Behavior Of Gases

Type of physical science: Classical physics

Field of study: Thermodynamics

Gases, unlike liquids or solids, exhibit simple behavior over a wide range of conditions. Equations describing the interrelationships that exist among the variables used to characterize a gas can be developed and used to investigate the properties of the molecules composing the gas.

Overview

A gas is one of the three common states of matter. Gases are distinguished from solids and liquids by their low density and ease of compressibility. Both gases and liquids are classified as fluids, a form of matter having indefinite shape. While a liquid occupies a fixed volume, however, a gas will expand or contract to occupy uniformly the volume of its container.

A sample of gas at equilibrium may be characterized by a set of variables describing the state of the gas. These state variables include the pressure or force exerted by the gas per unit area, the volume occupied by the gas, the temperature of the gas, and the quantity of gas in the sample. For a mixture of different gases, a complete description of the state of the gas must specify the composition of the gas mixture, that is, the amount of each type of gas in the mixture.

For example, dry air is a mixture of approximately 78 percent nitrogen, 21 percent oxygen, and 1 percent argon, with smaller amounts of trace gases such as carbon dioxide.

In studying the behavior of gases, several simple relationships are found to exist among the variables describing the gas. For a fixed quantity of gas at constant temperature, the product of pressure times volume (pV) remains approximately constant over a wide range of pressures (Boyle's law). Similarly, the volume occupied by a fixed quantity of gas at constant pressure decreases with decreasing temperature (Charles's law). If an absolute temperature scale is used--such as the Kelvin scale, where the lowest possible temperature is assigned a value of 0 degrees--then the ratio of volume to temperature (V/T) measured under conditions of constant pressure and quantity of gas remains approximately constant over a wide range of temperatures.

The above observations on the behavior of gases can be summarized in a simple equation, pV = nkT, known as the ideal gas law. In this equation, the product of pressure times volume (pV) is proportional to the temperature of the gas (T) and the number of gas molecules present (n). The constant of proportionality in the equation, k, is called the Boltzmann constant and is a fundamental constant in statistical thermodynamics.

An alternative way of presenting the ideal gas law is to express the number of gas molecules in terms of the number of moles of molecules in the gas. A mole of molecules is equal to the number of atoms found in a 12-gram sample of isotopically pure carbon, which is roughly 6 x 10²³. This number is called Avogadro's number. Giving the quantity of gas molecules in terms of moles of molecules represents a way of avoiding having to deal with large numbers. If quantity of gas is given in terms of the number of moles of molecules present in the gas, the ideal gas law can be written as pV = nRT, where n is the number of moles of gas, and R, a new constant of proportionality, is called the gas constant.

The ideal gas law can be applied to both pure gases and mixtures of gases. For a mixture of gases at equilibrium, the temperature and volume for each component of the mixture will be the same. The total pressure of the gas mixture can therefore be considered to be the sum of the partial pressures exerted by each component of the mixture, which will depend on the relative number of molecules of each gas in the system and which can be calculated using the ideal gas law (Dalton's law).

The underlying assumptions implicit in the ideal gas law are that the fraction of the total volume of a gas physically occupied by the gas molecules themselves is small and that the forces of interaction acting between gas molecules are also small. The validity of these assumptions depends on the density of the gas, that is, the ratio (m/V), the mass of the gas divided by the volume occupied by the gas. In general, the accuracy of the ideal gas law improves when the density of a gas decreases, since decreasing density both increases the average separation between gas molecules, minimizing the interaction forces acting between them, and decreases the fraction of the total volume of the gas occupied by molecules. In fact, all gases and mixtures of gases obey the ideal gas law as a limiting law under conditions where the density of the gas approaches zero.

Deviations from ideal gas behavior are expected to occur as the density of a gas increases, either by compressing the gas or by lowering the temperature of the gas. The amount of deviation from ideal gas behavior can be expressed in terms of the compressibility factor of the gas (Z), defined by the equation Z = pV/nkT. For an ideal gas, the compressibility factor is equal to 1 under all conditions. For a real gas at moderate densities, the most important interaction force acting between gas molecules is a weak, long-range attractive force. This attractive force acts to lower the pressure of the gas to a value below that predicted by the ideal gas law and therefore to lower the compressibility factor for the gas. Nevertheless, at high densities the dominant interaction force is a strong, short-range repulsive force, which acts to increase the pressure of the gas above that expected from the ideal gas law and therefore to increase the compressibility factor.

