The vapor pressure of water at 30 Celsius is 31.82 mmHg and the density is 0.9956 g/mL. What is the new pressure after adding enough sorbitol to make a 2.7 m solution?

Question

The vapor pressure of water at 30 Celsius is 31.82 mmHg and the density is 0.9956 g/mL. What is the new pressure after adding enough sorbitol to make a 2.7 m solution?

Answer 1

2.7 m = 2.7 moles sorbitol/kg water.

P_soln = X_solvent*P^o_{pure solvent}

Answer 2

To find the new pressure after adding sorbitol to make a 2.7 m solution, we need to use Raoult's law. Raoult's law describes the vapor pressure of an ideal solution, which is the solute (sorbitol) dissolved in the solvent (water).

Raoult's Law states that the vapor pressure of an ideal solution is equal to the mole fraction of the solvent multiplied by the vapor pressure of the pure solvent. Mathematically, it can be expressed as:

P_total = X_solvent * P_solvent

Where:
P_total is the total vapor pressure of the solution,
X_solvent is the mole fraction of the solvent,
P_solvent is the vapor pressure of the pure solvent.

In this case, the solvent is water and the solute is sorbitol.

First, we need to find the mole fraction (X_solvent) of water in the solution. The mole fraction is the ratio of the moles of solvent to the total moles of solute and solvent.

Given that the solution is a 2.7 m (molality) solution, 2.7 moles of sorbitol is dissolved in 1 kg of water.

To find the moles of water, we can use the density of water:

density = mass/volume

Since we know the density of water is 0.9956 g/mL, we can calculate the volume of water in kg:

1 kg water = 1000 g / 0.9956 g/mL ≈ 1004.64 mL

Next, we need to convert this volume to liters:

1004.64 mL = 1004.64 mL * 1 L/1000 mL ≈ 1.00464 L

Now, let's calculate the number of moles of water:

moles_water = mass_water / molar_mass_water

The molar mass of water is approximately 18.015 g/mol.

moles_water = 1000 g / 18.015 g/mol ≈ 55.50 mol

Therefore, the moles of sorbitol (solute) in the solution is 2.7 mol, and the moles of water (solvent) is 55.50 mol.

The mole fraction of water (X_solvent) can be calculated as follows:

X_solvent = moles_water / (moles_water + moles_sorbitol)

X_solvent = 55.50 mol / (55.50 mol + 2.7 mol) ≈ 0.953

Finally, we can calculate the new vapor pressure (P_total) using Raoult's law:

P_total = X_solvent * P_solvent

P_total = 0.953 * 31.82 mmHg ≈ 30.37 mmHg

Therefore, the new pressure after adding enough sorbitol to make a 2.7 m solution is approximately 30.37 mmHg.