Equation of State

Type of physical science: Classical physics

Field of study: Fluids

An equation of state relates the volume with the amount, pressure, and temperature of a pure substance or mixture. The equation of state may predict these properties when unknown and may provide a base for understanding the nature of the substance or mixture.

Overview

An equation of state relates the amount of a substance or mixture to the space occupied, the volume, at the temperature and pressure of the material. Amount can be thought about as mass or number of molecules. The mass of liquid water in 1 liter is 1,000 grams. The mass of air in 1 liter is about 1.33 grams. A molecule is the smallest particle of a substance that it is possible to have, and can be imagined as the result of repeated cuttings of a small portion of a substance in half. If an ice cube were cut into halves, then fourths, eighths, sixteenths, and so on, eventually an ultimate particle would be reached that could not be divided again without destroying the nature of the water itself. Such an ultimate particle is a molecule of water.

Because a molecule is so tiny, a new quantity--a mole--is used to represent a very large number of molecules. A mole is a quantity such as a dozen , or a century . A mole is 602,000 million, million, million particles. One mole of air molecules occupies a volume of 22.4 liters near the ocean (at 1 atmosphere pressure) at the freezing temperature of water.

An example of an equation of state can be seen by visualizing a bicycle pump. The more air put in the tire, the greater the volume of the tire. Doubling the amount of air in the tire causes the volume of the tire to double, and tripling the amount of air causes the volume to triple.

This is an example of a direct linear proportion between amount and volume. The equation of state is the expression of the proportion. The volume (V) of a gas equals the moles (n) of the gas multiplied by a constant of proportionality (c) (V = cn); temperature and pressure remain unchanged.

Another example of an equation of state can be seen by visualizing a deep-sea diving bell. The top of the bell is equipped with a movable piston that moves down as the depth of the bell increases. The pressure of the ocean water forces the piston down. On the surface, only 1 atmosphere of air forces the piston down. At a depth of 10 meters, the pressure is 2 atmospheres (one from the air and one from the water), and the volume of the bell shrinks to exactly one-half of what it was on the surface. When the bell is at a depth of 31 meters, the pressure is 4 atmospheres and the volume of the bell is one-fourth of what it was on the surface. This is an example of an inverse linear proportion. The equation of state is the generalization that the volume (V) of a gas equals a constant of proportionality (c) divided by the pressure (p) (V = c/p); temperature and the moles of gas remain unchanged.

There is another direct linear proportion between volume and temperature. It can be illustrated with a hot air balloon. In order to show the direct proportion, temperature must be measured in the Kelvin scale. The Kelvin scale is set at 273 degrees at the temperature of ice-water, and 373 degrees in boiling water at 1 atmosphere pressure.

These three equations of state are the same for any of the gases in the air, as a mixture or pure. The three equations can be combined into one, the ideal gas equation: the volume (V) of any gas equals a constant of proportionality (R) multiplied by the temperature (T) and by the moles of gas (n), and divided by the pressure (p)

(V = nRT/P; R = 0.082). The kinetic molecular theory can be used to explain the equation, and can be visualized as a set of tiny marbles in rapid motion. Each marble represents a molecule; each moves in a different direction. The molecules collide, then bounce away, perpetually. Ideal gas molecules never run down; they never become liquids. There is always much more space between them than the tiny space the molecules occupy themselves.

Real gases have properties that the ideal gas equation cannot predict, and are different from tiny marbles in motion. The ideal gas is a model. A model is part true and part false, just as a statue or a portrait has the appearance of a person but does not move, laugh, or cry.

Real gas molecules get very close together when compressed at high pressures. Then they begin to coalesce. The spaces between real gas molecules shrink more than they would if they were ideal. When this occurs and the temperature is low enough, the real gas becomes a liquid, and a meniscus forms. The meniscus is the "skin" that marks the surface of a glass of water or a bowl of soup. Water bugs do not sink but remain on the surface of a pond because the "skin" of liquid water has some strength. The meniscus, or "skin," marks the difference between a liquid and a gas. The ideal gas model cannot show a meniscus.

If a certain amount of liquid with its gas is enclosed in the proper volume and sealed, it can show the critical state, a state all real substances have that an ideal gas does not have. If the liquid and gas sample is enclosed within a steel block that has a window, the critical state can be observed. As the steel block is heated, the pressure increases when the temperature increases because the volume and amount are fixed. After a while, bubbles can be observed, as if the liquid were about to boil. Then, vertical streams of the substance are seen. The meniscus appears to thicken; it becomes cloudy, then golden, as a sunset in a cloudy sky. Next, the view becomes dark. As heating continues, the view clears but now there is no meniscus. At these temperatures, there is no difference between liquid and gas; the substance is called a fluid. The state when the meniscus disappears is the critical state; its temperature is the critical temperature.

