Question

The vapor pressures of two liquids \(P\) and \(Q\) are \(80\) torr and \(60\) torr respectively. The total vapor pressure of the solution obtained by mixing \(3\) mol of \(P\) and \(2\) mol of \(Q\) would be:

• A. \(68\) torr
• B. \(140\) torr
• C. \(72\) torr
• D. \(20\) torr

Hint:

For binary ideal solutions: \[ P_{total}=X_AP_A^\circ + X_BP_B^\circ \] Always calculate mole fractions first before applying Raoult's law.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This problem deals with the vapor pressure of a binary liquid-liquid solution.
According to Raoult's Law, for an ideal solution, the partial vapor pressure of each volatile component is equal to the product of its mole fraction in the liquid phase and its vapor pressure in the pure state.
Dalton's Law of partial pressures states that the total vapor pressure of the solution is the sum of the partial pressures of all components.
Ideally, molecules of P and Q interact with each other in the mixture the same way they interact with themselves in pure form.
Step 2: Key Formula or Approach:
The total pressure (\(P_{total}\)) is given by:
\[ P_{total} = P_{P} + P_{Q} \]
Using Raoult's Law:
\[ P_{P} = X_{P} \cdot P_{P}^{\circ} \]
\[ P_{Q} = X_{Q} \cdot P_{Q}^{\circ} \]
Therefore:
\[ P_{total} = X_{P} \cdot P_{P}^{\circ} + X_{Q} \cdot P_{Q}^{\circ} \]
Where:
\(X\) denotes mole fraction.
\(P^{\circ}\) denotes pure vapor pressure.
Step 3: Detailed Explanation:
1. Calculate total moles:
Moles of P (\(n_{P}\)) = 3 mol.
Moles of Q (\(n_{Q}\)) = 2 mol.
Total moles (\(n_{total}\)) = \(3 + 2 = 5\) mol.
2. Calculate mole fractions (\(X\)):
Mole fraction of P (\(X_{P}\)) = \(\frac{n_{P}}{n_{total}} = \frac{3}{5} = 0.6\).
Mole fraction of Q (\(X_{Q}\)) = \(\frac{n_{Q}}{n_{total}} = \frac{2}{5} = 0.4\).
Note that \(X_{P} + X_{Q} = 0.6 + 0.4 = 1.0\), which is correct.
3. Apply Raoult's Law:
Pure vapor pressure of P (\(P_{P}^{\circ}\)) = 80 torr.
Pure vapor pressure of Q (\(P_{Q}^{\circ}\)) = 60 torr.
Partial pressure of P (\(P_{P}\)) = \(0.6 \times 80 = 48\) torr.
Partial pressure of Q (\(P_{Q}\)) = \(0.4 \times 60 = 24\) torr.
4. Calculate Total Pressure:
\(P_{total} = 48 + 24 = 72\) torr.
The calculated pressure is 72 torr, which reflects the weighted average contribution of both liquids to the total pressure above the mixture.
Step 4: Final Answer:
The total vapor pressure of the solution is 72 torr.