Question

1. the vapor pressure of pure water at 35 °c is 42.2 mm hg. what is the vapor pressure of a mixture of 15.0 g sucrose, and 85.0 g water ? the molar mass of sucrose, c₁₂h₂₂o₁₁, is 342.3 g/mol and h₂o is 18.02 g/mol.

a. 7.45 mm hg
b. 21.8 mm hg
c. 35.9 mm hg
d. 40.6 mm hg
e. 41.8 mm hg

Explanation:

Step1: Calculate moles of sucrose

Moles of sucrose = mass / molar mass = \( \frac{15.0\ g}{342.3\ g/mol} \approx 0.0438\ mol \)

Step2: Calculate moles of water

Moles of water = \( \frac{85.0\ g}{18.02\ g/mol} \approx 4.717\ mol \)

Step3: Calculate mole fraction of water

Mole fraction of water (\( X_{H_2O} \)) = \( \frac{moles\ of\ water}{moles\ of\ water + moles\ of\ sucrose} = \frac{4.717}{4.717 + 0.0438} \approx 0.9907 \)

Step4: Apply Raoult's Law

Vapor pressure of solution (\( P_{solution} \)) = \( X_{H_2O} \times P^0_{H_2O} \)
\( P_{solution} = 0.9907 \times 42.2\ mm\ Hg \approx 41.8\ mm\ Hg \)? Wait, no, wait, recalculate mole fraction:
Wait, 4.717 + 0.0438 = 4.7608
\( X_{H_2O} = 4.717 / 4.7608 ≈ 0.9908 \)
42.2 0.9908 ≈ 41.8? But wait, maybe I made a mistake. Wait, no, let's check again. Wait, sucrose is a non-volatile solute, so Raoult's law applies. Wait, but maybe my calculation of moles of water is wrong? 85 / 18.02: 184=72, 85-72=13, 13/18.02≈0.721, so 4.721? Wait, 18.024.717≈85, yes. Then moles of sucrose: 15/342.3≈0.0438. So total moles: 4.717 + 0.0438 = 4.7608. Mole fraction of water: 4.717 / 4.7608 ≈ 0.9908. Then vapor pressure: 42.2 0.9908 ≈ 41.8? But option E is 41.8, but wait, maybe I messed up. Wait, no, wait, maybe the mole fraction is calculated as moles of solvent over total moles. Wait, but let's check the options. Wait, maybe I made a mistake in the mole fraction. Wait, no, let's recalculate:

Wait, 85g water: moles = 85 / 18.02 ≈ 4.71698 mol

15g sucrose: moles = 15 / 342.3 ≈ 0.04382 mol

Total moles = 4.71698 + 0.04382 = 4.7608 mol

Mole fraction of water: 4.71698 / 4.7608 ≈ 0.9908

Vapor pressure: 42.2 0.9908 ≈ 42.2 - 42.2(1 - 0.9908) = 42.2 - 42.20.0092 ≈ 42.2 - 0.388 ≈ 41.81 mm Hg. But option E is 41.8, but wait, maybe the correct answer is D? Wait, no, maybe I messed up the mole fraction. Wait, no, let's check the options again. Wait, maybe I made a mistake in the calculation. Wait, 4.717 / (4.717 + 0.0438) = 4.717 / 4.7608 ≈ 0.9908. 42.2 0.9908 ≈ 41.8. But option E is 41.8. But wait, maybe the problem is that I miscalculated the mole fraction. Wait, no, let's check with another approach. The mole fraction of solute is 0.0438 / 4.7608 ≈ 0.0092, so the vapor pressure lowering is \( \Delta P = X_{solute} \times P^0 \), so \( \Delta P = 0.0092 \times 42.2 ≈ 0.388 \), so \( P = 42.2 - 0.388 ≈ 41.8 \), which is option E. But wait, maybe the answer is D? Wait, no, let's check the calculations again. Wait, 85g water: moles = 85 / 18.02 = 4.71698 mol. 15g sucrose: 15 / 342.3 = 0.04382 mol. Total moles: 4.71698 + 0.04382 = 4.7608 mol. Mole fraction of water: 4.71698 / 4.7608 = 0.9908. Vapor pressure: 0.9908 42.2 = 42.2 0.9908. Let's calculate 42.2 0.99 = 41.778, 42.2 0.0008 = 0.03376, so total 41.778 + 0.03376 = 41.81176, which is approximately 41.8, so option E. But wait, maybe the correct answer is D? Wait, no, maybe I made a mistake in the mole fraction. Wait, no, the formula is correct. Raoult's law: \( P = X_{solvent} \times P^0 \). So the answer should be E? But wait, let's check the options again. The options are A:7.45, B:21.8, C:35.9, D:40.6, E:41.8. So according to the calculation, it's E. But wait, maybe I messed up the moles of water. Wait, 85 divided by 18.02: 18.02*4=72.08, 85-72.08=12.92, 12.92/18.02≈0.717, so 4.717 mol. Correct. Moles of sucrose: 15/342.3≈0.0438. Correct. So the calculation seems right. So the answer is E.

Answer:

E. 41.8 mm Hg