Equations of state for real gases have terms to account for the size of molecules and for the attraction between molecules that cause the molecules to coalesce. The volume of a gas decreases as the temperature is lowered or the pressure is raised. The effect of the size of the molecules can be visualized by imagining two marbles that come closer and closer until they touch; as one marble rolls around the other, the volume that must be excluded from the volume that can be decreased is seen. In addition, the pressure must have a certain value subtracted from it because the attraction reduces the impact of the collisions that cause pressure. The value subtracted from the pressure depends on how close the molecules are and is inversely proportional to the volume of the gas squared. Pressure (p) equals a constant (R) multiplied by the temperature (T) and divided by volume (V) less excluded volume (b) minus the pressure drop from attraction (a/V²) (p = RT/V - b - a/V², for one mole). This is known as the van der Waals equation of state; it shows some of the shrinking caused as gas molecules coalesce; b and a are different for different substances.

Gases are often stored compressed in steel tanks. Chlorine, used as a disinfectant for water, is one example. A 4-liter tank contains 570 grams of chlorine, 8 moles. On a warm day, the tank shows 29 atmospheres on its pressure gauge. The ideal gas equation predicts a pressure of 49 atmospheres; van der Waals' equation predicts 30 atmospheres.

At very high pressures, the excluded volume subtracted for two-molecule collisions is too large. This fact can be visualized by imagining three marbles that come together simultaneously. The third marble has penetrated the excluded volume of the pair. To do a better job, the pressure can no longer be an inverse linear proportion of volume but becomes related to a set of values each inverse to volume to a different extent; one is linear (1/V), another is quadratic (1/V times itself), another is cubic (1/V times itself times itself). An equation of state that does this is called a hard-sphere equation of state because it is best modeled as a whole collection of marbles squeezed close. Equations of state that are hard sphere and include changes for attraction usually are best.

There are other equations of state that are written to reproduce the experimental data without any reference to models. They have many adjustable constants that differ for each substance. These equations of state are empirical. They fit the experimental results better but cannot be used to understand the nature of a substance or mixture.

Applications

Equations of state abound in the use of the atmosphere. About 10 kilometers above the surface of the earth, the temperature drops to about 220 Kelvins and the pressure drops to about three-tenths of an atmosphere. Equations of state can be used to predict the volumes of meteorological balloons that operate at that altitude. Air pollution commonly occurs where thermal inversions occur. There, the temperature of the air several hundred meters high is greater than the air near the surface. As the ground radiates heat, carbon dioxide, water vapor, and pollutants may absorb that heat, and in the absence of natural circulation, the air may become poorly ventilated. Equations of state are useful to predict the extent of thermal inversion and the conditions for its formation, as well as its dispersal.

Freons are substances composed of carbon, chlorine, and fluorine; they cause the hole in the ozone layer that develops over Antarctica every year. Freons are used as refrigerants in refrigerators and air conditioners in the home, in industries, and in automobiles. Freons are useful because when they expand at warm temperatures, they absorb heat energy and provide a cooling effect. They must be compressed in a cycle in order to be used repeatedly; during the compression, heat energy is passed to the outside. New materials must replace the Freons to reduce ozone depletion. Equations of state will be used to predict compressed states and expanded states for the new materials, when only a few have been measured, and to estimate the heat energy absorbed or released when a new material expands or compresses.

Supercritical fluids are substances or mixtures above the critical temperature at high pressures. Carbon dioxide has a pressure of 74 atmospheres at its critical temperature of 304 Kelvins. Many substances dissolve in a supercritical fluid that do not dissolve in the same fluid at temperatures below critical, thus supercritical fluids can be used to extract desirable substances from mixtures that occur in nature. Supercritical carbon dioxide can extract caffeine from green coffee beans. Supercritical propane has been used to extract fuel from oil sands and vegetable oil from seeds. Equations of state predict the properties of the mixtures in extraction and the heat energies that are part of the process.