A large number of alternative equations of state has been developed to describe the behavior of gases under conditions where deviations from the ideal gas law occur. The two most popular alternative equations are the virial equation and van der Waals equation. In the virial equation, the product pV is set equal to a series of terms involving either the pressure or volume. In principle, such an expression can be made as accurate as desired by increasing the number of terms involved in the series. In the van der Waals equation, specific corrections are applied to the ideal gas law to account for both short-range and long-range forces of interaction between molecules. Both the virial equation and van der Waals equation predict that gases behave ideally at low density, as expected.

The van der Waals equation is particularly useful in providing a qualitatively correct model for critical behavior in gases. At low temperatures, a gas can be converted into a liquid by increasing the pressure exerted on the gas. Yet, if the temperature is above Tc, the critical temperature of the gas, liquefaction will not occur. Instead, as pressure is increased, the behavior of the gas changes continuously from that of an ideal gas to that of a nearly incompressible liquid. The van der Waals equation is the simplest gas equation that correctly describes the qualitative behavior of a gas in the vicinity of the critical point and that provides an expression for the critical temperature, pressure, and volume for a gas.

The critical constants for a gas can be used to search for similarities in the behavior of different gases. This is done by defining a set of reduced variables for a gas. The reduced variables are equal to the actual variables divided by the corresponding critical constant. For example, the reduced temperature of a gas, Tr, is equal to the actual temperature divided by the critical temperature. When gas behavior is expressed in terms of reduced variables, it is found that real gases in the same state of reduced temperature and volume tend to have the same reduced pressure. This observation is the basis for the principle of corresponding states, a statement that all gases behave in approximately the same manner when reduced variables are used. The law of corresponding states works best for gases composed of spherically shaped molecules having no strong long-range force of interaction.

Another interesting property of gases is the effect of a sudden change in pressure on the temperature of a gas. If a gas is allowed to expand adiabatically--that is, under conditions where there is no heat flow between the gas and its surroundings--the temperature of the gas after the expansion will in general be different from that found before the expansion. This observed temperature change is called the Joule-Thomson effect. The magnitude of the temperature change depends on the value of the Joule-Thomson coefficient for the gas, which itself depends on both pressure and temperature and which is different for different gases. A gas with a positive value for the Joule-Thomson coefficient cools when expanded, while a negative value for the Joule-Thomson coefficient implies an increase in temperature upon expansion of the gas. For a fixed pressure, the point at which the sign of the Joule-Thomson coefficient changes is called the inversion temperature. For an ideal gas, the Joule-Thomson coefficient is zero under all conditions, implying that the origin of the Joule-Thomson effect is intermolecular interaction forces in the gas.

In principle, it should be possible to derive all the observed properties of gases from a consideration of the behavior of gases on a microscopic, or molecular, level. In the kinetic theory of gases, a simple expression for the distribution of molecular velocities in a gas is derived. From this expression, a temperature can be defined and the ideal gas law derived for a collection of weakly interacting molecules. The main conclusions from kinetic theory are that the average kinetic energy of a gas is proportional to the temperature of the gas and that the average velocity of a gas molecule is inversely proportional to the square root of the mass of the molecule. Kinetic theory can be used to explain the transport properties of gases, such as effusion (escape of a gas to vacuum through a small hole), diffusion (the spontaneous mixing of gases in the absence of convection), viscosity (resistance of a gas to flow), and thermal conductivity (transport of thermal energy by a gas).

Applications

The simple behavior observed for gases in the limit of low density forms the basis of a number of common experimental techniques. For example, the ideal gas law predicts that a given number of gas molecules will, at constant temperature and pressure, occupy a fixed volume independent of the identity of the gas. Therefore, by determining the density of a gas at a particular pressure and temperature, the molecular weight of the gas can be found. This technique is used to find the molecular weight of unknown gases and can be used to determine molecular weights of liquids or solids if they can be converted into gas. Since gas densities depend on temperature and pressure as well as the identity of the gas, they are usually reported for a specific temperature and pressure. This standard temperature and pressure (STp) is defined as a temperature of 0 degrees Celsius (the normal melting point temperature for ice, equal to 273 Kelvins) and a pressure of 1 atmosphere, approximately the pressure of the earth's atmosphere at sea level.

