Applications of the Kinetic-Molecular Theory: Equations of State for Gases

James Richard Fromm

The state of any amount of substance is something which chemists often find it necessary to specify clearly. Such a specification for even the simplest substance must include at least n, the amount of substance present; p, the pressure, and T, the absolute temperature. In most cases it is found convenient to specify a standard state of a system as well, so that different chemical systems can be compared under the same conditions. As a standard, the choice of n = 1 is universal; however, the pressure is commonly chosen as either p = 100 kPa (1 bar) or as p = 101.325 kPa (1 atm). The temperature is commonly chosen as either T = 298.15 K (25oC) or as T = 273.15 K (0oC). Throughout all of the sections of this work the values of p = 100 kPa and T = 298.15 K will be used as standard. For gases, other works may employ "STP" (standard temperature and pressure) as p = 101.325 kPa and T = 273.15 K.

Ideal Gas Law

The simplest known equation of state, which is an equation linking at least the three properties of temperature, pressure, and volume of a chemical system, is the ideal gas law, pV = nRT. This equation accurately describes an ideal gas, but it describes a real gas such as oxygen or carbon dioxide accurately only at pressures below atmospheric. As the pressure increases or the temperature decreases, real gases are found to deviate significantly from the behavior expected of ideal gases.

Many more complex equations of state have been proposed to describe the behavior of gases, liquids, and solids but their physical interpretation is often not obvious. We will consider only one of them, the van der Waals equation.

van der Waals Equation

In 1873, Johannes van der Waals, a physics professor at the University of Amsterdam, developed an equation to account more accurately for the behavior of real gases. It was considered a sufficiently important development to justify the award of the Nobel Prize in 1910.

The van der Waals equation is the second most simple equation of state; only the ideal gas law is simpler. It is used to describe the behavior of gases when pressures are higher, or temperatures are lower, than those at which the ideal gas law is sufficiently accurate. The van der Waals equation describes the relationship between the physical quantities of pressure, temperature, and volume more accurately than does the ideal gas law. It does so, however, at the cost of a more complex equation and the use of a unique set of two van der Waals coefficients for each different gas. The usual form of the van der Waals equation is:

(p + (n2a/V2))(V - nb) = nRT

In the van der Waals equation, the term n2a/V2 reflects the fact that the attractive forces between molecules are not zero. The measured pressure is thus less than it should be because the attractive forces act to reduce it. The term nb reflects the fact that the volume of the molecules of a real gas is not zero, and so the volume in which the molecules may move is less than the total measured volume. A table of the van der Waals coefficients for several common gases is given below.

Table: Van der Waals Coefficients of Selected Gases
Gas atm  dm6/mol dm3/mol
Ideal 0.0 0.0
He 0.034 0.0237
Ar 1.345 0.0322
O2 1.360 0.0318
N2 1.390 0.0391
CO2 3.592 0.0427
CH4 2.253 0.0428
H2 0.244 0.0266

Example. The volume of one mole of oxygen molecules is 31.8 mL according to the van der Waals coefficient values tabulated. If the molecular diameter is taken as 0.370 nm, an approximate molar volume would be NAd3, or 30.5 mL. The volume actually occupied by one mole of oxygen gas at 25oC, according to the ideal gas law, is 24465 mL. The molecules of the gas actually occupy only 0.13% of the total volume occupied by the gas at 25oC. The vast majority of the volume of the gas is empty space through which the molecules move.

Copyright 1997 James R. Fromm