Page 269 - Physical Chemistry

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Chapter 8 For example, differentiating the van der Waals equation (8.2), we get
Real Gases
2
0P RT 2a 0 P 2RT 6a
a b and a b
2
3
0V m T 1V b2 2 V m 0V m T 1V b2 3 V 4 m
m
m
Application of the conditions (8.12) then gives
RT c 2a and RT c 3a
1V m,c b2 2 V 3 m,c 1V m,c b2 3 V 4 m,c (8.13)
Moreover, the van der Waals equation itself gives at the critical point
RT c a
P (8.14)
c
2
V m,c b V m,c
Division of the first equation in (8.13) by the second yields V b 2V /3, or
m,c m,c
V m,c 3b (8.15)
2
3
Use of V 3b in the first equation in (8.13) gives RT /4b 2a/27b , or
m,c c
T 8a>27Rb (8.16)
c
2
Substitution of (8.15) and (8.16) into (8.14) gives P (8a/27b)/2b a/9b , or
c
P a>27b 2 (8.17)
c
We thus have three equations [(8.15) to (8.17)] relating the three critical constants P ,
c
V , T to the two parameters to be determined, a and b. If the van der Waals equation
m,c c
were accurately obeyed in the critical region, it would not matter which two of the
three equations were used to solve for a and b. However, this is not the case, and
the values of a and b obtained depend on which two of the three critical constants are
used. It is customary to choose P and T , which are more accurately known than V .
c c m,c
Solving (8.16) and (8.17) for a and b, we get
2 2
b RT >8P , a 27R T >64P vdW gas (8.18)
c
c
c
c
Some van der Waals a and b values calculated from Eq. (8.18) and P and T data
c c
of Table 8.1 are:
Gas Ne N H O HCl CH OH n-C H
2 2 3 8 18
6
2
6
10 a/(cm atm mol ) 0.21 1.35 5.46 3.65 9.23 37.5
3
1
b/(cm mol ) 16.7 38.6 30.5 40.6 65.1 238
From (8.15), V 3b. Also, V 2.7V (Sec 8.3), where V is the liq-
m,c m,c m,nbp m,nbp
uid’s molar volume at its normal boiling point. Therefore b is roughly the same as
V (as noted in Sec. 8.2). V is a bit more than the volume of the molecules
m,nbp m,nbp
themselves. Note from the tabulated b values that the larger the molecule, the greater
the b value. Recall that the van der Waals a is related to intermolecular attractions. The
greater the intermolecular attraction, the greater the a value.
Combination of (8.15) to (8.17) shows that the van der Waals equation predicts
for the compressibility factor at the critical point
3
Z PV >RT 0.375 (8.19)
c
c
c m,c
8
This may be compared with the ideal-gas prediction P V /RT 1. Of the known Z
c m,c c c
values, 80% lie between 0.25 and 0.30, significantly less than predicted by the van der
Waals equation. The smallest known Z is 0.12 for HF; the largest is 0.46 for
c
CH NHNH .
3 2