Freezing-point depression

The freezing point is the temperature at which the liquid solvent and solid solvent are at equilibrium, so that their vapor pressures are equal. When a non-volatile solute is added to a volatile liquid solvent, the solution vapour pressure will be lower than that of the pure solvent. As a result, the solid will reach equilibrium with the solution at a lower temperature than with the pure solvent.[2] This explanation in terms of vapor pressure is equivalent to the argument based on chemical potential, since the chemical potential of a vapor is logarithmically related to pressure. All of the colligative properties result from a lowering of the chemical potential of the solvent in the presence of a solute. This lowering is an entropy effect. The greater randomness of the solution (as compared to the pure solvent) acts in opposition to freezing, so that a lower temperature must be reached, over a broader range, before equilibrium between the liquid solution and solid solution phases is achieved. Melting point determinations are commonly exploited in organic chemistry to aid in identifying substances and to ascertain their purity.

In the liquid solution, the solvent is diluted by the addition of a solute, so that fewer molecules are available to freeze (a lower concentration of solvent exists in a solution versus pure solvent). Re-establishment of equilibrium is achieved at a lower temperature at which the rate of freezing becomes equal to the rate of liquefying. The solute is not occluding or preventing the solvent from solidifying, it is simply diluting it so there is a reduced probability of a solvent making an attempt at freezing in any given moment.

At the lower freezing point, the vapor pressure of the liquid is equal to the vapor pressure of the corresponding solid, and the chemical potentials of the two phases are equal as well.

Freezing-point depression can also be used as a purity analysis tool when analyzed by differential scanning calorimetry. The results obtained are in mol%, but the method has its place, where other methods of analysis fail.

In the laboratory, lauric acid may be used to investigate the molar mass of an unknown substance via the freezing-point depression. The choice of lauric acid is convenient because the melting point of the pure compound is relatively high (43.8 °C). Its cryoscopic constant is 3.9 °C·kg/mol. By melting lauric acid with the unknown substance, allowing it to cool, and recording the temperature at which the mixture freezes, the molar mass of the unknown compound may be determined.[9]

This is also the same principle acting in the melting-point depression observed when the melting point of an impure solid mixture is measured with a melting-point apparatus since melting and freezing points both refer to the liquid-solid phase transition (albeit in different directions).

In principle, the boiling-point elevation and the freezing-point depression could be used interchangeably for this purpose. However, the cryoscopic constant is larger than the ebullioscopic constant, and the freezing point is often easier to measure with precision, which means measurements using the freezing-point depression are more precise.

FPD measurements are also used in the dairy industry to ensure that milk has not had extra water added. Milk with a FPD of over 0.509 °C is considered to be unadulterated.[10]

If the solution is treated as an ideal solution, the extent of freezing-point depression depends only on the solute concentration that can be estimated by a simple linear relationship with the cryoscopic constant ("Blagden's Law").

${\displaystyle \Delta T_{f}\propto {\frac {\text{Moles of dissolved species}}{\text{Mass of solvent}}}}$

${\displaystyle \Delta T_{f}=K_{f}bi}$

where:

• ${\displaystyle \Delta T_{f}}$ is the decrease in freezing point, defined as the freezing point ${\displaystyle T_{f}^{0}}$ of the pure solvent minus the freezing point ${\displaystyle T_{f}}$ of the solution, as the formula above results in a positive value given that all factors are positive. From the ${\displaystyle \Delta T_{f}}$ calculated using the formula above, the freezing point of the solution can then be calculated as ${\displaystyle T_{f}=T_{f}^{0}-\Delta T_{f}}$.
• ${\displaystyle K_{f}}$, the cryoscopic constant, which is dependent on the properties of the solvent, not the solute. (Note: When conducting experiments, a higher k value makes it easier to observe larger drops in the freezing point.)
• ${\displaystyle b}$ is the molality (moles of solute per kilogram of solvent)
• ${\displaystyle i}$ is the van 't Hoff factor (number of ion particles per formula unit of solute, e.g. i = 2 for NaCl, 3 for BaCl₂).

Some values of the cryoscopic constant K_f for selected solvents:[11]

The simple relation above doesn't consider the nature of the solute, so it is only effective in a diluted solution. For a more accurate calculation at a higher concentration, for ionic solutes, Ge and Wang (2010)[13][14] proposed a new equation:

${\displaystyle \Delta T_{\text{F}}={\frac {\Delta H_{T_{\text{F}}}^{\text{fus}}-2RT_{\text{F}}\cdot \ln a_{\text{liq}}-{\sqrt {2\Delta C_{p}^{\text{fus}}T_{\text{F}}^{2}R\cdot \ln a_{\text{liq}}+(\Delta H_{T_{\text{F}}}^{\text{fus}})^{2}}}}{2\left({\frac {\Delta H_{T_{\text{F}}}^{\text{fus}}}{T_{\text{F}}}}+{\frac {\Delta C_{p}^{\text{fus}}}{2}}-R\cdot \ln a_{\text{liq}}\right)}}.}$

In the above equation, T_F is the normal freezing point of the pure solvent (273 K for water, for example); a_liq is the activity of the solvent in the solution (water activity for aqueous solution); ΔH^fus_TF is the enthalpy change of fusion of the pure solvent at T_F, which is 333.6 J/g for water at 273 K; ΔC^fus_p is the difference between the heat capacities of the liquid and solid phases at T_F, which is 2.11 J/(g·K) for water.

The solvent activity can be calculated from the Pitzer model or modified TCPC model, which typically requires 3 adjustable parameters. For the TCPC model, these parameters are available[15][16][17][18] for many single salts.