Equations of state are being applied to learn about the nature of the solids at high pressures deep in the earth. The earth's crust below nonmountainous land extends 30 kilometers on the average; below, a more dense mantle continues for 3,000 kilometers. The mantle rests on a molten, mostly iron core that has a density twice that of the mantle above it. The pressure is more than 1 million atmospheres and the temperature is about 4,000 degrees Celsius at the boundary between the mantle and the core. The pressure is about 20,000 atmospheres, 120 kilometers into the mantle from the crust; temperatures there are near 1,200 degrees Celsius, where lava melts. The mantle is considered to be solid, but some minerals may liquefy and exist, in part, dissolved in supercritical carbon dioxide that contains some water, methane, and hydrogen sulfide. Equations of state that are used to model solids consist of a pressure from the forces that hold the molecules in place, added to a pressure from the vibrations of these molecules, and added to a pressure from deformations. Deformations are changes in the volume of a solid caused by a change in pressure. Sudden changes in pressure occur from ruptures of underground cavities, from earthquakes, or from underground explosions; they may transmit deformations up to 100 kilometers. It is difficult to measure such deformations when they occur deep underground. Earthquakes commonly occur at depths of more than 10 kilometers. The scattering of synchrotron X rays, extra strong X rays produced by electrons accelerated magnetically around a ring, provide one source of such data. Such experiments allow deformations of metals and rock samples at high pressures and temperatures to be studied in the laboratory. Equations of state can then be determined, tested, and comparisons made with seismic data.

A 1 million-ton nuclear explosion in the air yields a peak overpressure of 7 atmospheres. Underground weapons tests are usually at depths of less than 2.5 kilometers. They approach an earthquake in magnitude but the overpressure occurs at a point rather than along a line. Equations of state for solids may be used to predict the amount of deformations and the range of underground explosions.

Very dense matter under high pressure occurs in the planets and stars. Equations of state are used to estimate properties and to build models for very dense matter. Venus is solid, as are the earth and Mars, but has an atmosphere of carbon dioxide 100 kilometers thick; the pressure is 100 atmospheres and the temperature is 400 degrees Celsius at the surface of Venus.

Jupiter, Saturn, Uranus, and Neptune are very different; they consist mostly of hydrogen and helium (98 percent), the same as the stars, and are many times larger than the earth. At the top of their cloud formations, pressures are near 1 atmosphere, with temperatures near -150 degrees Celsius. Clouds of methane, ammonia, and ice exist 200 kilometers into the planets at pressures from 10 to 100 atmospheres and temperatures near 30 degrees Celsius, close to that of the surface of the earth. Pressures continue to increase deeper within the planets, reaching 1,000 atmospheres and 300 degrees Celsius, about 500 kilometers in, and more than 1 million atmospheres near the center.

Using optical data from spacecraft missions such as Voyager 1 and 2, and from high-energy X-ray scattering in the laboratory at high pressures, and using the mixtures that exist in solar matter, equations of state have been constructed that simulate the planets. The equations of state can be used to extend calculations of properties deep within the planets where direct measurements are unavailable. Equations of state for hydrogen and helium are known to 1,000 atmospheres and show a transition to metallic hydrogen and to atomic hydrogen.

Density is the mass of a substance or mixture divided by its volume. The molecules of a substance consist of atoms joined together. An atom is analogous to a miniature solar system with a positively charged nucleus in the center that has most of the mass and with negatively charged electrons in orbit around the nucleus. At the very high temperatures and densities in stars, up to 10 million degrees Celsius and more than 1 billion times the density of the earth, matter exists only as nuclei and electrons. Models of very dense matter predict equations of state that show pressure and density are exponentially related. Pressure equals density times itself a certain number of times. Such equations of state are used to estimate the mass and radius of the core of a star as it ages and becomes more dense and to identify possible processes occurring in the stars.

Context

In 1680, Robert Boyle discovered the first equation of state in his experiments: the inverse linear proportion between pressure and volume. By the end of the eighteenth century, Jacques-Alexandre-Cesar Charles and Joseph-Louis Gay-Lussac had established direct linear proportions between volume and temperature and between pressure and temperature.

The relationship between volume and number of molecules was harder to establish. In 1738, Daniel Bernoulli published a qualitative version of the kinetic theory of gases but few accepted it. Amedeo Avogadro suggested in 1811 that equal volumes of all gases at the same temperature and pressure contained the same number of particles, but there was confusion among the scientific community about the meaning of the particles. In 1860, Stanislao Cannizzaro clarified Avogadro's particle as a molecule. About the same time, Rudolf Clausius resurrected Bernoulli's kinetic theory. The ideal gas model and equation of state began to be used at this time.

In 1881, in a classic paper, "On the Continuity Between Gases and Liquids," Johannes Diderik van der Waals published the first real gas equation of state. He used Thomas Andrews' experimental results about the critical state of carbon dioxide. Since then, many others have used van der Waals' equation and modified it to develop new equations. Such modifications are used daily around the world.

The first solid equation of state was derived by G. Mie in 1903. F. Birch applied F. D.

Murnaghan's deformation theory to write an equation of state in 1952 that continues to be used.

High-density equations of state became of great importance in the 1980's and 1990's as the success of Voyager 2 provided new information about the nature of the planets.

