluid dynamics, Chemistry

Field of study: Thermodynamics

The science of chemical thermodynamics studies systems that are at equilibrium. Examples of such systems include chemical reactions, such as the formation of ammonia from hydrogen and nitrogen, and phase changes, such as that of solid water to liquid water. Fugacity is a property of chemicals specific to thermodynamics that allows the precise description of such equilibrium systems.

Overview

Chemistry is the science that studies the changes that matter undergoes. There are two kinds of changes of particular interest to the chemist. The first is the chemical reaction. In this event, some substances are changed into different substances—as, for example, when hydrogen and oxygen react to create water. The second kind of change is the phase change. In this event, a given substance changes from one form of that substance to another form of the same substance. An example would be the conversion of solid water to liquid water, that is, melting.

It is a fact of experience that some processes actually occur, but their in-detail reverse does not occur. Hydrogen burns spontaneously in an atmosphere of oxygen to give water, but water does not spontaneously decompose to produce hydrogen and oxygen gas. Water can be decomposed by passing an electric current through it, but that is not the in-detail reverse of the burning reaction.) Similarly water cooled to 10 degrees Celsius will freeze spontaneously, but ice at 10 degrees Celsius will not melt. Some systems can be maintained in a state of equilibrium. For example, ice and liquid water in contact with each other in an insulated container such as a Thermos flask will persist for considerable periods of time without any change in the amounts of ice or liquid. In this case, the ice has no tendency to melt, and the liquid has no tendency to freeze. (Of course, the warmth of the room will eventually melt the ice, because nothing is a perfect insulator.)

In order to describe these kinds of phenomena, the science of chemical thermodynamics was born. Chemical thermodynamics is the science that studies the energy changes accompanying chemical reactions and phase changes.

Chemical thermodynamics is founded upon two different experimental observations. The first observation is that energy is conserved. Any chemical or physical process can be accompanied by interactions with the world around the system, these interactions being in the form of work done or in the form of heat flows. The first universal observation is that any work done by a process plus any heat that flows because of the process is exactly balanced by a change in the energy of the system. This observation has been codified in the first law of thermodynamics, which states that work and heat add up to the energy change within the system. Since chemical thermodynamics is a modern science, it sums up this observation in the equation "energy change = work + heat."

The second observation on which thermodynamics is founded is the fact that nature has a direction. This is a common observation. In the fall, leaves change color and fall from the tree. One can either study the phenomenon as a scientist or simply marvel at the beauty. Yet if leaves were to leap from the ground, reattach to the tree, and turn green, one would be inclined to panic over the unnatural event. The process happens in one direction, and the in-detail reverse is impossible. Many less dramatic examples can be given. A bar of metal heated at one end will, over time, become uniform in temperature. No bar of metal of its own accord will spontaneously become hot at one end and cold at the other. Since chemistry is the science that studies change in matter, it is essential that chemical thermodynamics expresses the directionality of nature in terms of an equation.

Chemical systems are unique. They are made of incredibly large numbers of atoms and molecules. If one had one sheet of copier paper for each molecule of water in eighteen milliliters of water (a few tablespoons), the stack of paper would be 6,470 light-years high. No other physical systems have such an underlying complexity. This complexity underlies the directionality of nature. Expressing the directionality of nature in some sort of equation turned out to be a difficult task that was ultimately summarized in the second law of thermodynamics. The second law, being unique to chemical systems, will have concepts associated with it for which there are no counterparts in everyday experience. Such concepts include entropy, free energy, and fugacity.

The most general statement of the second law of thermodynamics states that in any chemical reactions or phase changes that actually can occur, the entropy of all systems involved must increase. In applying this principle, one must be careful to include everything that is changing. This includes not only the chemicals or phases in which one is interested but also the immediate environment that is affected by the change. Like most general statements of laws, the second law in this form is too general to be of much use. For this reason, auxiliary versions of the second law have been formulated, and these auxiliary versions introduce new properties such as free energy and fugacity.

The concept of "escaping tendency" describes a system's ability or tendency to change. For example, if one had a system in which the temperature was not uniform, heat would flow from the regions of high temperature to the regions of low temperature. Thus, temperature would be a measure of the escaping tendency of heat. When the escaping tendency was the same everywhere throughout the system, that is, when the temperature was the same, there was no heat flow. This idea was extended to other kinds of escaping tendencies.

