* (s) = * (l) + RT s+ l ln xA .
(39)
A
A
As with total thermodynamic properties (eq 25), the Gibbs energies, enthalpies, and entro-
pies of fusion are related by
*
*
*
∆lsGH 2O (T , p) = ∆ls HH 2O (T , p) - T∆lsSH 2O (T , p).
(40)
From this expression, we can obtain directly
* (s) - * (l) ∆ls HH 2O (T, p) ∆lsSH 2O (T, p)
*
*
A
A
=
-
(41)
ln xA
.
RT s+ l
RT s+ l
R
Recalling that for a binary solution
xA + xB = 1
one can calculate the molar Gibbs energy of fusion for two systems: a pure solvent and a
solution.
For the pure solvent system,
*
l *
* (s) - * (l) ∆ s HH 2O (T *, p) ∆ sSH 2O (T *, p)
l
=
-
A
A
ln 1 = 0 =
(42)
RT s+ l,*
RT s+ l,*
R
and for the solution system,
* (s) - * (l) ∆ s HH 2O (T, p) ∆ sSH 2O (T *, p)
l *
l *
=
-
A
A
(43)
ln(1 xB ) =
RT s+ l
RT s+ l
R
where T s+ l,* is the freezing point of the pure solvent system. Subtracting one from the
other, we obtain
*
*
∆ls HH 2O (T, p)
∆ls HH 2O (T *, p)
-
(44)
ln(1 xB ) =
.
RT s +l
s +l,*
RT
With small error, it can be assumed that the enthalpy of fusion changes insignificantly with
small changes in temperature, that is,
*
*
∆ls HH 2O (T, p) ≈ ∆ls HH 2O (T *, p).
(45)
This leads immediately to
*
∆ls HH 2O (T, p) 1
1
ln(1 - xB ) =
-
s +l
(46)
T s +l,*
T
R
If it is further assumed that ln(1 - xB ) ≈ -xB and that T s+ l ≈ T s+ l,* .
An approximate expression for the freezing-point depression is therefore provided:
RT*2
∆T =
x
(47)
∆l H * (T, p) B
s H2O
where ∆T = T s+ l,* - T s+ l .
16
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