Page 373 - Physical Chemistry

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Chapter 12 We now specialize to ideally dilute solutions. Here x is very small, and terms in
B
2
Multicomponent Phase Equilibrium x and higher powers in (12.10) are negligible compared with the x term. (If x
B
B
B
4
2
2
10 , then x 10 .) Thus
B
ln g x x ideally dil. soln. (12.11)
A A
B
For a very dilute solution, the freezing-point change T T* will be very small
f
f
and T will vary only slightly in the integral in (12.7). The quantity H m,A (T) will
fus
thus vary only slightly, and we can approximate it as constant and equal to H m,A at
fus
T* . Substituting (12.11) into (12.7), taking H m,A /R outside the integral, and using
f
fus
2
(1/T ) dT 1/T, we get for (12.7)
¢ H m,A 1T*2 1 1 ¢ H m,A T T*
f
f
f
fus
fus
x a b a b (12.12)
B
R T* T f R T*T
f
f f
The quantity T T* is the freezing-point depression T :
f
f
f
¢T T T* (12.13)
f
f
f
2
Since T is close to T*, the quantity T*T in (12.12) can be replaced with (T*) with
f
f
f
f
f
negligible error for ideally dilute solutions (Prob. 12.10); Eq. (12.12) becomes
2
¢T x R1T*2 >¢ H m,A (12.14)
fus
f
B
f
We have x n /(n n ) n /n , since n V n . The solute molality is m
B
A
A
A
B
B
B
B
B
n /n M , where M is the solvent molar mass. Hence for this very dilute solution, we
A
A
B
A
have x M m , and (12.14) becomes
B
B
A
M R1T*2 2
A
f
¢T m B
f
¢ H m,A
fus
¢T k m ideally dil. soln., pure A freezes out (12.15)*
B
f
f
where the solvent’s molal freezing-point-depression constant k is defined by
f
2
k M R1T *2 >¢ H m,A (12.16)
f
fus
A
f
Note from the derivation of (12.15) that its validity does not require the solute to be
nonvolatile.
For water, H at 0°C is 6007 J/mol and
fus m
1
1
3
118.015 10 kg>mol218.3145 J mol K 21273.15 K2 2
k 1.860 K kg>mol
f
6007 J mol 1
Some other k values in K kg/mol are benzene, 5.1; acetic acid, 3.8; camphor, 40.
f
An application of freezing-point-depression data is to find molecular weights of
nonelectrolytes. To find the molecular weight of B, one measures T for a dilute so-
f
lution of B in solvent A and calculates the B molality m from (12.15). Use of m
B
B
n /w [Eq. (9.3)], where w is the solvent mass, then gives n , the number of moles
B
B
A
A
of B in solution. The molar mass M is then found from M w /n [Eq. (1.4)], where
B
B
B
B
w is the known mass of B in solution. Since (12.15) applies only in an ideally dilute
B
solution, an accurate determination of molecular weight requires that T be found
f
for a few molalities. One then plots the calculated M values versus m and extrapo-
B
B
lates to m 0. Practical applications of freezing-point depression include the use
B
of salt to melt ice and snow and the addition of antifreeze (ethylene glycol,
HOCH CH OH) to the water in automobile radiators.
2
2
Some organisms that live in below-0°C environments use freezing-point depres-
sion to prevent their body fluids from freezing. Freezing-point-depressing solutes syn-
thesized by organisms in response to cold include glycerol [HOCH CH(OH)CH OH],
2
2