Free energy is one such measure of escaping tendency. Free energy is a specialized notion limited to thermodynamic analysis of chemical systems. Unlike the notion of temperature, free energy has no counterpart in day-to-day existence to assist in its visualization. Free energy is defined by equations, and methods have been created for its measurement through experiment. Its use is best given in terms of an example. Consider the phase change from solid water to liquid water. This phase change can be taken to occur at some temperature. Now, it is common knowledge that water freezes (or ice melts) at 0 degrees Celsius. Therefore, the melting will not occur at temperatures below 0 degrees Celsius, and it will occur at temperatures above 0 degrees Celsius; 0 degrees Celsius, neither melting nor freezing will be favored, but rather an equilibrium between both ice and liquid water will be achieved. The free energy is defined such that below 0 degrees Celsius, the free energy of the solid is lower than that of the liquid, above 0 degrees Celsius, the free energy of the solid is greater than that of the liquid, and at 0 degrees Celsius, the free energy of the solid is the same as that of the liquid. That phase will be favored which has the lower free energy. Two phases will be in equilibrium if the free energies of the two phases are the same. Thus, the behavior of the free energy as a measure of escaping tendency provides chemists with an alternate statement of the second law that is more manageable.

Other properties can measure escaping tendency. Once again, consider an example. It is known that if one places liquid water in a closed container at some temperature, some of the water evaporates and appears in the gas above the liquid. The pressure of the water in the gas phase is called its "vapor pressure." If one had placed ice in a similar closed container at some temperature, some of the water would evaporate, giving rise to a vapor pressure of the ice. Now imagine two such containers at the same temperature, one filled with liquid and one with ice. (Such an experiment might be possible only in the mind; it could not be performed at 10 degrees Celsius, for example, because ice cannot exist at that temperature.) Each container would have a vapor pressure. Since evaporation is a process in which the water is escaping from the liquid or solid, it would be expected that the phase with the higher vapor pressure, the higher tendency to evaporate, would be the phase with the higher escaping tendency. Thus, the vapor pressure is a new measure of escaping tendency.

What is fugacity? There is a simple relationship between free energy as a measure of escaping tendency and vapor pressure as escaping tendency that can be expressed in terms of a simple mathematical equation. However, this simple relationship is true only under certain conditions, namely low pressures, pressures well below the atmospheric pressure. Many phase changes and chemical reactions are studied at high pressures, where this simple relationship is no longer valid. In order to save the simple mathematical relationship, the thermodynamicist invents a new abstract quantity called "fugacity" to take the place of pressure. There is nothing tangible that can be pointed to and called "the fugacity." It is an abstract property of a substance that is defined by a mathematical relationship to free energy; is experimentally determinable; measures a substance's escaping tendency; under some conditions at least, is the same thing as the vapor pressure of the substance; and permits simple application of the second law of thermodynamics to chemical systems.

The equivalence of fugacity and pressure at low pressures gives rise to one other view of fugacity. At low pressures, gases are said to behave "ideally." This means that they obey simple mathematical laws. It is this simplicity that is at the root of the simple relationship between pressure and free energy noted above. At high pressures, gases behave nonideally, destroying the simple mathematical relationships. Fugacity is invented to take the place of pressure under these nonideal conditions. From this viewpoint, fugacity is the pressure that a real gas would have if it were behaving ideally. In effect, fugacity is a pressure corrected for nonideality. Since there are many objections to this view, it should always be realized that fugacity is an abstract quantity that is defined in terms of another abstract quantity, the free energy, and that it is best understood in the context of chemical and phase equilibria.

Applications

Thermodynamics has been characterized as a utilitarian science that creates mathematical relationships between the difficult to measure and the easy to measure. If one browses through a treatise on chemical thermodynamics, one will discover that it is strewn with mathematical equations. Thermodynamics starts with the first and second laws, which are universal, and with definitions; then, using the rules of algebra and calculus, thermodynamics derives mathematical connections between different observable properties of a system.

