Question

What is vapour pressure of a solution containing 1 mol of a non-volatile solute in 36 g of water? ($P_1^0 = 400\ \text{mm Hg}$)

• A. 334 mm Hg
• B. 267 mm Hg
• C. 240 mm Hg
• D. 284 mm Hg

Hint:

An alternative time-saving path is using Raoult's Law rewritten directly for the solvent's perspective: $P_{\text{soln}} = x_{\text{solvent}} \times P^0$. Here, $x_{\text{solvent}} = \frac{2}{3}$, so $P_{\text{soln}} = \frac{2}{3} \times 400 = \frac{800}{3} = 267\ \text{mm Hg}$ in just one single step!

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks us to find the final vapor pressure of an aqueous solution ($P_1$) prepared by mixing 1 mole of a non-volatile solute with 36 grams of pure water, given that the vapor pressure of pure water ($P_1^0$) is $400\ \text{mm Hg}$.

Step 2: Key Formula or Approach:
According to Raoult's Law for a solution containing a non-volatile solute, the relative lowering of vapor pressure is equal to the mole fraction of the solute ($x_2$):
$$\frac{P_1^0 - P_1}{P_1^0} = x_2 = \frac{n_2}{n_1 + n_2}$$ Where:
$P_1^0 =$ Vapor pressure of pure solvent ($400\ \text{mm Hg}$).
$P_1 =$ Vapor pressure of the final solution.
$n_2 =$ Number of moles of solute ($1\ \text{mol}$).
$n_1 =$ Number of moles of solvent (water).

Step 3: Detailed Explanation:
First, let's determine the number of moles of water ($n_1$). The molar mass of water ($\text{H}_2\text{O}$) is $18\ \text{g\ mol}^{-1}$:
$$n_1 = \frac{\text{Mass of water}}{\text{Molar mass of water}} = \frac{36\ \text{g}}{18\ \text{g\ mol}^{-1}} = 2\ \text{moles}$$ Now, calculate the mole fraction of the non-volatile solute ($x_2$):
$$x_2 = \frac{n_2}{n_1 + n_2} = \frac{1}{2 + 1} = \frac{1}{3}$$ Substitute this mole fraction and the pure vapor pressure into Raoult's Law equation:
$$\frac{400 - P_1}{400} = \frac{1}{3}$$ Cross-multiply to solve for $P_1$:
$$3(400 - P_1) = 400$$ $$1200 - 3P_1 = 400$$ $$3P_1 = 1200 - 400$$ $$3P_1 = 800$$ $$P_1 = \frac{800}{3} \approx 266.67\ \text{mm Hg}$$ Rounding off to the nearest whole integer value present in our options yields 267 mm Hg, matching option (B).

Step 4: Final Answer:
The vapor pressure of the solution is 267 mm Hg, which corresponds to option (B).