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8.20 The Berthelot equation of state for gases is process (5.13), note that V → q as P → 0. Use a modification
m
of the process (5.13) in which step (c) is replaced by two steps,
2
1P a>TV m 21V m b2 RT
a contraction from infinite molar volume to molar volume V m
3
2
(a) Show that the Berthelot parameters are a 27R T /64P c followed by a volume change from V to V , to show that
id
c
m
m
and b RT /8P . (b) What value of Z is predicted? (c) Write
c
c
c
id
the Berthelot equation in reduced form. id V m RT V m
A m 1T, P2 A m 1T, P2 a P¿ b dV¿ m RT ln
3
8.21 For C H , V 148 cm /mol. Use the reduced van der q V¿ m V m
2 6 m,c
Waals equation (8.29) to answer Prob. 8.7. Note that the result id
where the integral is at constant T and V RT/P. This formula
m
is very different from that of Prob. 8.7b.
is convenient for use with equations like the Redlich–Kwong
8.22 For gases obeying the law of corresponding states, the and van der Waals, which give P as a function of V . Formulas
m
id
id
second virial coefficient B is accurately given by the equation for S S and H H are not needed, since these differ-
m
m
m
m
id
(McGlashan, p. 203) ences are easily derived from A A using ( A / T) S m
m
m
V
m
and A m U m TS m H m PV m TS . (b) For the
m
BP c >RT c 0.597 0.462e 0.7002T c >T id
Redlich–Kwong equation, show that, at T and P, A A
m
m
id
1/2
Use this equation and Table 8.1 data to calculate B of Ar at 100, RT ln (1 b/V ) (a/bT ) ln (1 b/V ) RT ln (V /V ).
m
m
m
m
id
id
200, 300, 500, and 1000 K and compare with the experimental (c) From (b), derive expressions for S S and U U for
m
m
m
m
values in Sec. 8.2. a Redlich–Kwong gas.
8.27 Use the corresponding-states equation for B in Prob. 8.22,
Section 8.8
8.23 Use the virial equation in the form (8.5) to show that at data in Prob. 8.24, and the results of Prob. 8.23 to estimate
id
id
m
2
6
T and P H H and S S for C H at 298 K and 1 bar and com-
m
m
m
pare with the experimental values.
dB † 1 dC †
2
2
id
H m H m RT c P P p d Section 8.9
dT 2 dT
8.28 Use (8.32) to verify the Taylor series (8.8) for 1/(1 x).
dB †
†
id
S m S m R ca B T bP 8.29 Derive the Taylor series (8.36) for ln x.
dT
x
8.30 Derive the Taylor series (8.37) for e .
1 dC †
2
†
aC T bP p d 8.31 Derive the Taylor series (8.38) for cos x by differentiat-
2 dT
ing (8.35).
1
† 2
†
id
G m G m RT 3B P C P p 4 8.32 Use (8.35) to calculate the sine of 35° to four significant
2
figures. Before beginning, decide whether x in (8.35) is in de-
8.24 (a) Use the results of Prob. 8.23 and Eqs. (8.9) and (8.6) grees or in radians.
id
to show that, for a van der Waals gas at T and P, H H
m m
2
id
(2a/RT b)P

and S S (a/RT )P

. (b) For 8.33 This problem is only for those familiar with the notion
m m
C H , T 305.4 K and P 48.2 atm. Calculate the values of of the complex plane (in which the real and imaginary parts of
2 6 c c
id
id
H H and S S predicted by the van der Waals equa- a number are plotted on the horizontal and vertical axes). The
m m m m
tion for C H at 298 K and 1 bar. (At 1 bar, powers of P higher radius of convergence c in (8.34) for the Taylor series (8.32)
2 6
than the first can be neglected with negligible error.) Compare can be shown to equal the distance between point a and the sin-
with the experimental values 15 cal/mol and 0.035 cal/(mol K). gularity in the complex plane that is nearest to a (see Sokol-
nikoff and Redheffer, sec. 8.10). Find the radius of convergence
8.25 Although the overall performance of the Berthelot equa-
2
for the Taylor-series expansion 1/(x 4) about a 0.
tion (Prob. 8.20) is quite poor, it does give pretty accurate esti-
id
id
mates of H H and S S for many gases at low pres- 8.34 Use a programmable calculator or computer to calculate
m m m m
n
x
sures. Expand the Berthelot equation into virial form and use the truncated e Taylor series m x /n! for m 5, 10, and 20 and
n 0
x
the approach of Prob. 8.24a to show that the Berthelot equation (a) x 1; (b) x 10. Compare the results in each case with e .
2
id
gives at T and P: H H (3a/RT b)P

and S
id
m m m
3
S (2a/RT )P

. (b) Neglect terms after P and use the General
m
results for Prob. 8.20a to show that the Berthelot equation pre- 8.35 The normal boiling point of benzene is 80°C. The den-
dicts H H 81RT P/64T P RT P/8P and S S sity of liquid benzene at 80°C is 0.81 g/cm . Estimate P , T ,
3
3
id
id
2
m m c c c c m m c c
3
3
27RT P/32T P . (c) Use the Berthelot equation to calculate and V for benzene.
c c m,c
id
id
H H and S S for C H at 298 K and 1 bar and com-
m m m m 2 6 8.36 The vapor pressure of water at 25°C is 23.766 torr.
pare with the experimental values. See Prob. 8.24b for data. Calculate G° for the process H O(l) → H O(g); do not as-
(d) Use the Berthelot equation to calculate S id S for SO 298 2 2
m m 2 sume ideal vapor; instead use the results of Prob. 8.24a and data
(T 430.8 K, P 77.8 atm) at 298 K and 1 atm.
c c in Sec. 8.4 to correct for nonideality. Compare your answer
8.26 (a) Let V be the molar volume of a real gas at T and P with that to Prob. 7.67 and with the value found from G°
298
f
m
and let V id be the ideal-gas molar volume at T and P. In the values in the Appendix.
m