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# Determination of molar mass (by the depression of freezing point)

## Molality

Molality is a unit of concentration defined by:

$\mu$ $=$ $\frac{n}{m}$ with: $n$ = number of moles of solute $m$ = mass of solvent (in kg)

## The law of Raoult

The melting (or freezing) point of a solution $tf_{solution}$ is lower than that of the pure solvent $tf_{solvent}$ . The law of Raoult connects this depression of the temperature of the solution to the molality of the solute:

$\Delta T_f$ $=$ $K_f \cdot \mu$ where: $\Delta T_f$ $=$ $tf_{solution} - tf_{solvent}$ $\mu$ = molality $K_f$ = cryoscopic constant depending only on the solvent.

## Measure of a melting point

Click on the image ! In the test tube, we will introduce the solution of the subtance whose molar mass we want to determine.

Images: →

## Determination of molar masses

1) Cryoscopic constant $K_f$ In $m \;g$ of a given solvent is introduced $y \;g$ of a solute of known molar mass . The melting points of the solution and of the pure solvent are measured (2 steps !) We calculate: Number of moles of solute = $n$ $=$ $\frac{y}{M}$ $\mu$ = $\frac{n}{m}$ $=$ $\frac{y}{m\cdot M}$ $K_f$ = $\frac{\Delta T}{\mu}$ 2) Molar mass $M$ In $m_1\; g$ of the same solvent are introduced $y_1 \; g$ of a solute whose molar mass $M_1 \;$ is to be found. The melting point of this solution is determined. We calculate: $\mu$ $=$ $\frac{\Delta T}{K_f}$ Number of moles of solute = $n_1$ $=$ $m_1 \cdot \mu$ $M_1 =\frac{y_1 }{n_1 }$

## Example

A solution $(S)$ of $2.40\;g$ of a substance supposed to be biphenyl ($C_{12}H_{10}$) is introduced in $75.0\;g$ benzene. The melting point of this solution is $4.39\;^oC$ Moreover, the melting point of pure benzene was determined : $5.45\;^oC$ and that of a solution $(S_1)$ of $7.24\;g$ of $C_2Cl_4H_2$ in $115.3\;g$ benzene: $3.55\;^oC$ It is asked to verify the molecular weight of biphenyl

1) Determination of $K_f$ $n_{C_2Cl_4H_2}$ $=$ $\frac{7.24}{168}$ $\mu_{S_1 }$ $=$ $\frac{7.24}{168\cdot 0.1153}$ $K_f$ $=$ $\frac{5.45-3.55}{\mu}$ $=$ $5.08 \frac{^oC}{mole}$ 2) Determination of the molar mass $\mu_S$ $=$ $\frac{5.45-4.39}{5.08}$ $=$ $0.208 \frac{mol}{kg}$ Number of moles of biphenyl = $n_1$ $=$ $0.075 \cdot 0,208$ $=$ $0.0156$ $M_1$=$\frac{2.40}{0.0156}$ $\approx$ $154 \frac{g}{mol}$ This corresponds to the molar mass of $C_{12}H_{10}$ !