In the second half of the twentieth century, interest in fluid equations of state reemerged, prompted by two developments. First, the critical state was studied in exacting detail for several substances and mixtures. The critical state has been found to be flatter than can be explained by any modification of van der Waals' equation; flatter means larger changes in volume with slight changes in pressure and temperature. Large heat energies are absorbed or released with those changes in volume. Second, the development of the computer has enabled large quantities of data to be compared with many equations of state. Equations have been developed that include up to fifty adjustable constants. They reproduce the data very well but cannot predict properties for new substances and mixtures, nor aid in understanding near critical and supercritical mixtures. Research is under way for new equations of state with few adjustable constants for use near and above the critical state. Such equations will be applied to energy, food processing, and the atmosphere in the twenty-first century.

Principal terms

ATMOSPHERE: the layer of air around the earth's surface that supports life; a unit of pressure equal to the force per unit area that the air exerts at the surface of the earth at sea level

FLUID: a liquid or gas; there is no difference between a liquid and a gas above a substance's critical temperature

KELVIN: a unit of temperature equal to one one-hundredth (0.01) of the difference between the freezing point and the boiling point of water, where the freezing point of water is set equal to 273 Kelvins

MOLE: an amount equivalent to 602,000 million, million, million particles

MOLECULE: the tiniest part of a substance that retains its nature

PRESSURE: the force that a substance exerts on a square centimeter of a surface it contacts; the air has a pressure of 1 atmosphere at sea level; the collisions of many molecules simultaneously cause pressure

STATE: the volume (V) occupied by a substance for a particular amount (mass) at a specified temperature (T) and pressure (p), all must be specified to define a state

SUBSTANCE: any pure material such as water, oxygen, nitrogen, ammonia, and carbon dioxide are common substances; seawater and air are mixtures of substances

VOLUME: the amount of space occupied by a substance (visualize a point moving to form a line, the line moving perpendicular to itself to form an area called a square, then the square moving perpendicular to itself to form a volume called a cube)

Bibliography

Basta, Nicolas. "Supercritical Fluids." HIGH TECHNOLOGY 4 (June, 1984): 75-79. Many examples of supercritical extraction are described.

Boyle, Robert. THE SCEPTICAL CHYMIST. Reprint. New York: Dutton, 1967. A reprint of Robert Boyle's classic. Boyle champions the use of experiments and observations to decide what is known.

Farber, Eduard. GREAT CHEMISTS. New York: Interscience, 1961. More than one hundred well-written biographies are included, among them Boyle, Dalton, Gay-Lussac, Cannizzaro, and van der Waals.

Hildebrand, Joel H. AN INTRODUCTION TO MOLECULAR KINETIC THEORY. New York: Reinhold-Wiley, 1963. This is a thorough development of both equations of state and the kinetic theory. Includes equations. Many insightful comments by Hildebrand make it worthwhile reading.

Jeans, James Hopwood, Sir. AN INTRODUCTION TO THE KINETIC THEORY OF GASES. Cambridge, England: Cambridge University Press, 1940. Jeans briefly surveys the history of the kinetic theory and offers a description of the three states of matter. The most fascinating section is his famous "Mechanical Illustration of the Kinetic Theory of Gases." Jeans uses the analogy of billiard balls in motion on a table with moving sides. A classic.

Morrison, Philip, and Phyllis Morrison. POWER OF TEN. New York: W. H. Freeman, 1982. A remarkable collection of pictures and commentary. The authors take the reader outward in powers of ten starting with the reader and ending with the universe.

Peters, Edward I. INTRODUCTION TO CHEMICAL PRINCIPLES. 5th ed. Philadelphia: W. B. Saunders, 1990. The prologue delightfully examines the size of bubbles under water. Chapter 2 has a good discussion of the states of matter and some definitions. Chapters 3 and 4 explain direct proportions.

Sengers, Jan V., and Anneke Levelt Sengers. "The Critical Region." CHEMICAL AND ENGINEERING NEWS 45 (June 10, 1968): 104-105. Contains large photographs of a transition through the critical state of carbon dioxide from a temperature above the critical one to a temperature below. The photographs enhance the accompanying description.

Zhukov, V. N., and V. A. Kalinin. EQUATIONS OF STATE FOR SOLIDS AT HIGH PRESSURES AND TEMPERATURES. Translated by Albin Tybulewicz. New York: Plenum Press, 1971. Much of the mathematics is beyond the beginning level, but the information about the earth's structure is easy to read and is fascinating.

The Chemistry of Air Pollution

Diffusion in Gases and Liquids

The Behavior of Gases

Liquefaction of Gases

The Atomic Structure of Liquids

The Physics of Weather