Question

Mole fraction of the component A in vapour phase is ${x_1}$ and mole fraction of component $A$ in liquid mixture is ${x_2}$ , then ( $p_A^0 =$ vapour pressure of pure $A$ ; $p_B^0 =$ vapour pressure of pure $B$ ), the total pressure of liquid mixture is:
(A) $p_A^0\dfrac{{{x_2}}}{{{x_1}}}$
(B) $p_A^0\dfrac{{{x_1}}}{{{x_2}}}$
(C) $p_B^0\dfrac{{{x_1}}}{{{x_2}}}$
(D) $p_B^0\dfrac{{{x_2}}}{{{x_1}}}$

Answer 1

Raoult’s law has been named after François-Marie Raoult, a French chemist who while conducting an experiment found out that when substances were mixed in a solution, the vapor pressure of the solution decreased simultaneously. Raoult’s law was established in the year 1887 and is also considered the law of thermodynamics.
Use Raoult’s law which states that for solution of volatile liquids, the partial vapour pressure of each component of the solution is directly proportional to its mole fraction present in solution.
Let for component 1, ${p_1} = p_1^0 \times {x_1}$
Where, $p_1^0$ is the vapour pressure of pure component 1 at the same temperature.
And Dalton’s law for vapour phase.

Complete answer:
It is given in the question that the mole fraction of the component $A$ in vapour phase is ${x_1}$ . So using Dalton's law of partial pressure.
Now for component $A$ ,
${p_A} = {p_T} \times {x_1}$
And the mole fraction of component $A$ in liquid mixture is given ${x_2}$
Now applying Raoult’s law which states that for solution of volatile liquids, the partial vapour pressure of each component of the solution is directly proportional to its mole fraction present in solution.
${p_A} = p_A^0 \times {x_2}$
And so from equating both equation as they are both equal to ${p_A}$ , we will get the following equation
${p_T} \times {x_1} = p_A^0 \times {x_2}$
Now, the total pressure will be ${p_T} = \dfrac{{{x_2}}}{{{x_1}}} \times p_A^0$
Which is our option A.

Note:
Dalton’s law of partial pressures is a gas law that states that the total pressure exerted by a mixture of gases is equal to the sum of the partial pressures exerted by each individual gas in the mixture. For example, the total pressure exerted by a mixture of two gases A and B is equal to the sum of the individual partial pressures exerted by gas A and gas B. It is to be noted that the above formulas are only valid for the same temperature and for vapour phase use Dalton's partial pressure concept and for components in liquid mixture use the concept of Raoult’s law.