# Depression of freezing point

We know that a liquid freezes only when its vapour pressure becomes equal to that of its solid form. But since the vapour pressure of a liquid is lowered by the addition of a solute, hence the freezing point of the solution is lowered which is known as depression of freezing point $(\Delta T_f)$ It also depends upon the number of moles of the added non-volatile solute. Thus,

$\Delta T_f \propto m\hspace{5mm} \text{or} \hspace{5mm} \Delta T_f = K_f \times m$

where $K_f$ = molal freezing point lowering constant cryoscopic constant and ‘m’ =molality of the solute.

According to Van’t Hoff, $K_f =0.002 T^2 / L_f$ where ‘T’=  freezing point of solvent in Kelvin, $L_f$ =latent heat of fusion (in calories / g of solvent).

The equation, $\Delta T_f = K_f x m$ , may be written as:

$\Delta T_f = \dfrac{1000 \times K_f \times w}{m \times W} \\[3mm] \text{or} \hspace{3mm}m = \dfrac{1000 \times K_f \times w}{\Delta T_f \times W}$

Where ‘w’ is the mass of solute, ‘W’ is the mass of solvent, ‘m’ = molar mass of the solute, $K_f$ is the molal depression constant and $\Delta T_f$ is depression in freezing point.

Since molality is related with the molecular mass of the solute, hence with the help of these equations molecular mass of the solute may be calculated.

Van’t Hoff’s Factor: In case of solutions of electrolytes it is generally observed that the lowering of vapour pressure is higher than the expected value. In the same way the $\Delta T_b \text{and} \Delta T_f$  for the electrolytes are found to be higher than the expected value. This is the reason that observed molecular mass of the electrolyte is found to be lower than the actual value. This is due to increase in number of ions (particles) in solution.

On the other hand, in the case of electrolytes which associate in solution give reverse value of the above, therefore molecular mass is found to be more than actual value. Van’t Hoff established a relationship between observed value of colligative property and calculated value of that. He introduced a factor ‘i’ which is known as Van’t Hoff’ s factor or constant.

Thus,

$i = \dfrac{M_c}{M_0} = \dfrac{\text{Experimental value of colligative property}}{\text{calculated value of colligative property}}$

As $M \propto \dfrac{1}{\text{Colligative Property}}$

Where $M_c$ = calculated (normal) molecular mass and $M_0$ = observed molecular mass

$i = \dfrac{\text{Colligative molarity}}{\text{Molarity}} = \dfrac{m_c}{m}$

Thus,

$m_c = i m$

Therefore we can write that, $\begin{pmatrix} \Delta T_b = iK_b m \\ \Delta T_f = iK_f m \\ PV = i nST \end{pmatrix}$

For any strong electrolyte of the type $AB, AB_2 \text{and} AB_3$ the ‘i’ for the solute is 2, 3 and 4 respectively.

With the knowledge of ‘i’ degree of dissociation or association can also be calculated.

$\alpha_{\text{Degree of dissociation}} = \dfrac{i- 1}{n- 1}$

n =No. of species formed after dissociation.

$\alpha_{\text{association}} = (1- i) . \dfrac{n}{n- 1}$

n = No. of species formed after association.

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