Derive the relation between depression in freezing point and molar mass of the solute.

Explanation

The depression in the freezing point of a solution is directly proportional to the molal concentration ( expressed in mol `kg^(-1)`) of the solution.
Thus
`Delta T_(r) prop m`
where mis the molality of the solution .
`DeltaT_(f) = k_(f)m,` where `K_(r)` is a constant of proportionality.
If m = 1 molal, `DeltaT_(r)`= `K_(f)`.Hence `K_(r)` is called the cryoscopic constant or molal depression constant.
The molality of a solution is given by,
`m = ("Number of moles of the solute ")/("Weight of the solvent in kg ")`
If `W_(1)` grams of a solvent contain `W_(2)` gram of a solute of the molar mass `M_(2)` , them the molality m of the solution is given by .
`m = (W_(2) xx 1000)/(W_(1)M_(2)) = (underset(kg)(W_(2)))/(underset(kg) (W_(1)) xx underset("kg mol"^(-1))(M_(2)))mol kg^(-1)`
`therefore DeltaT_(r) = k_(r)xx(W_(2) xx 1000)/(W_(1)M_(2))`
If the weights are expressed in kg then,
`DeltaT_(r) = K_(r) xx (W_(r))/(W_(1)M_(2))`
Hence, from the measurement of the depression in the freezing point of the solution, the molar mass of the substance can be determined.