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9.23 Use Fig. 9.8 to find the partial molar volumes at ethanol through the point (x , z ), we have z mx b
B
B

mole fraction 0.400. (V* z ) b. Therefore, b V A V* , which is what
V
m,A
A
m,A
we wanted to prove. (d)Verify that the tangent line’s intercept at
9.24 The densities of H O and CH OH at 25°C and 1 atm are
2 3 x
V
B
m,B
3
0.99705 and 0.78706 g/cm , respectively. For solutions of these B 1gives V* .
two compounds at 25°C and 1 atm, V/n-vs.-x data are: 9.27 Use Fig. 9.13 to find the differential heats of solution of
mix H 2 O
H SO and of H O at x 0.4 and at x 0.333.
3
( mix V/n)/(cm /mol) 0.34 0.60 0.80 2 4 2 H 2 SO 4 H 2 SO 4
9.28 Prove that in a two-component solution, H
x 0.1 0.2 0.3 diff,B
H 2 O .
3
( mix V/n)/(cm /mol) 0.94 5 1.01 0.98 10¢ mix H>0n B 2 T,P,n A
9.29 The NBS tables (Sec. 5.9) give at 25°C H° NaCl(s)
f
x 0.4 0.5 0.6 411.153 kJ/mol and give the following apparent H° data in
H 2 O f
kJ/mol for aqueous NaCl solutions at 25°C:
3
( mix V/n)/(cm /mol) 0.85 0.61 5 0.31
n /n 9 15 25 50
x 0.7 0.8 0.9 H 2 O NaCl
H 2 O
H° 409.279 408.806 408.137 407.442
Use the intercept method (Fig. 9.8) to find the partial molar vol- f NaCl(aq)
umes at x H 2 O values of (a) 0; (b) 0.4; (c) 0.6. The value 409.279 kJ/mol as the apparent enthalpy of forma-
9.25 Let V be the volume of an aqueous solution of NaCl at tion of 1 mol of NaCl in 9 mol of H O means that when 1 mol
2
25°C and 1 atm that contains 1000 g of water and n moles of of NaCl(s) is dissolved in 9 mol of H O(l) at 25°C and 1 bar,
2
B
NaCl. One finds that the following empirical formula accu- the enthalpy change is
rately reproduces the experimental data: 409.279 kJ/mol ( 411.153 kJ/mol) 1.874 kJ/mol
3>2
2
V a bn B cn B kn B for n A M A 1 kg Calculate and plot H/n versus x NaCl and use the intercept
mix
3
3
a 1002.96 cm , b 16.6253 cm >mol method to find H NaCl H* m,NaCl and H H 2 O H* m,H 2 O at x NaCl
0.05, where H* is for solid NaCl.
m,NaCl
3>2
3
3
c 1.7738 cm >mol , k 0.1194 cm >mol 2
9.30 Look up apparent H° data (see Prob. 9.29 for an ex-
f
(a) Show that the NaCl partial molar volume V B is planation) for HCl solutions in the NBS thermodynamics tables
and find H for HCl and for H O at x 0.30, T 298 K,
V B b 13c>22n 1>2 diff 2 HCl
B 2kn B for n A M A 1 kg
and P 1 bar.
(b) Find V NaCl for a solution with NaCl molality m B
1.0000 mol/kg. (c) Use (9.16) to show that the partial molar 9.31 Show that for a two-component solution of B and C,
volume of the water in the solution is
mix H n B H diff,B n C H diff,C

2
1
V A 1M A >1000 g21a cn 3>2 kn B 2 for n A M A 1 kg
2
B
(d) Show that the results for (a) and (c) can be written as Section 9.5
9.32 True or false? (a) Intermolecular interactions are negli-
V B b 13c>221m B kg2 1>2 2km B kg gible in an ideal solution. (b) If B is a component of an ideal so-
1 lution, m cannot be greater than m*. (c) If B is a component of
2
2
3>2
V A 1M A >1000 g21a cm kg 3>2 km B kg 2 B B
2
B
B
B
a solution, m cannot be greater than m*. (d) A solution of
Since V A , V B , and m are all intensive quantities, we need not water plus ethanol is nearly ideal.
B

specify n in these equations. (e) Find V H 2 O for a solution with 9.33 Would a liquid mixture of the two optical isomers of

A
m 1.0000 mol/kg. (f ) Find V q . CHFClBr be an ideal solution? Explain.
B
NaCl
9.26 Prove the validity of the intercept method (Fig. 9.7) of
determining partial molar volumes in a two-component solution Section 9.6
as follows. (All the equations of this problem are for fixed T and
P.) (a) Let z mix V/n, where n n n .Verify that V 9.34 True or false? (a) At constant T and P, mix G must be
B
A
(n n )z n V* m,A n B V* m,B .(b)Take( / n ) of the negative for an ideal solution. (b) At constant T and P, mix G
A
B
A
A n B must be negative for every solution. (c) At constant T and P,
m,A
equation in (a)to show that V A n10z>0n A 2 n B z V* . S H/T for an ideal solution. (d) For equilibrium be-
[Eq. (1.35)] and the mix mix
B
(c) Use 10z>0n A 2 n B (dz/dx ) 10x B >0n A 2 n B tween a solution and its vapor, m of the solution must equal m
result for (b)to show that dz/dx (V* m,A V A z)/x . Also,
B
B
. of the vapor. (e) For equilibrium between an ideal solution and
B
v
l
explain why the n subscript can be omitted from 10z>0x B 2 n B an ideal vapor, x must equal x . ( f) In an ideal solution, the
Let y mx b be the equation of the tangent line to the z- B B
B
versus-x B curve at the point with x x and z z , and let partial molar volume of a component equals the molar volume
B B of the pure substance. (See Prob. 9.44.)
V and V be the partial molar volumes at x . Recall that,
B
A
B
for the straight line y mx b, the slope is m and the inter- 9.35 State the two approximations that are made when
B
cept at x 0is b. The slope m is given by the result for (c)as Raoult’s law is derived from the ideal-solution chemical

B
m (V* m,A V z )/x . Also, since the tangent line passes potentials.
B
A