Question

Find the difference between B.P. and M.P. of a solution of concentration of 0.2 m $ {\text{NaCl}} $ (Assuming 100% ionization): [ $ {{\text{K}}_{\text{f}}} $ and $ {{\text{K}}_{\text{b}}} $ of water are 0.5 and 2 K. kg. mol respectively]. Boiling point and freezing point of water are 373 and 273 K.
(A) 0.6
(B) 1.0
(C) 101
(D) 274

Answer 1

Hint: The elevation in boiling point and the depression in freezing point are two kinds of colligative properties as these two properties depend only upon the number of particles of the solute and not on the nature of the solute.
The boiling point elevation is the difference between the boiling points of the solution and pure solvent whereas the freezing point depression is the difference between the freezing points of the pure solvent and that of the solution.

Complete step by step solution:
The term ‘elevation in boiling point’ refers to the increase in the boiling point of the solution than that of the pure solvent. It has been found that this increase in boiling point depends upon the concentration of the solute dissolved in a solution and hence it is considered a colligative property.
The elevation in boiling point is found to be related to the molality of the solution by the following expression:
$\Delta T_{b}= K_{b}\times m$
Here, $\Delta T_{b}$ represents the elevation in boiling point which is equal to $ \left( {{{\text{T}}_{\text{b}}}{\text{ - }}{{\text{T}}_{\text{b}}}^{\text{0}}} \right) $ , where $ {{\text{T}}_{\text{b}}} $ is the boiling point of the solution and $ {{\text{T}}_{\text{b}}}^0 $ is the boiling point of the pure solvent.
 $ {{\text{K}}_{\text{b}}} $ represents the molal elevation constant and m represents the molality of the solution.
The term ‘depression in freezing point’ refers to the decrease in the freezing point of the solution than that of the pure solvent. It has been found that this decrease in freezing point depends upon the concentration of the solute dissolved in a solution and hence it is considered a colligative property.
The depression in freezing point is found to be related to the molality of the solution by the following expression:
$\Delta T_{f}= K_{f}\times m $
Here, $\Delta T_{f}$ represents the depression in freezing point which is equal to $ \left( {{{\text{T}}_{\text{f}}}^0{\text{ - }}{{\text{T}}_{\text{f}}}} \right) $ , where $ {{\text{T}}_{\text{f}}} $ is the freezing point of the solution and $ {{\text{T}}_{\text{f}}}^0 $ is the freezing point of the pure solvent.
 $ {{\text{K}}_{\text{f}}} $ represents the molal depression constant and m represents the molality of the solution.
Given, molality $ {\text{m = 0}}{\text{.2,}}{{\text{K}}_{\text{b}}}{\text{ = 2Kkg/mol,}}{{\text{K}}_{\text{f}}}{\text{ = 0}}{\text{.5Kkg/mol,}}{{\text{T}}_{\text{b}}}^{\text{0}}{\text{ = 373K,}}{{\text{T}}_{\text{f}}}^{\text{0}}{\text{ = 273K}} $ .
We need to find out the difference in $ {{\text{T}}_{\text{b}}} $ and $ {{\text{T}}_{\text{f}}} $ .
Substitute these values in the two expressions,
 $
  \Delta T_{b}= K_{b}\times m
\Rightarrow T_{b}-373= 2\times 0.2
\Rightarrow T_{b}= 373+0.4
\Rightarrow T_{b}= 373.4K
  $
And,
 $
 \Delta T_{f}= K_{f}\times m
\Rightarrow 273-T_{f}= 0.5\times 0.2
\Rightarrow T_{f}= 273-0.10
\Rightarrow T_{f}= 272.9K
 $
Thus the boiling point $ {{\text{T}}_{\text{b}}}{\text{ = 373}}{\text{.4K}} $ and the freezing point (same as the substance’s melting point) $ {{\text{T}}_{\text{f}}}{\text{ = 272}}{\text{.9K}} $ .
So, the required difference between the boiling and melting points of the solution is
  $
  {\text{ = }}{{\text{T}}_{\text{b}}}{\text{ - }}{{\text{T}}_{\text{f}}} \\
  {\text{ = 373}}{\text{.4K - 272}}{\text{.9K}} \\
  {\text{ = 100}}{\text{.5K}} \\
  $
The closest option to this answer is option C.

Note:
These two expressions can also be used to calculate the molecular masses of the solute as both these expressions involve the molality which is actually the number of moles of solute dissolved in 1000g of the solvent. The colligative property of elevation in boiling point is measured by the Landsberger method and that of depression in freezing point is measured by the Beckmann method.