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8.37 (a) Use the virial equation (8.5) to show that RT/(P a/V ). To obtain an initial estimate V m0 of V ,we
2
m
m
2
m
RT 2 dB † dC † dD † neglect a/V to get V m0 b RT/P.An improved estimate is
2
m JT a P P p b V m1 b RT/(P a/V 2 m0 ). From V ,weget V , etc. Use suc-
m1
m2
C P,m dT dT dT
cessive approximations to find the van der Waals V for CH at
m
4
†
2
lim m JT 1RT >C P,m 21dB >dT2 0 273 K and 100 atm, given that T 190.6 K and P 45.4 atm
c
c
PS0 for CH . (The calculation is more fun if done on a programma-
4
Thus, even though the Joule–Thomson coefficient of an ideal ble calculator.) Compare with the V in Fig. 8.1.
m
gas is zero, the Joule–Thomson coefficient of a real gas does 8.42 Use Fig. 8.10 to find V for CH at 286 K and 91 atm.
4
m
not become zero in the limit of zero pressure. (b) Use (8.4) to See Prob. 8.41 for data.
show that, for a real gas, ( U/ V) → 0 as P → 0.
T
8.43 In Prob. 7.33, the Antoine equation was used to find
8.38 Use the virial equation (8.4) to show that for a real gas H of H O at 100°C. The result was inaccurate due to ne-
m
2
vap
id
lim 1V m V m 2 B1T2 glect of gas nonideality. We now obtain an accurate result. For
3
PS0 H O at 100°C, the second virial coefficient is 452 cm /mol.
2
(a) Use the Antoine equation and Prob. 7.33 data to find dP/dT
8.39 At low P, all terms but the first in the m series in Prob.
JT
8.37 can be omitted. (a) Show that the van der Waals equation for H O at 100°C, where P is the vapor pressure. (b) Use the
2
m
m
(8.9) predicts m (2a/RT b)/C at low P. (b) At low tem- Clapeyron equation dP/dT H /(T V ) to find vap H of
m
JT P,m
2
peratures, the attractive term 2a/RT is greater than the repulsive H O at 100°C; calculate V using the truncated virial equa-
m
term b and the low-P m is positive. At high temperature, b tion (8.7) and the saturated liquid’s 100°C molar volume, which
JT 3
2a/RT and m 0. The temperature at which m is zero in the is 19 cm /mol. Compare your result with the accepted value
JT JT
P → 0 limit is the low-pressure inversion temperature T . 40.66 kJ/mol.
i,P→0
For N , use data in Sec. 8.4 and the Appendix to calculate the 8.44 Some V versus P data for CH (g) at 50°C are
2 m 4
van der Waals predictions for T and for m at 298 K and
i,P→0 JT
low P. Compare with the experimental values 638 K and 0.222 P/atm 5 10 20 40 60
K/atm. (Better results can be obtained with a more accurate 3
V /(cm /mol) 3577 1745 828 365 206
equation of state—for example, the Redlich–Kwong.) m
8.40 For each of the following pairs, state which species has For the virial equation (8.4) with terms after C omitted, use a
the greater van der Waals a, which has the greater van der spreadsheet to find the B and C values that minimize the sums
Waals b, which has the greater T , and which has the greater of the squares of the deviations of the calculated pressures from
c
H at the normal boiling point. (a) He or Ne; (b) C H or the observed pressures.
m
2
vap
6
C H ; (c) H O or H S. 8.45 True or false? (a) The parameter a in the van der Waals
8
3
2
2
8.41 The van der Waals equation is a cubic in V , which makes equation has the same value for all gases. (b) The parameter a
m
it tedious to solve for V at a given T and P. One way to find in the van der Waals equation for N has the same value as a in
m 2
V is by successive approximations. We write V b the Redlich–Kwong equation for N .
m m 2