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A $\pu{4.305 g}$ sample of a nonelectrolyte is dissolved in $\pu{105 g}$ of water. The solution freezes at $\pu{-1.23^\circ C}$. Calculate the molar mass of the solute. $k_f$ for water is $\pu{1.86 C/m}$.
My steps:
$$1.23 = 1.86 \times \frac{x}{1.05},$$
where $x = y / 4.305$.
$$0.66 = \frac{x}{1.05}$$ $$x = 0.69$$ $$0.69 = y / 4.305$$ $$y = \pu{2.989 g}$$
But I know this is not correct.
The freezing point depression is based on the molal concentration (moles of solute per kg of solvent).
$$\Delta T_f = -k_f \cdot m$$
You know the freezing point depression of the solution and the cryoscopic constant, so you can calculate the molality:
$$m = -\dfrac{\Delta T_f}{k_f}=-\dfrac{\pu{-1.23 ^\circ C}}{1.86 \frac{\textrm{kg}\cdot ^\circ \text{C}}{\textrm{mol}}}=\pu{0.66 molal}$$
Since molality is the number of moles of solute divided by the mass of the solvent in kg, you can calculate the moles of the solute:
$$n_\textrm{solute}= \pu{0.66 m} \cdot \pu{ 0.105 kg}= \ ...$$
Divide your mass of solute by the moles of solute and you will get the molar mass (and not 2.989 g).
