Obtain the relationship between freezing point depression of a solution containing nonvolatile-nonelectrolyte solute and its molar mass.

#### Solution

1. The freezing point depression (ΔT_{f}) is directly proportional to the molality of solution.

Thus, ΔT_{f} = K_{f} m ….(1)

2. Suppose we prepare a solution by dissolving W_{2} g of solute in W_{1} g of solvent.

Moles of solute in W_{1} g of solvent = `"W"_2/"M"_2`

where, M_{2} is the molar mass of solute.

Mass of solvent = W_{1} g = `("W"_1 g)/(100 "g"//"kg") = "W"_1/1000` kg

3. The molality is expressed as,

m = `"Moles of solute"/"Mass of solvent in kg"`

m = `("W"_2//"M"_2 "mol")/("W"_1//1000 "kg")`

m = `(1000 "W"_2)/("M"_2 "W"_1) "mol kg"^-1` ....(2)

4. Substituting equation (2) in equation (1), we get,

`triangle "T"_"f" = (1000 "K"_"f" "W"_2)/("M"_2 "W"_1)`

Hence,

`"M"_2 = (1000 "K"_"f" "W"_2)/(triangle "T"_"f" "W"_1)`