What is molal depression constant `(K_(f))` ? Derive equation relating `K_(f)` with molar mass of solute.

Explanation

Decrease in freezing point of a solution prepared by dissolving one gram molar mass of non - volatile solute into one kilogram of solvent is called molal depresion constant.
Let `T_(f)^(0)` be the freezing point of pure solvent and `T_(f)` be its freezing point when non - volatile solute is dissolved in it. The decrease in freezing point. `Delta T_(f)=T_(f)^(0)-T_(f)` is known as depression in freezing point.
For dilute solution (ideal solution) is directly proportional to molality, m of the solution.
Thus, `Delta T_(f)prop m`
`Delta T_(f)=K_(f).m " "` ....(i)
The proportionality constant `K_(f)`, which depends on the nature of the solvent is known as Freezing Point Depression Constant or Molal Depression Constant or Cryoscopic Constant. The unit of `K_(f)` is K kg `mol^(-1)`.

If `w_(2)` gram of the solute having molar mass as `M_(2)`, present in `w_(1)` gram of solvent, produces the depression in freezing point `Delta T_(f)` of the solvent then molality of the solute is given by the equation.
`m=(w_(2)//M_(2))/(w_(1)//1000)`
Substituting this value of molality in equation (i) we get,
`Delta T_(f)=(K_(f)xx w_(2)//M_(2))/(w_(1)//1000)`
`Delta T_(f)=(K_(f)xx w_(2)xx1000)/(w_(1)xx M_(2))`
`M_(2)=(K_(f)xx w_(2)xx 1000)/(Delta T_(f)xx w_(1))`