Solve the following problem:
A very small amount of sugar is dissolved in water, and the solution is in equilibrium with pure ice. The equation of phase equilibrium is
g’ = g’’ + RT In (1 - x),
Where g’ is the molar Gibbs function of pure ice, s" is the molar Gibbs function of pure water, and x is the mole fraction of sugar in solution.
(a) For an infinitesimal change in x at constant pressure, show that
-s’ dT = -s’’ dT + R In (1-x) dT + RTd In (1-x).
(b) Substituting for R In (1 - x) the value obtained from the equation of phase equilibrium, show that the equation in part (a) reduces to
(h’’ - h’/T)dT = RT d In (1 - x)
(c) Taking into account that x « 1 and calling h" - h’ the latent heat of fusion IF, show that the depression of the freezing point is
ΔT = (RT2/ IF)x .