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as chl. How good is the fit? (c) Use the results of Prob. 10.15 to where v v v , m i is the electrolyte’s stoichiometric
predict g I,chl and g I,hep at x chl 0 and at x chl 0.4. molality (10.48), M is the solvent’s molar mass, a is the
A
A
solvent’s activity on the mole-fraction scale, and m A m*
10.18 Repeat Prob. 10.17b using first the two-parameter A
RT ln a was used. Show that
Redlich–Kister equation and then the four-parameter Redlich– A
Kister equation. Compare these fits with the three-parameter fit. 1 P* A
f ln
10.19 Use the acetone–chloroform g data of Example 10.1 of M A vm i P A
I
E
Sec. 10.3 to calculate G values. Then use a spreadsheet Solver where an ideal vapor is assumed. So f can be found from
m
to do two-, three-, and four-parameter Redlich–Kister fits vapor-pressure measurements.
E
(Prob. 10.15) to G and comment on the quality of the fits. Use
m
the best fit to predict g values. 10.32 (a) Use (10.107) in Prob. 10.31 and (10.51) in the
q
I
Gibbs–Duhem equation (10.55) to show that
Section 10.4 d ln g df 31f 12>m i 4 dm i const. T, P
10.20 Which of these activity coefficients must go to 1 as the (b) Use (10.107), (10.39), (8.36), and (10.48) to show that f →
solvent mole fraction x goes to 1: g , g , g , g II,A ?
1 as x → 1. (c) Show that integration of the result from (a)
m,i
A
II,i
c,i
A
10.21 For a 1.50 mol/kg 25°C sucrose solution in water, g gives
m
1.292 for the solute sucrose. For this solution, find g , a , and
II II m f1m i 2 1
a for sucrose.
m ln ; 1m2 f1m2 1 dm i const. T, P (10.108)
m i
0
10.22 Use Eq. (10.27) and the expressions in Prob. 10.4 for
Convention II standard-state solute properties to derive the fol- Values of f can be found from vapor pressures (Prob. 10.31),
lowing solute molality-scale standard-state properties: and (10.108) then allows g ; to be found. The infinity in the in-
q
q
V° m,i V i , H° m,i H i , S° m,i 1S° i R ln m i >m°2 q tegrand in (10.108) can be avoided as shown in Prob. 10.33.
10.33 The Debye–Hückel theory shows that at very high dilu-
10.23 Equate the concentration-scale expression (10.31) for 1/2
m to (10.24). Then take the limit as x → 1 to show that tions of electrolyte i, f 1 m i (McGlashan, sec. 20.4), so
A
i
m° m° RT ln V* c°, where V* is the solvent’s molar the integrand in (10.108) in Prob. 10.32 becomes infinite as
m,A
m,A
II,i
c,i
volume. Use this result to show that g c,i (x /V* c )g II,i m → 0 and the integral is hard to evaluate graphically. Show
i
i
m,A i
1/2
(r m /c )g , where r is the density of the pure solvent. Then that if we define w m , then this integral becomes
i
i
A
A
i
i
m,i
w
2
show that a c,i (r m°/c°)a . For H O at 25°C and 1 bar, 2 [(f 1)/w ] dw . Since f 1 w , there is no infinity.
i
i
0
i
2
A
m,i
3
r 0.997 kg/dm , so here a 0.997a . 10.34 For a solution of a single nonelectrolyte i, the practical
A
c,i
m,i
osmotic coefficient is defined by Eq. (10.107) of Prob. 10.31
f
Section 10.5 with set equal to 1, and Eq. (10.108) of Prob. 10.32 is valid if
n
10.24 For each of these electrolytes, give the values of n , n , g ; is replaced by m,i . A statistical-mechanical treatment of non-

z , and z : (a) KCl; (b) MgCl ; (c) MgSO ; (d) Ca (PO ) . electrolyte solutions shows that f 1 c 1 m i c 2 m i p
2

4 2
3
2
4
(e) Which of the electrolytes (a) to (d) are 1:1 electrolytes? [Pitzer (1995), p. 250], where the c’s are functions of T and P.
10.25 Write the expression for g in terms of g and g for Therefore at high dilutions, f 1 is proportional to m and the
i

each of the electrolytes in Prob. 10.24. integrand in (10.108) remains finite as m i S 0 for a nonelec-
trolyte, unlike the integrand in (10.23). Robinson and Stokes
10.26 Express m for ZnCl in terms of m and m .
i 2 used vapor-pressure data to find that practical osmotic coeffi-
10.27 Calculate n for each electrolyte in Prob. 10.24. cients in aqueous sucrose solutions at 25°C obey the relation

2
3
10.28 Show that if n n , then n n n . f 1 am>m° b1m>m°2 c1m>m°2 d1m>m°2 4

10.29 Express a for MgCl (aq) in terms of m . where a 0.07028, b 0.01847, c 0.004045, d
i
i
2
0.000228, m is the sucrose molality, and m° 1 mol/kg. (a) Use
Section 10.6 (10.108) with g ; replaced by g m,i to show that for sucrose
3 2 4 3 5 4
10.30 At 25°C, the vapor pressure of a 4.800 mol/kg solution ln g m 2am>m° b1m>m°2 c1m>m°2 d1m>m°2
3
2
4
of KCl in water is 20.02 torr. The vapor pressure of pure water (b) Calculate g and g for sucrose in a 6.00 mol/kg aqueous
m II
at 25°C is 23.76 torr. For the solvent in this KCl solution, find 25°C solution.

(a) f; (b) g and a if x is calculated using H O, K (aq), and
A
2
A
A

Cl (aq) as the solution constituents; (c) g and a if x is cal- Section 10.7
A
A
A
culated using H O and KCl(aq) as the solution constituents. 10.35 Calculate the ionic strength I in a solution that con-
2
m
10.31 The (solvent) practical osmotic coefficient for a solu- tains 0.0100 mol KCl, 0.0050 mol MgCl , 0.0020 mol MgSO ,
4
2
tion of a strong electrolyte is defined as and 100 g H O.
2
A
ln a A m* m A 10.36 For a solution of a single strong electrolyte, show that
1
f (10.107) I z z nm .
M A vm i RTM A vm i m 2 i