Page 310 - Physical Chemistry

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9.36 Find mix G, mix V, mix S, and mix H for mixing 100.0 g 9.46 Let phases a and b, each composed of liquids 1 and 2,
of benzene with 100.0 g of toluene at 20°C and 1 atm. Assume be in equilibrium with each other. Show that if substances 1 and
a
b
b
a
an ideal solution. 2 form ideal solutions, then x x and x x . Therefore, the
2
1
1
2
two phases have the same composition and are actually one
9.37 Benzene (C H ) and toluene (C H CH ) form nearly phase. Hence liquids that form ideal solutions are miscible in
6
6
3
5
6
ideal solutions. At 20°C the vapor pressure of benzene is 74.7 all proportions.
torr, and that of toluene is 22.3 torr. (a) Find the equilibrium par-
tial vapor pressures above a 20°C solution of 100.0 g of benzene Section 9.7
plus 100.0 g of toluene. (b) Find the mole fractions in the vapor 9.47 Consider the constant-T-and-P dilution process of
phase that is in equilibrium with the solution of part (a).
adding n A,2 n A,1 moles of solvent A to an ideally dilute solu-
9.38 At 100°C the vapor pressures of hexane and octane are tion (solution 1) that contains n moles of solute i and n A,1 moles
i
1836 and 354 torr, respectively. A certain liquid mixture of of A to give an ideally dilute solution of n moles of i and n A,2
i
these two compounds has a vapor pressure of 666 torr at 100°C. moles of A. Experimental vapor-pressure data for highly dilute
Find the mole fractions in the liquid mixture and in the vapor solutions show that G for this process is given by
phase. Assume an ideal solution.
¢G n i RT 1ln x i,2 ln x i,1 2
9.39 A solution of hexane and heptane at 30°C with hexane RT 1n A,2 ln x A,2 n A,1 ln x A,1 2 (9.66)
mole fraction 0.305 has a vapor pressure of 95.0 torr and a
vapor-phase hexane mole fraction of 0.555. Find the vapor where x , x , x , and x A,1 are the final and initial mole frac-
i,1
i,2
A,2
pressures of pure hexane and heptane at 30°C. State any ap- tions of the solute and the solvent in the solution. (a) Use
proximations made. Eq. (9.23) to show that for this process
9.40 (a) Use Raoult’s law to show that for an ideal solution of ¢G n i 1m i,2 m i,1 2 n A,2 m A,2 n A,1 m A,1
B and C, the B mole fraction in the vapor phase in equilibrium 1n A,2 n A,1 2m* (9.67)
with the solution is A
l
x B P* >P* where m , m , m , and m A,1 are the final and initial chemical
i,1
i,2
A,2
B
v
x B C potentials of the solute and the solvent in the solution.
l
*
1 x B 1P B >P* 12 Comparison of the coefficient of n in (9.67) with that in (9.66)
C
i
gives
(b) At 20°C the vapor pressure of benzene (C H ) is 74.7 torr
6
6
and that of toluene (C H CH ) is 22.3 torr. For solutions of ben- m i,2 m i,1 RT 1ln x i,2 ln x i,1 2 const. T, P (9.68)
3
5
6
zene plus toluene (assumed ideal) in equilibrium with vapor at
v
l
20°C, plot x versus x for benzene. Repeat for toluene. The only way (9.68) can hold is if
B
B
9.41 At 20°C and 1 atm, the density of benzene is 0.8790 m i RT ln x i f i 1T, P2 (9.69)
3
3
g/cm and that of toluene is 0.8668 g/cm . Find the density of a where f (T, P) is some function of T and P, which cancels in
i
solution of 33.33 g of benzene and 33.33 g of toluene at 20°C m m at constant T and P. (b) Use Eqs. (9.66) to (9.68) to
and 1 atm. Assume an ideal solution. i,2 i,1
show that
9.42 (a) Show that mix P n A,2 m A,2 n A,1 m A,1 n A,2 1m* RT ln x A,2 2
C 0 for an ideal solution. (b) At
25°C and 1 atm, C P,m 136 J/(mol K) for benzene (C H ) and A
6
6
C P,m 156 J/(mol K) for toluene (C H CH ). Find C of a so- n A,1 1m* RT ln x A,1 2
A
6
P
5
3
lution of 100.0 g of benzene and 100.0 g of toluene at 25°C and const. T, P (9.70)
1 atm. Assume an ideal solution.
The only way (9.70) can hold is if
9.43 Draw tangents to the mix G/n curve of Fig. 9.16 to find
m m* and m m* at x 0.50 and at x 0.25. Compare m A m* RT ln x A (9.71)
A
B
B
B
A
B
A
your results with those calculated from m m* RT ln x .
i
i
i
9.44 From Eq. (9.42) for the chemical potentials, derive the Section 9.8
following equations for partial molar properties of a component 9.48 A solution of ethanol (eth) and chloroform (chl) at 45°C
of an ideal solution: with x 0.9900 has a vapor pressure of 177.95 torr. At this
eth
high dilution of chloroform, the solution can be assumed to be
S i S* m,i R ln x i , V i V* m,i , H i H* m,i
essentially ideally dilute. The vapor pressure of pure ethanol at
These results are consistent with mix V 0 and mix H 0.
45°C is 172.76 torr. (a) Find the partial pressures of the gases
9.45 Consider an ideal gas mixture at T and P; show that for in equilibrium with the solution. (b) Find the mole fractions in
component i, m m*(T, P) RT ln x . Therefore an ideal gas the vapor phase. (c) Find the Henry’s law constant for chloro-
i
i
i
mixture is an ideal solution. Of course, an ideal solution is not form in ethanol at 45°C. (d) Predict the vapor pressure and
necessarily an ideal gas mixture. Note also the different choice vapor-phase mole fractions at 45°C for a chloroform–ethanol
of standard state for an ideal-solution component and an ideal- solution with x eth 0.9800. Compare with the experimental
v
gas-mixture component. values P 183.38 torr and x eth 0.9242.