Depression in Freezing Point

Freezing point is defined as the temperature at which the solid and liquid states of a substance have the same vapour pressure -- i.e. the solid and liquid phases are in equilibrium. Water's freezing point, for example, is $0^\circ$C, the temperature at which ice and liquid water coexist in equilibrium.

When a nonvolatile solute is added to water at its freezing point, the freezing point of the resulting solution is lower than $0^\circ$C. This lowering, when a solute is added to a solvent, is called depression in freezing point, $\Delta T_f$:

Reading the vapour-pressure-versus-temperature picture (Figure 9.12): the pure solvent's freezing point $T_f^\circ$ is where the solid-solvent curve and the pure-liquid-solvent curve cross (at $1$ atm, $0^\circ$C for water). The solution's liquid curve sits below the pure solvent's throughout (lower vapour pressure, exactly as in section 9.9.2), so it crosses the (unchanged) solid curve at a lower temperature $T_f<T_f^\circ$ -- this is the solution's freezing point.

Experimentally, this depression is directly proportional to the molal concentration of solute:

where $K_f$ is the molal freezing-point-depression constant (also called the cryoscopic constant) of the solvent. As with $K_b$, if $m=1$, then $\Delta T_f=K_f$ -- the depression produced by exactly one mole of solute per kilogram of solvent.

Table 9.4 -- $K_f$ values for common solvents (K kg mol$^{-1}$): water 1.86; ethanol 1.99; benzene 5.12; chloroform 4.79; carbon disulphide 3.83; ether 1.79; cyclohexane 20.0; acetic acid 3.90.

Determining molar mass from freezing point depression. For a solution of $w_B$ g solute in $w_A$ g solvent, molality is

so

which rearranges to give the solute's molar mass:

$M_B = \frac{K_f\,w_B}{\Delta T_f\,w_A}\times1000 \qquad (9.30)$ …