## The Ideal Cas

is frequently used as an aid in explaining the fundamental concepts of thermodynamics. It is a volumetric equation of state that reasonably well describes most real gases at sufficiently low pressure.

R in this equation is the gas constant. Its numerical value is obtained from the results of the experiment in the constant-volume gas thermometer described in Section 1.1.1. Specifically, the slope of the line in Figure 1.1 yields the numerical value of R. This quantity turns out to be the product of Avogadro's number <6.023 x 1023 molecules/mole) and Boltzmann's constant (1.38 x 10~23 J/molecule-K), and so is a universal constant. Its value, in various units, is:

R = 8.314 J/mole-K or 1.986 calories/mole-K or 82.06 cm3-atm/moIe-K

The first value is the SI* or metric value. The second value given above is convenient in many calculations where thermochemical properties from tables or plots are in the now-obsolete calorie or kilocalorie (kcal) units. The third form above is convenient in direct application of Equation (2.1) in which p and v are expressed in commonly used units (neither cubic centimeters nor atmospheres are SI units).

Although the gas constant R obtained from the slope of the line in Figure 1.1 is empirical, the fact that it is the product of two universal constants of physics suggests a deeper meaning. Indeed it does. The ideal gas law, together with the value of R, can be derived in statistical thermodynamics from the translational motion (i.e., kinetic energy) of a collection of particles subject to two conditions:

1. The actual volume occupied by the particles in a container must be small compared to the volume of the container. It is somewhat misleading to characterize a molecule as having a volume as if it had distinct boundaries like a soccer ball. In reality, a repulsive potential emanates from a molecule (Figure 2.1) and interacts with the repulsive fields of colliding molecules to cause the two to scatter from each other. This interaction of repulsive force fields gives each particle an effective volume. The requirement that the effective volume of the molecule be small compared to the container volume permits ideal gas behavior to be approached by all gases at sufficiently low pressure (which is equivalent to low density).

2. The above condition means that particles are essentially geometric points that collide with each other as well as impinge on container walls. In between collisions, the particles do not experience an attractive force from the presence of other particles. That long-range attractive forces (Figure 2.1) must exist is evident because all substances eventually condense into liquid or solid states at sufficiently high pressure and low temperature. However, for the gas to be ideal, these attractive interparticle forces must be negligible. This means that, on average, the particles must be far apart, which again implies low gas density.

If either of conditions 1 or 2 is not satisfied, the EOS does not follow the form given by Equation (2.1), and the gas is termed a real, or nonideal, gas.

* SI stands for (in French), "Système International."

FIGURE 2.1 Potential energy of interaction between gas molecules. rHS is the radius of the atom or molecule represented as a hard sphere.

The ideal gas has been referred to above as a collection of "particles" rather than specifically as "atoms" or "molecules." The reason is that particles other than atoms or molecules can exist in an ideal-gas state. Well-therm ali zed neutrons in the moderator of a nuclear reactor, for example, possess all of the properties of an ideal gas even though they are not analyzed using Equation (2.1). Similarly, a sufficiently-dilute collection of electrons exhibits classical ideal-gas behavior.

The microscopic thermodynamic view of the ideal gas (now considering ordinary gases of atoms or molecules) sheds light on the distinction between the volumetric equation of state, v{Tj)), and the thermal equation of state, say w(7». In the ideal gas, the only molecular motion that affects the p-v-T behavior is translation in the space of the container. The thermal equation of state, on the other hand, depends on the kinetic energy of translation motion as well as on forms of eneigy internal to the molecule. These include rotation of the molecule around its symmetry axes, vibration of the atoms in the molecule relative to each other and, at very high temperatures, electronic excitation. All of these forms of internal eneigy influence the thermal EOS but not the p-v-T behavior of the gas.

Problems 2.5, and 2.11 provide exercises utilizing the ideal gas law.

2.3 NONIDEAL (REAL) GASES

As described above, deviations from ideal gas behavior are, on a microscopic level, due to the intermolecular forces or, equivalently, the potential energy between molecules in the gas. Figure 2.1 shows a generic intermolecular potential function with both a short-range repulsive interaction (positive potential energy) and a longer range attractive interaction (negative potential energy). At large separation distances, the potential energy between the two species approaches zero, which is the condition for ideal-gas behavior.

Characterization of nonideality can be accomplished with the simple cylinder-piston apparatus shown in Figure 2.2. The pressure of a fixed quantity of -gas held at constant temperature is varied and the volume measured. The ratiopv/RT is plotted against pressure. For an ideal gas, this quantity should be equal to unity for all pressures. The graph in Figure 2.2 shows the behavior of a gas at two temperatures. At 7"j, deviation from ideality is positive, or pv/RT > 1. At the lower temperatureT2, the deviation is negative, and pv/RT < 1. In all cases, however, the data extrapolate to the ideal gas value pv/RT = 1 as p 0.

Measure V(p, T) (i.e., the EOS)

(at Tj - Positive Deviation from Ideality)

(at Tj - Positive Deviation from Ideality)