Question

If \( P_0 \) and \( P_S \) are the vapour pressures of the solvent and solution respectively and \( X_0 \) and \( X_S \) are mole fractions of solvent and solute respectively, then

• A. \( P_0 = X_S P_S \)
• B. \( P_S = X_0 P_0 \)
• C. \( P_0 = X_0 P_S \)
• D. \( P_S = X_S P_0 \)

Hint:

Raoult’s Law can help you determine the relationship between the vapor pressure and mole fractions in a solution.

Answer 1

The Correct Option is B

1. Relevant Principle: This question relates to the colligative property of vapor pressure lowering, which is described by Raoult's Law for ideal solutions with a non-volatile solute.
2. Raoult's Law Statement: Raoult's Law states that the partial vapor pressure of a solvent above a solution (\(P_{\text{solvent}}\)) is equal to the vapor pressure of the pure solvent (\(P_{\text{pure solvent}}\)) multiplied by the mole fraction of the solvent in the solution (\(X_{\text{solvent}}\)). \[ P_{\text{solvent}} = X_{\text{solvent}} \times P_{\text{pure solvent}} \]
3. Applying to Given Symbols:
   • The vapor pressure of the solution (\(P_S\)) is the partial vapor pressure of the solvent above the solution (assuming a non-volatile solute).
   • The vapor pressure of the pure solvent is given as \(P_0\).
   • The mole fraction of the solvent in the solution is given as \(X_0\).
   • The mole fraction of the solute is given as \(X_S\). (Note: \(X_0 + X above a solution and its mole fraction in the solution is described by Raoult's Law. This law applies ideally when the solute is non-volatile.
   • State Raoult's Law: Raoult's Law states that the partial vapor pressure of a solvent over a solution (\(P_S\)) is equal to the vapor pressure of the pure solvent (\(P_0\)) multiplied by the mole fraction of the solvent (\(X_0\)) in the solution.
   • Mathematical Expression: According to Raoult's Law: \[ P_S = P_0 \times X_0 \] or equivalently \[ P_S = X_0 P_0 \]
   • Definitions from the Question:
     • \(P_0\) = Vapor pressure of the pure solvent
     • \(P_S\) = Vapor pressure of the solution (effectively, the partial vapor pressure of the solvent above the solution)
     • \(X_0\) = Mole fraction of the solvent
     • \(X_S\) = Mole fraction of the solute
   • Compare with Options: We need to find the option that matches the formula \(P_S = X_0 P_0\).
     • (a) \(P_0 = X_S P_S\): Incorrect.
     • (b) \(P_S = X_0 P_0\): Correct. This directly matches Raoult's Law.
     • (c) \(P_0 = X_0 P_S\): Incorrect.
     • (d) \(P_S = X_S P_0\): Incorrect. Uses the mole fraction of the solute instead of the solvent.

The correct option is (b) \(P_S = X_0 P_0\).

Answer 2

According to Raoult's Law for an ideal solution:

$$ P_S = X_0 \cdot P_0 $$ where:
$P_S$ = vapour pressure of the solution,
$P_0$ = vapour pressure of the pure solvent,
$X_0$ = mole fraction of the solvent. 

Correct answer: (b) $P_S = X_0 P_0$