Question

The cryoscopic constant for acid is $ 3.6 $ $ {\text{K Kg/mol}} $ . A solution of \[{\text{1 g}}\] of a hydrocarbon in \[{\text{100 g}}\] of acetic acid freezes at $ {\text{16}}{\text{.1}}{{\text{4}}^{{\text{ 0}}}}{\text{C}} $ instead of the usual $ {\text{16}}{\text{.6}}{{\text{0}}^{{\text{ 0}}}}{\text{C}} $ . The hydrocarbons contain $ 92.3\% $ carbon. If the molecular formula is $ {{\text{C}}_{\text{x}}}{{\text{H}}_{\text{y}}} $ . Find the value of x and y.
A.X = 2, y = 4
B.X = 3, y = 8
C.X = 4, y = 10
D.X = 6, y = 6

Answer 1

Hint:The cryoscopic constant is defined as the depression in the freezing point of a solution when one mole of a solute is dissolved in one kilogram of the solvent.
Formula: $ {{\Delta }}{{\text{T}}_{\text{f}}}{\text{ = }}{{\text{K}}_{\text{f}}}{{ \times m}} $ . ……………… equation 1,
Where, “m” is the molality of the solvent which is equal to,
   $ {\text{m = }}\dfrac{{{\text{mass of the solute in g}}}}{{{\text{molecular mass of the solute}}}} \times \dfrac{1}{{{\text{mass of the solvent in kg}}}} $

Complete step by step answer:
Let the molecular mass of the solute be M and as there is \[{\text{1 g}}\] of a hydrocarbon, so the number of moles, n = $ \dfrac{{\text{1}}}{{\text{M}}} $ .
The mass of the acetic acid used for the solution is \[{\text{100 g}}\] which is equivalent to:
    $ {\text{m = }}\dfrac{{\text{1}}}{{{\text{M}} \times 0.1}} $ = $ \dfrac{{{\text{10 }}}}{{\text{M}}} $ .
The depression in the freezing point is equal to = $ {\text{16}}{\text{.6}}{{\text{0}}^{{\text{ 0}}}}{\text{C}} $ - $ {\text{16}}{\text{.1}}{{\text{4}}^{{\text{ 0}}}}{\text{C}} $ = $ {0.46^{{\text{ 0}}}}{\text{C}} $
Putting the values of the molarity, the depression in the freezing point and the cryoscopic constant in the equation we get,
    $ {\text{0}}{\text{.46 = 3}}{{.6 \times }}\dfrac{{{\text{10}}}}{{\text{M}}} $
Therefore, the molecular weight of the solute is,
 \[{\text{M = }}\dfrac{{{\text{360}}}}{{46}}\] = \[{\text{78 g/mol}}\] .
Now, $ 92.3\% $ of \[{\text{78 g/mol}}\] = the weight of carbon = \[{\text{72 g}}\] .
Hence, the weight of hydrogen in the molecules is \[{\text{78 - 72 = 6 g}}\]
Since the atomic weight of carbon is 12, so \[{\text{72 g}}\] will be the weight of 6 carbon atoms while the weight of 6 hydrogen atoms is 6.

Hence, the value of x = 6 and y = 6. The correct answer is option D.

Note:
The cryoscopic constant is a colligative property of a solution that is dependent on the molar mass of the solute in the solution, the molar mass of the solvent on the solution, the enthalpy of fusion of the solvent, and finally the freezing point of the solvent.