Some of the most important mathematical relationships describe phase changes and equilibria among phases. For example, there is the observed fact that if one dissolves a solute in a solvent, the freezing point of that solution will be lower than the freezing point of the pure solvent. This is the basis for the use of antifreeze in automobile radiators. It is possible to determine experimentally the mathematical connection between the drop in the freezing point and the concentration of the solution, but the laws of thermodynamics contain this relationship implicitly. It is possible to demonstrate mathematically, starting with the laws of thermodynamics, that the observed relationship between freezing point and concentration should be obeyed. This demonstration uses the fact that the freezing temperature is the temperature at which the solvent in the solution (water, for example) is in equilibrium with the solid solvent (ice). The key step in this demonstration uses the second law in terms of fugacity. The fugacity (escaping tendency) of the water in the ice is equal to the fugacity of the water in the solution. Once this is set down, the remainder of the demonstration, although mathematically involved, flows toward a conclusion that can be compared with experiment.

Another interesting example is the way the melting point of ice depends upon the pressure exerted upon the ice. Experiment shows that, for ice, the higher the pressure, the lower the freezing point. Ice is subjected to high pressures underneath the blades of ice skates. At these high pressures, the ice is actually above its melting point and thus melts, providing a lubricating layer of liquid water that makes skating possible. The laws of thermodynamics show that the freezing point should go down as the pressure goes up and even predict the amount by which it goes down for a certain pressure. The first step in such a demonstration is the recognition that the freezing point of ice is the temperature at which ice is in equilibrium with liquid water. In this case, the fugacity of ice equals the fugacity of liquid water. Once this fact is expressed mathematically, the demonstration proceeds apace.

As another example of the use of fugacity, consider the system in which one has pure water brought to a temperature and pressure at which solid (ice), liquid, and gas are simultaneously present in a closed container. Once again, one has a phase equilibrium. The fugacity is a tool for analyzing this equilibrium; when the equilibrium exists, the fugacities of each of the three forms of water must be equal to one another. There is no interest in the numerical value of the fugacity. Merely stating that the fugacities of the three forms are equal allows a demonstration that the three forms of water can exist simultaneously only at a unique temperature and pressure that is called the "triple point." For water, the triple point occurs at 0.0098 degrees Celsius and a pressure of 0.00611 bar.

The measurement of fugacity is not normally an end in itself. In the first place, it can only be calculated in an indirect fashion from other experimental data. The one case that is occasionally encountered is in obtaining the fugacities of gases. The behavior of gases is normally described in terms of an equation of state that is a mathematical relationship connecting the volume of a gas with the temperature and pressure at which the volume is measured. By means of suitable equations, it is possible to use this volume-temperature-pressure data to calculate the fugacity of the gas. At low pressures, the fugacity and the pressure have the same value. As the pressure increases, the numerical values for the pressure and the fugacity diverge. This divergence measures the extent to which the gas deviates from ideal gas behavior. However, there are other means of measuring this divergence from ideal behavior, and the fugacity adds little to them.

Fugacity (and free energy) plays an important role in demonstrating that certain experimental observations should have been expected given the validity of the laws of thermodynamics. Yet aside from the example of gases, the measurement of fugacity is not normally an end in itself. It is primarily an abstract tool that allows a concise application of the second law to systems that are of interest to the chemist. As an abstract tool, it escapes visualization as a concrete thing. It is an excellent example of how modern scientists create useful entities based on experimental facts that, while abstract, allow description of the behavior of the material world.

Context

The science of thermodynamics is a mature science. It has its roots in the seventeenth and eighteenth centuries, when its fundamental laws were establised, and its modern expression was already neatly formulated in the late nineteenth century by G. Willard Gibbs in America and by others. The notions of escaping tendency and fugacity were defined by Gilbert Lewis around 1900 and given popularity in the 1923 textbook by Lewis and Merle Randall. The notion of fugacity has been a part of thermodynamics ever since, and elementary textbooks on physical chemistry and thermodynamics include expositions on the subject.

