Question

Two non-reactive gases A and B present in a container with partial pressures $200$ and ${{180mm}}$ of ${{Hg}}$. When a third non-reactive gas C is added then total pressure becomes ${{1atm}}$ then mole fraction of C will be:
A. $0.75$
B. $0.5$
C. $0.25$
D. cannot be calculated

Answer 1

Hint:The dissolved non-volatile solute lowers the vapor pressure. Dalton’s law and Raoult’s law can be used to solve this question. Raoult’s law states that the vapor pressure of a solution is directly proportional to the mole fraction of solvent. The solution which obeys Raoult’s law is called the ideal solution.

Complete step by step answer:
Here, three non-reactive gases are mixed together. Thus its total pressure will be the sum of the partial pressures of all these non-reactive gases. This is derived from Dalton’s law of partial pressures.
It is given that the partial pressure of gas A, \[{{{p}}_{{A}}} = 200{{mm}}\] of ${{Hg}}$
Partial pressure of gas B, ${{{p}}_{{B}}} = 180{{mm}}$ of ${{Hg}}$
Partial pressure of gas C can be expressed as ${{{p}}_{{C}}}$.
Based on Dalton’s law of partial pressures, total pressure can be expressed as:
Total pressure, ${{{P}}_{{t}}} = {{{p}}_{{A}}} + {{{p}}_{{B}}} + {{{p}}_{{C}}}$
Substituting the values of partial pressures of A and B in the above equation, we get
${{{P}}_{{t}}} = {{200}} + {{180}} + {{{p}}_{{C}}}$
Also, we know that the total pressure is given as ${{1atm}}$ which is equal to $760{{mm}}$ of ${{Hg}}$.
Thus ${{760}} = {{200}} + {{180}} + {{{p}}_{{C}}}$
Partial pressure of C, ${{{p}}_{{C}}} = 760 - 380 = 380{{mm}}$ of ${{Hg}}$.
Now according to Raoult’s law,
Raoult’s law can be mathematically expressed as:
Partial pressure, ${{p}} = {{x}}{{{P}}_{{t}}}$, where ${{{P}}_{{t}}}$is the total vapor pressure and ${{x}}$is the mole fraction.
Thus the partial pressure of C, ${{{p}}_{{C}}} = {{x}}{{{P}}_{{t}}}$
i.e. ${{380}} = {{x}} \times {{760mm}}$ of ${{Hg}}$
On simplification, we get
Mole fraction of C, ${{x = }}\dfrac{{380}}{{760}} = 0.5$
Thus the mole fraction of C is $0.5$.

Hence, the correct option is B.

Note:
Vapor pressure of a liquid is much different in a solution than it is in pure liquid. Vapor pressure is the pressure acted over a substance at which vapors are formed. When a plot of vapor pressure of solution, ${{{P}}_{{{soln}}}}$, against mole fraction of solvent, ${{{X}}_{{{solvent}}}}$, is represented, it gives a straight line with a slope of ${{{P}}_{{{solvent}}}}$.