Ideal gas behavior is also the basis of the gas thermometer. This thermometer makes use of the fact that the product of pressure times volume for a fixed quantity of gas obeying the ideal gas law is directly proportional to the temperature of the gas. Since all gases behave like ideal gases in the limit of low density, this provides a means of defining a practical temperature scale that is independent of the substance used as the working gas in the thermometer. In practice, helium is often used in gas thermometers, since it closely follows the ideal gas law over a wide range of pressures and temperatures. Gas thermometers can be used to measure temperatures as low as 1 Kelvin.

Nonideal properties of gases can also have practical applications. For example, the Joule-Thomson effect can be used as a means of liquefying gases. If the temperature of a gas is below the inversion temperature, then the gas will cool when expanded adiabatically. By carrying out successive cycles of adiabatic expansion followed by constant temperature compression, the temperature of a gas can be lowered to the point where condensation occurs.

This is the basis of the Linde refrigerator (developed by Karl Paul Gottfried von Linde), used to liquefy air, nitrogen, oxygen, and other gases.

Since nonideal behavior is the result of interaction forces acting between gas molecules, deviations from ideal gas behavior can be used to investigate these forces of interaction. For example, based on statistical mechanics, a relationship can be found between the terms appearing in the virial equation of state and the interaction energy for a pair of gas molecules. The terms appearing in the virial equation or the van der Waals equation can also be used to estimate the size of a gas molecule. This in fact represents one of the first experimental methods developed for the determination of the sizes of molecules.

The predictions of kinetic theory for the behavior of gases also have a number of interesting applications. Since the average velocity of a gas molecule is inversely proportional to the mass of the molecule, light molecules should on average have higher velocities than heavy molecules at the same temperature. One consequence of this is that light gases can reach the escape velocity for a planet or moon more easily than heavy molecules. The absence of significant amounts of hydrogen and helium in the earth's atmosphere is believed to be caused by the slow diffusion of these gases into space. More massive planets, such as Jupiter and Saturn, have a higher escape velocity and have been able to retain most of their initial mass of hydrogen and helium, while for smaller bodies, such as Earth's moon, diffusion has acted to remove essentially all the atmosphere initially present.

The fact that the rate of diffusion is faster for light molecules than for heavy molecules can be used as the basis of a separation technique. This is particularly useful for separation of different isotopic forms of an element, which normally cannot be separated chemically. For example, separation of different isotopes of uranium can be achieved by conversion of uranium into uranium hexafluoride. Differences in the rate of diffusion of molecules containing different isotopes of uranium are sufficient to produce enriched samples of uranium used as fuel in nuclear reactors.

Context

Because small changes in the pressure and temperature of a fixed quantity of gas result in measurable changes in volume, gases were the first state of matter whose physical behavior was extensively studied. In the seventeenth century, Robert Boyle, an Irish chemist, first made the observation that for a fixed quantity of gas at constant temperature, the product of pressure times volume is constant. Shortly thereafter, the French physicist Guillaume Amontons noted that the volume of a fixed amount of gas held at constant pressure decreased by a fixed amount for a given change in temperature independent of the identity of the gas used. His work was neglected for almost a century until the law was rediscovered independently in the early 1800's by Jacques-Alexandre-Cesar Charles and Joseph-Louis Gay-Lussac, who were interested in the properties of lighter-than-air balloons. John Dalton, an English chemist, soon extended their observations to mixtures of gases and demonstrated that the total pressure of a mixture of gases could be expressed in terms of the partial pressures of each component of the gas. William Thomson (later Lord Kelvin) also used the observations of Charles and Gay-Lussac as the basis of defining the absolute temperature scale that now bears his name.

The key observation allowing the above laws to be combined into a single equation was put forth in 1811 by Italian physicist Amedeo Avogadro, who suggested that all gas samples at the same pressure, volume, and temperature contain the same number of gas molecules.

Avogadro's hypothesis was neglected for more than forty years, until his countryman Stanislao Cannizzaro demonstrated its use in determining the molecular weights of gases. Once Avogadro's hypothesis was accepted, the ideal gas law followed.

