242. Derive an expression relating the elevation of boiling point to the amount of solute present in the solution.

 Answer:

the vapour pressure of a liquid increases with increase of temperature. It boils at the temperature at which its vapour pressure is equal to the atmospheric pressure. For example, water boils at 373.15 K (100° C) because at this temperature the vapour pressure of water is 1.013 bar (1 atmosphere)

Let ${\mathrm{T}}_{\mathrm{b}}^0$ be the boiling point of pure solvent and
T_b be the boiling point of solution. The increase in
the boiling point $∆{\mathrm{T}}_{\mathrm{b}} = {\mathrm{T}}_{\mathrm{b}} - {\mathrm{T}}_{\mathrm{b}}^0$  is known as
elevation of boiling point.

for dilute solutions the elevation of boiling point (ΔTb) is directly proportional to the molal concentration of the solute in a solution. Thus
ΔTb ∝ m 
or ΔTb = K_b m
Here m (molality) is the number of moles of solute dissolved in 1 kg of solvent and the constant of proportionality, Kb is called Boiling Point Elevation Constant or Molal Elevation Constant (EbullioscopicConstant). The unit of Kb is K kg mol-1. If w2 gram of solute of molar mass M2 is dissolved in w1 gram of solvent, then molality, m of the solution is given by the expression:

$$\begin{gathered} \mathrm{m} = \frac{{\displaystyle \raisebox{1ex}{${\mathrm{w}}_2$}\!\left/ \!\raisebox{-1ex}{${\mathrm{M}}_2$}\right.}}{{\displaystyle \raisebox{1ex}{${\mathrm{w}}_1$}\!\left/ \!\raisebox{-1ex}{$1000 $}\right.}} = \frac{1000\times {\mathrm{w}}_2}{{\mathrm{M}}_2\times {\mathrm{w}}_1} \\ \mathrm{Substituting} \mathrm{the} \mathrm{value} \mathrm{of} \mathrm{molality} \mathrm{in} ∆{\mathrm{T}}_{\mathrm{b}}\mathrm{m} \mathrm{we} \mathrm{get} \\ ∆{\mathrm{T}}_{\mathrm{b}} =\frac{ {\mathrm{K}}_{\mathrm{b}}\times 1000\times {\mathrm{w}}_2}{{\mathrm{M}}_2\times {\mathrm{w}}_1} \\ {\mathrm{M}}_2 = \frac{ {\mathrm{K}}_{\mathrm{b}}\times 1000\times {\mathrm{w}}_2}{∆{\mathrm{T}}_{\mathrm{b}}\times {\mathrm{w}}_1} \end{gathered}$$

Thus, in order to determine M2, molar mass of the solute, known mass of solute in a known mass of the solvent is taken and ΔTb is
determined experimentally for a known solvent whose Kb value is known.