Page 265 - Physical Chemistry

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Chapter 8 If P is not high, terms beyond C/V or C P in (8.4) and (8.5) are usually negli-
m
Real Gases
gible and can be omitted. At high pressures, the higher terms become important. At
very high pressures, the virial equation fails. For gases at pressures up to a few atmo-
spheres, one can drop terms after the second term in (8.4) and (8.5), provided T is not
very low; Eq. (8.5) becomes
V RT>P B low P (8.7)
m
where (8.6) was used. Equation (8.7) gives a convenient and accurate way to correct
for gas nonideality at low P. Equation (8.7) shows that at low P, the second virial
coefficient B(T) is the correction to the ideal-gas molar volume RT/P. For example,
for Ar(g) at 250.00 K and 1.0000 atm, the truncated virial equation (8.7) and the pre-
ceding table of Ar B values gives V RT/P B 20515 cm /mol 28 cm /mol
3
3
m
3
20487 cm /mol.
Multiplication of the van der Waals equation (8.2) by V /RT gives the compres-
m
sion factor Z PV /RT of a van der Waals (vdW) gas as
m
PV m V m a 1 a
Z vdW gas
RT V b RTV m 1 b>V m RTV m
m
where the numerator and denominator of the first fraction were divided by V . Since
m
1/(1 b/V ) is greater than 1, intermolecular repulsions (represented by b) tend to
m
make Z greater than 1 and P greater than P . Since a/RTV is negative, intermolec-
id m
ular attractions (represented by a) tend to decrease Z and make P less than P .
id
b is approximately the liquid’s molar volume, so we will have b V for the gas
m
and b/V 1. We can therefore use the following expansion for 1/(1 b/V ):
m m
1
2
3
1 x x x p for 0x0 6 1 (8.8)
1 x
You may recall (8.8) from your study of geometric series. Equation (8.8) can also be
derived as a Taylor series (Sec. 8.9). The use of (8.8) with x b/V gives
m
PV m a 1 b 2 b 3
Z 1 ab b p vdW gas (8.9)
RT RT V m V 2 m V 3 m
The van der Waals equation now has the same form as the virial equation (8.4). The
van der Waals prediction for the second virial coefficient is B(T) b a/RT.
3
3
2
2
At low pressures, V is much larger than b and the b /V , b /V , ... terms in (8.9)
m m m
can be neglected to give Z 1 (b a/RT)/V .Atlow T (and low P), we have a/RT
m
b, so b a/RT is negative, Z is less than 1, and P is less than P (as in the low-P parts
id
of the 200-K and 500-K CH curves in Fig. 8.1b). At low T, intermolecular attractions
4
(van der Waals a) are more important than intermolecular repulsions (van der Waals
b) in determining P. At high T (and low P), we have b a/RT 0, Z 1, and P
P (as in the 1000-K curve in Fig. 8.1b). At high T, the molecules smash into each
id
other harder than at low T, which increases the influence of repulsions on P.
A comparison of equations of state for gases [K. K. Shah and G. Thodos, Ind.
Eng. Chem., 57(3), 30 (1965)] concluded that the Redlich–Kwong equation is the best
two-parameter equation of state. Because of its simplicity and accuracy, the
Redlich–Kwong equation has been widely used, but has now been largely superseded
by more accurate equations of state (Sec. 8.4).
Gas Mixtures
So far we have considered pure real gases. For a real gas mixture, V depends on the
mole fractions, as well as on T and P. One approach to the P-V-T behavior of real gas
mixtures is to use a two-parameter equation of state like the van der Waals or