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l
solution at 35°C with x eth 0.200 has a vapor pressure of 10.14 Assume that acetone and chloroform form a simple
304.2 torr and a vapor composition of x v eth 0.138. (a) Calculate solution (Prob. 10.13) at 35.2°C. Use the value of I 1CHCl 3 2 at
g and a for chloroform and for ethanol in this solution. (b) Cal- x1CHCl 3 2 0 in Fig. 10.3 to evaluate W. Then calculate g for
1
I
I
culate m m* for each component of this solution. (c) Calculate each component at x(CHCl ) 0.494 and compare with the
i
i
3
G for the mixing of 0.200 mol of liquid ethanol and 0.800 mol experimental values.
of liquid chloroform at 35°C. (d) Calculate mix G for the cor-
responding ideal solution. 10.15 Accurate modeling of activity coefficients in mixtures
requires an equation with more than the one parameter used in
10.8 For solutions of water (w) and H O (hp) at 333.15 K, some
2 2 the simple-solution model of Prob. 10.13. A widely used repre-
liquid and vapor compositions and (total) vapor pressures are sentation of the excess Gibbs energy G of a solution of liquids
E
x l 0.100 0.300 0.500 0.700 0.900 B and C is the Redlich–Kister equation
w
E
E
x w v 0.301 0.696 0.888 0.967 0.995 G m G >1n B n C 2
P/kPa 3.00 5.03 8.33 12.86 17.77 x B x C RT3A 1 A 2 1x B x C 2
2
The pure-liquid 333.15 K vapor pressures are P* 19.92 kPa A 3 1x B x C 2 p A n 1x B x C 2 n 1 4
w
l
and P* 2.35 kPa. For a solution at 333.15 K with x 0.300, where the A coefficients are functions of T and P, and are found
w
hp
calculate (a) g and a of water and of H O ; (b) a and g of by fitting G data or activity-coefficient data. Usually, three
E
2
II
II
2
I
I
water and of H O if the solvent is taken as water; (c) mix G to terms in brackets are enough to give an accurate fit to the data.
2
2
form 125 g of this solution from the pure components at con- Note that the Redlich–Kister equation is not symmetric in B
stant T and P.
and C, so the values of the A coefficients depend on which liq-
10.9 Activity coefficients of Zn in solutions of Zn in Hg at uid we call B. The activity coefficients are found from G m E by
25°C were determined by measurements on electrochemical using the last equation in Prob. 10.5. (a) Verify that if only one
cells. Hg is taken as the solvent. The data are fitted by g II,Zn term in brackets is used, the Redlich–Kister equation becomes
1 3.92x Zn for solutions up to saturation. (a) Show that the simple-solution model of Prob. 10.13. (b) If the designa-
tions of B and C are interchanged, how are the values of the A
1
ln g Hg 12.922 33.92 ln 11 x Zn 2 ln 11 3.92x Zn 24
coefficients changed? (c) Use the equation of Prob. 10.5c to
for this composition range. Use a table of integrals. (b) Calcu- show that if three terms in brackets are used, the Redlich–Kister
late g , a , g , and a for x 0.0400. equation gives
II,Zn II,Zn Hg Hg Zn
10.10 Use activity-coefficient data in Sec. 10.3 to calculate 2 3 4
ln I,B 1A 1 3A 2 5A 3 2x C 14A 2 16A 3 2x C 12A 3 x C
E
G /n versus composition for acetone–chloroform solutions at
2 3 4
35.2°C, where n is the total number of moles. Compare the ln I,C 1A 1 3A 2 5A 3 2x B 14A 2 16A 3 2x B 12A 3 x B
results with Fig. 10.3b. 10.16 Special techniques exist that allow quick and accurate
q
q
10.11 (a) Show that g g /g . (b) Use the result for (a) measurement of the infinite-dilution activity coefficients I,B
II,i I,i I,i
q
q
and Fig. 10.3a to find the relation between g and g for chlo- and I,C (where I,B lim x B S0 I,B 2 for a solution of liquids B
II
I
roform in the solvent acetone at 35.2°C. and C. These two activity coefficients can then be used to find
the values A and A in the two-parameter version of the
10.12 (a) A certain aqueous solution of sucrose at 25°C has a 1 2
Redlich–Kister equation (Prob. 10.15), thereby allowing activ-
vapor pressure of 23.34 torr. The vapor pressure of water at
ity coefficients to be estimated over the full composition range.
25°C is 23.76 torr. Find the activity of the solvent water in this (a) Use the equations of Prob. 10.15c to show that
sucrose solution. (b) A 2.00 mol/kg aqueous sucrose solution
1
1
q
q
has a vapor pressure of 22.75 torr at 25°C. Find the activity and A 1 1ln I,B ln I,C 2, A 2 1ln I,C ln I,B 2
q
q
2
2
activity coefficient of the solvent water in this solution.
q 1.12 9 ,
4
4
(b) For solutions of CCl and SiCl at 50°C, I,CCl 4
10.13 For two liquids whose molecules have similar sizes and q 1.14 0 , and the two-parameter Redlich–Kister equation
I,SiCl 4
shapes, an approximation that sometimes works is to assume fits well. Calculate A and A for this mixture and predict the
1
2
the liquids form a simple solution. A simple solution of B and 0.4.
activity coefficients at x SiCl 4
C is one where the activity coefficients depend on composition
according to 10.17 (a) Some activity coefficients in solutions of 1-chloro-
butane (chl) and heptane (hep) at 323.20 K are
2 2
RT ln I,B Wx C , RT ln I,C Wx B
x chl 0.100 0.300 0.500 0.700 0.900
where W is a function of T and P, but not the mole fractions.
(a) Use the Gibbs–Duhem equation to show that if a two- g I,chl 1.340 1.169 1.081 1.032 1.004
2
component solution has RT ln I,B W1T, P2x C , then it must g I,hep 1.005 1.039 1.093 1.173 1.311
2
have RT ln I,C Wx B . (b) Show that for a simple solution of
B C, Calculate G for these five solutions. (b) Use a spreadsheet
E
m
E
G n B n C W1T, P2 Solver to fit the parameters in the three-parameter
E
n B n C Redlich–Kister equation (Prob. 10.15) to this G data. Take B
m