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(c) the Redlich–Kwong equation; (d) the virial equation, given properties of the gas: T , P , and v. For propane v 0.153. (a)
c
c
6
3
2
6
7
that for ethane B 179 cm /mol and C 10400 cm /mol at Show that a(T ) 1.08 10 atm cm mol 2 for propane at
2
3
6
2
30°C, and B 157 cm /mol and C 9650 cm /mol at 50°C. 25°C. (b) Use the Soave–Redlich–Kwong equation to find the
vapor pressure and saturated liquid and vapor molar volumes of
8.8 For a mixture of 0.0786 mol of C H and 0.1214 mol of
2 4 propane at 25°C. The Redlich–Kwong spreadsheet of Fig. 8.6
3
CO in a 700.0-cm container at 40°C, calculate the pressure
2 can be used if the T 1/2 factors in the denominators of all for-
using (a) the ideal-gas equation; (b) the van der Waals equation,
data in Table 8.1, and the C H critical data T 282.4 K, P mulas are deleted.
2 4 c c
49.7 atm; (c) the experimental compression factor Z 0.9689. 8.16 The Peng–Robinson equation is
2
8.9 Show that if all terms after C/V are omitted from the vir- RT a1T2
m
1
ial equation (8.4), this equation predicts Z . P V m b V m 1V m b2 b1V m b2
3
c
8.10 (a) Calculate the van der Waals a and b of Ar from data
where
in Table 8.1. (b) Use Eq. (8.9) to calculate the van der Waals
second virial coefficient B for Ar at 100, 200, 300, 500, and b 0.07780RT c >P c
2
2
1000 K and compare with the experimental values in Sec. 8.2. a1T2 0.457241R T c >P c 251 k31 1T>T c 2 1>2 46 2
8.11 Problem 4.22 gives U m,intermol a/V for a fluid that k 0.37464 1.54226v 0.26992v 2
m
obeys the van der Waals equation. Taking U m,intermol 0 for the
gas phase, we can use a/V m,nbp,liq to estimate U of vaporiza- where v is defined in Prob. 8.15. (a) Use data in Prob. 8.15 to 6
m
7
3
tion at the normal boiling point (nbp). The temperature and show that for propane at 25°C, a(T ) 1.13 10 atm cm
2
liquid density at the normal boiling point are 77.4 K and mol . (b) Use the Peng–Robinson equation to predict the
3
3
0.805 g/cm for N and 188.1 K and 1.193 g/cm for HCl. Use vapor pressure and saturated liquid and vapor molar volumes of
2
the van der Waals constants listed in Sec. 8.4 to estimate propane at 25°C. You will need the integral 2 1>2
tal values 1.33 kcal/mol for N , 3.86 kcal/mol for HCl, and 2 1 dx 2 1 1>2 ln 2x s 1s 4c2 1>2
H

of N , HCl, and H O. Compare with the experimen-
2
m,nbp
2
vap
2
9.7 kcal/mol for H O. 2 x sx c 1s 4c2 2x s 1s 4c2
2
8.17 To calculate H from a cubic equation of state, we
m
vap
Section 8.5 integrate ( U / V ) T( P/ T ) P [Eq. (4.47)] along
m
m T
8.12 Use the spreadsheet of Fig. 8.6 to find the Redlich– JKLMN in Fig. 8.5 to get V m
Kwong estimates of the vapor pressure and saturated liquid and
v
vapor molar volumes of propane at 20°C. ¢ vap U m U m U m l
8.13 Use a spreadsheet and Table 8.1 data to find the V v m 0P eos
Redlich–Kwong estimates of the vapor pressure and saturated c T a b P eos d dV m const. T
l 0T
liquid and vapor molar volumes of CO at 0°C. The experi- V m V m
2
3
3
mental values are 34.4 atm, 47.4 cm /mol, and 452 cm /mol.
where P eos and ( P / T ) V m are found from the equation of
eos
8.14 Use the van der Waals equation to estimate the vapor state. Then we use
pressure and saturated liquid and vapor molar volumes of ¢ vap H m ¢ vap U m P1V m V m 2
l
v
propane at 25°C.
where the vapor pressure P and the saturated molar volumes are
8.15 The Soave–Redlich–Kwong equation is
found from the equation of state, as in Sec. 8.5. (a) Show that
RT a1T2 the Redlich–Kwong equation gives
P
V m b V m 1V m b2 3a V m 1V m b2
v
l
v
l
¢ vap H m ln P1V m V m 2
where b 0.08664RT /P (as in the Redlich–Kwong equation) 1>2 l v
c c 2bT V m 1V m b2
and a(T) is the following function of temperature:
(b) Calculate H for propane at 25°C using the Redlich–
m
vap
2
2
a1T2 0.427481R T c >P c 251 m31 1T>T c 2 0.5 46 2 Kwong equation and the results of Example 8.1.
m 0.480 1.574v 0.176v 2 8.18 For diethyl ether, P 35.9 atm and T 466.7 K. The
c
c
The quantity v is the acentric factor of the gas, defined as lowest observed negative pressure that liquid diethyl ether can
be subjected to at 403 K is 14 atm. Use a spreadsheet to plot
the 403 K Redlich–Kwong isotherm; find the pressure mini-
v 1 log 10 1P vp >P c 2 0 T>T c 0.7
where P is the vapor pressure of the liquid at T 0.7T . The mum (point K in Fig. 8.5) for the superheated liquid and com-
vp c
acentric factor is close to zero for gases with approximately pare with 14 atm.
spherical molecules of low polarity. A tabulation of v values is
given in Appendix A of Poling, Prausnitz, and O’Connell. The Section 8.7
Soave–Redlich–Kwong equation has two parameters a and b, 8.19 Verify the reduced van der Waals equation (8.29) by sub-
but evaluation of these parameters requires knowing three stituting (8.18) for a and b and (8.19) for R in (8.2).