EQUATIONS OF STATE

B. EQUATIONS OF STATE

The ideal gas equation of state is also considered to be a thermally or mechanically perfect state equation. In Chapter 1 we presented a simple derivation of this equation by using microscopic thermodynamic considerations and neglecting intermolecular forces and the mo- lecular body volume. The resulting relation was

Pv 0 = RT . (1) The subscript 0 implies that the gas is ideal at the given conditions. If the measured gas volume

at given P and T values is identical to that calculated by using Eq. (1), the gas is said to be an ideal gas. However, if the measured specific volume at the same pressure and temperature dif- fers from that determined using Eq. (1), the gas is considered to be a real gas. The simplest way to present the real gas equation of state is by introducing a correction to the specific vol- ume by defining the compressibility factor Z, i.e.,

(2) The actual specific volume v (T,P) can be determined from experiments while theo-

Z(T,P) = v (T,P)/ v 0 (T,P).

retical volume can be determined using ideal gas law. From Eqs. (1) and (2) we obtain the re- lation

PvTP (,) = Z T P RT (,) , (3) which is called the real gas or imperfect equation of state. If Z = 1, Eq. (3) reduces to the ideal

gas equation of state. Equation (3) can be represented using reduced properties. Applying Eq. (2) at the critical point

Z(T c ,P c )= v (T c ,P c )/ v 0 (T c ,P c ),

(4) where T c and P c , respectively, denote the critical temperature and pressure. Table A-1 tabulates

these values for many substances. Applying Eq. (3) at the critical point, we obtain the follow- ing relation:

Pv c c =(,) Z T P RT c c c .

Figure 1: Experimental data for Z vs. P R with T R as a parameter for different gases (from G.J. Su, “Modified Law of Corresponding States,” Ind. Eng. Chem., 38, 803, 1946. With permis- sion.).

From Eqs. (3) and (5), we can express the compressibility factor

Z(T R ,P R )=P R v (T R ,P R ) Z(T c ,P c )/T R = f(P R ,T R )Z c ,

(6) where P R denotes the reduced pressure P/P c , and T R the reduced temperature T/T c . According

to Van Der Waals equation of state (later sections), Z c = 3/8 and is same for all substances. Then it is apparent from Eq. (6) that Z is only a function of T R ,v R. Fig. B.2a shows the com-

pressibility chart. In general, values of Z c lie in the range from 0.2–0.3.

Figure 1 contains experimental data for Z vs. P R with T R as a parameter for different gases. Compressibility charts (Chart B.2a) to determine the value of Z can be used at the ap- propriate reduced pressures and temperatures in order to ascertain whether a gas is real or ideal under specified conditions. Experiments can also be conducted to determine which equation of state the gas observes and to measure the compressibility factor. It is also possible to obtain an approximate criterion for real or ideal gas behavior using the intermolecular force potential

diagram presented in Chapter 1. When l » 3l 0 , the gas molecules move randomly in the absence of intermolecular attractive forces. If the specific volume of a solid v s or (liquid v f ) are known, the molecular number density is n´ = N Avog /v s (or = N Avog /v f ), and l ≈n´ –1/3 .