The science of thermodynamics has provided modern chemists with a valuable tool for discussing chemical processes. Phase changes and phase equilibria are beneficiaries of the application of these tools. For example, in the field of geology (and such associated technologies as mineral discovery and mining), the understanding of phase equilibria and phase changes at the high temperatures and pressures found in the deep reaches of the earth has led to an understanding of the mechanism of formation of many minerals. This in turn has allowed the identification of possible sites for the discovery of important commercial minerals. Along a different track, these same studies of mineral formations applied to lunar rock samples give clues about the history of the solar system. An understanding of phase equilibria is important in metallurgy for the concise description and prediction of the properties of alloys. The thermodynamic description of phase changes is critical for reporting the properties of ordinary materials subjected to extraordinary conditions. For example, there is the form of solid water ("ice") that exists at high pressures and at temperatures well above the boiling point of water (100 degrees Celsius).

Fugacity has played an auxiliary role in all these studies. It is a means through which more immediately usable connections among measured properties can be derived. Although thermodynamics is a mature science, the expression of its fundamental principles is constantly being refined. If one compares the early expositions of the subject with more recent ones, one discovers that escaping tendency and fugacity, as criteria for equilibria, are receding into the background. Modern expositions of the basic principles of thermodynamics have tended to emphasize chemical potentials in the place of the older concepts. While conveying the same information, the chemical potential is more convenient mathematically. In addition, the chemical potential expresses the conditions of equilibrium in a language that is shared with other scientific disciplines. Electricity flows from a point of high electrical potential (voltage) to a point of low electrical potential, and matter flows from a place of high gravitational potential to a place of low potential; it does not flow at all where its gravitational potential is everywhere the same. The chemical potential (which is expressible in terms of fugacities) behaves in a similar fashion. Chemical matter flows (reacts) from a state of high chemical potential to a state of low chemical potential. In spite of this drift in terminology, however, fugacity will always be part of the chemical thermodynamic literature.

Principal terms

CHEMICAL THERMODYNAMICS: The branch of chemistry that studies the energetics of chemical reactions and phase changes

ENTROPY: A thermodynamic property of matter that is rooted in the huge numbers of atoms or molecules in a chemical system and that is a measure of the directionality of nature

EQUILIBRIUM: A condition in which all observable properties of a chemical system have come to constant values

ESCAPING TENDENCY: A general abstract thermodynamic property of phases that measures the stability of one phase relative to another

FIRST LAW OF THERMODYNAMICS: The expression of a law of nature, obeyed by all chemical systems, that the energy of the universe is a constant

FREE ENERGY: An abstract thermodynamic property that serves as a measure of the tendency of substances to escape; that is, it serves as a measure of the stability of one phase of a substance relative to another

PHASE: One of the three states of matter, solid, liquid, or gas

PHASE CHANGE: Alteration of a substance from one phase to another: for example, solid to liquid, solid to gas, liquid to gas, and solid (type 1) to solid (type 2)

SECOND LAW OF THERMODYNAMICS: A law of nature, obeyed by all chemical systems, which sums up the fact that a given chemical reaction or phase change will occur in one direction but will not occur in the reverse direction

Bibliography

Angrist, Stanley W., and Loren G. Helpler. Order and Chaos: Laws of Energy and Entropy. New York: Basic Books, 1967. Prerequisite to an understanding of fugacity and escaping tendency is a knowledge of the laws of thermodynamics. This book provides a readable introduction to the laws.

Lewis, Gilbert Newton, and Merle Randall. Thermodynamics and Free Energies of Chemical Substances. New York: McGraw-Hill, 1923. Although old, this classic treatise contains a readable discussion of a rather technical topic. Lewis first introduced the term "fugacity" in a publication in 1900 and is considered one of the founders of modern chemical thermodynamics.

‗‗‗‗. Thermodynamics. Revised by Kenneth S. Pitzer and Leo Brewer. New York: McGraw-Hill, 1961. This revision of the classic illustrates how the language of thermodynamics has developed and how fugacity has come to be replaced by a more understandable concept, the chemical potential. The language of this text contains representative presentations of the topic.

Parker, Sybil P. Physical Chemistry Source Book. New York: McGraw-Hill, 1987. This compendium contains a brief and technical description of the concept. It is worth examining simply because it isolates fugacity from all the other thermodynamic concepts with which it is associated.

Sage, Bruce H. Thermodynamics of Multicomponent Systems. New York: Reinhold Publishers, 1965. This book is for the reader determined to see fugacity as a property measured in its own right. It is very technical, but the single chapter on fugacity does illustrate the ends to which one must go to measure the quantity for a real system.