At the same time that the ideal gas law was being developed, scientists began to investigate deviations from this simple behavior. James Prescott Joule and Thomson carried out a series of experiments in the early 1850's establishing the Joule-Thomson effect. At about the same time, the Irish chemist Thomas Andrews, based on his work with carbon dioxide, suggested that for every gas there existed a critical temperature above which the gas could not be liquefied by the application of pressure. Johannes Diderik van der Waals, a Dutch physicist, proposed a modified form of the ideal gas law in 1873 that accounted for the effects of intermolecular forces acting between molecules and that qualitatively explained the critical behavior of gases observed by Andrews.

From a different perspective, James Clerk Maxwell and Ludwig Boltzmann, by applying statistical arguments to the behavior of individual molecules, developed the kinetic theory of gas and gave a theoretical derivation of the ideal gas. Modifications of kinetic theory to account for intermolecular interaction forces soon made it possible to investigate these forces by studying the bulk behavior of gases.

In the twentieth century, the study of the behavior of gases continued. When statistical mechanics and quantum mechanics were combined, it was found that under certain conditions the spin characteristics of a collection of particles could affect their equation of state. Albert Einstein and Sir Jagadis Chandra Bose found that one type of behavior occurred for particles with integer spin, called bosons, while Enrico Fermi and Paul Adrien Maurice Dirac showed a different behavior for fermions, particles with half-integer spins. A particularly interesting application of the theory developed by Fermi and Dirac is in the description of electrons within a metal, which in many respects behave like a collection of gas molecules.

The behavior of gases in the vicinity of the critical point has also led to the development of a new physical theory, which in many cases can be generalized to cover other types of critical behavior. A classical theory of critical phenomena was first developed by Lev Davidovich Landau and generalized by Laszlo Tisza. Nevertheless, theoretical work by Lars Onsager in the mid-1940's suggested shortcomings in the classical theory, a result soon confirmed by experiment. A more complete theory for critical behavior, developed by Kenneth G. Wilson in the 1960's, has provided a correct description of critical behavior, both for gases and for other cases.

Principal terms

COMPRESSIBILITY: the degree to which the volume occupied by a substance decreases as the pressure acting on the substance increases

CRITICAL TEMPERATURE: the temperature below which a gas can be liquified by the application of pressure

EQUATION OF STATE: a mathematical relationship between the state variables describing a system

IDEAL GAS LAW: an equation of state obeyed by all gases in the limit of low density

JOULE-THOMSON EFFECT: the change in temperature observed when a gas is expanded under conditions where there is no net flow of heat into or out of the gas

LAW OF CORRESPONDING STATES: a statement that gases tend to behave in the same way when pressure, volume, and temperature are expressed in terms of reduced variables

MOLE: a unit for quantity equal to the number of atoms in a 12-gram sample of isotopically pure carbon, approximately 6 x 10²³

REDUCED VARIABLE: the ratio of the actual value of a variable describing a gas divided by the corresponding critical constant

STATE VARIABLE: a variable used to characterize the state of a system, such as a gas

Bibliography

Asimov, Isaac. A SHORT HISTORY OF CHEMISTRY. Garden City, N.Y.: Doubleday, 1965. A highly readable account of the history of chemistry, which places the development of the gas laws into the general context of the development of the chemical sciences.

Hill, Terrell L. LECTURES ON MATTER AND EQUILIBRIUM. New York: W. A. Benjamin, 1966. A good basic introduction to the properties of matter. The first four chapters discuss states of matter, ideal and nonideal gas behavior, and intermolecular forces.

Hirschfelder, Joseph, C. F. Curtiss, and R. B. Bird. MOLECULAR THEORY OF GASES AND LIQUIDS. New York: Wiley, 1954. While highly technical, this book represents the definitive discussion of the properties of gases and liquids.

Smith, Richard F. CHEMISTRY FOR THE MILLIONS. New York: Scribner's, 1972. An introductory discussion of basic chemical principles for the nonscientist. The role of the gas laws in the context of chemistry is clearly illustrated.

Tabor, David. GASES, LIQUIDS, AND SOLIDS. Baltimore: Penguin, 1969. A presentation of the behavior of ideal and nonideal gases, as well as of liquids and solids, from the perspective of statistical mechanics.

Two of the ideal gas laws

Equation of State

Fluid Mechanics and Aerodynamics

Liquefaction of Gases