Page 264 - Physical Chemistry

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A real gas then obeys PV ZnRT. Numerical tables of Z(P, T) are available for Section 8.2
many gases. Real-Gas Equations of State

8.2 REAL-GAS EQUATIONS OF STATE
An algebraic formula for the equation of state of a real gas is more convenient to use
than numerical tables of Z. The best-known such equation is the van der Waals
equation
a RT a
a P b1V b2 RT or P (8.2)
m
2
V m V b V 2 m
m
where the first equation was divided by V b to solve for P. In addition to the gas
m
constant R, the van der Waals equation contains two other constants, a and b, whose
values differ for different gases. A method for determining a and b values is given in
2
Sec. 8.4. The term a/V in (8.2) is meant to correct for the effect of intermolecular at-
m
tractive forces on the gas pressure. This term decreases as V and the average inter-
m
molecular distance increase. The nonzero volume of the molecules themselves makes
the volume available for the molecules to move in less than V, so some volume b is
subtracted from V . The volume b is roughly the same as the molar volume of the
m
solid or liquid, where the molecules are close together; b is roughly the volume ex-
cluded by intermolecular repulsive forces. The van der Waals equation is a major
improvement on the ideal-gas equation but is unsatisfactory at very high pressures and
its overall accuracy is mediocre.
A quite accurate two-parameter equation of state for gases is the Redlich–Kwong
equation [O. Redlich and J. N. S. Kwong, Chem. Rev., 44, 233 (1949)]:
RT a
P (8.3)
V b V 1V b2T 1>2
m
m
m
which is useful over very wide ranges of T and P. The Redlich–Kwong parameters a
and b differ in value for any given gas from the van der Waals a and b.
Statistical mechanics shows (see Sec. 21.11) that the equation of state of a real gas
not at very high pressure can be expressed as the following power series in 1/V :
m
B1T2 C1T2 D1T2
PV RT c 1 p d (8.4)
m
2
V m V m V 3 m
This is the virial equation of state. The coefficients B, C, . . . , which are functions
of T only, are the second, third, . . . virial coefficients. They are found from experi-
mental P-V-T data of gases (Probs. 8.38 and 10.64). Usually, the limited accuracy of
the data allows evaluation of only B(T) and sometimes C(T). Figure 8.2 plots the typ-
ical behavior of B and C versus T. Some values of B(T) for Ar are
3
B/(cm /mol) 251 184 86 47 28 16 1 7 12 22
T/K 85 100 150 200 250 300 400 500 600 1000
Statistical mechanics gives equations relating the virial coefficients to the potential
energy of intermolecular forces.
A form of the virial equation equivalent to (8.4) uses a power series in P:
3
†
2
†
PV RT 31 B 1T2P C 1T2P D 1T2P p 4 (8.5)
†
m
Figure 8.2
†
†
The relations between the coefficients B , C , . . . and B, C, . . . in (8.4) are worked out
in Prob. 8.4. One finds Typical temperature variation of
the second and third virial
2
†
†2
†
2
B B RT, C 1B C 2R T (8.6) coefficients B(T) and C(T).