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Q1.34:
Vapour pressure of water at 293 K is 17.535 mm Hg. Calculate the vapour pressure of water at 293 K when 25 g of glucose is dissolved in 450 g of water.

Solution :

Given Variables & Initial Setup

To determine the vapour pressure of the solution, we utilize Raoult's Law for solutions containing non-volatile solutes. The relevant parameters are:

  • Vapour pressure of pure solvent ($P_1^\circ$) = $17.535 \, \text{mm Hg}$
  • Mass of solute (Glucose, $C_6H_{12}O_6$), $w_2 = 25 \, \text{g}$
  • Molar mass of glucose ($M_2$) = $(6 \times 12) + (12 \times 1) + (6 \times 16) = 180 \, \text{g mol}^{-1}$
  • Mass of solvent (Water, $H_2O$), $w_1 = 450 \, \text{g}$
  • Molar mass of water ($M_1$) = $(2 \times 1) + 16 = 18 \, \text{g mol}^{-1}$

Step 1: Calculating the Number of Moles

Applying the formula $n = \frac{w}{M}$ [Per the definition of Molar Mass]:

Moles of glucose ($n_2$):

$n_2 = \frac{25 \, \text{g}}{180 \, \text{g mol}^{-1}} \approx 0.1389 \, \text{mol}$

Moles of water ($n_1$):

$n_1 = \frac{450 \, \text{g}}{18 \, \text{g mol}^{-1}} = 25.0000 \, \text{mol}$

Step 2: Determining the Mole Fraction of the Solute

The mole fraction of the solute ($x_2$) is defined as the ratio of moles of solute to the total number of moles in the solution:

$x_2 = \frac{n_2}{n_1 + n_2}$

$x_2 = \frac{0.1389}{25 + 0.1389} = \frac{0.1389}{25.1389} \approx 0.005525$

Step 3: Calculating Vapour Pressure Lowering

Per Raoult’s Law [Relative Lowering of Vapour Pressure for non-volatile solutes]:

$\frac{P_1^\circ - P_1}{P_1^\circ} = x_2$

Where $P_1$ is the vapour pressure of the solution. Substituting the known values:

$\frac{17.535 - P_1}{17.535} = 0.005525$

Step 4: Algebraic Isolation of $P_1$

$17.535 - P_1 = 0.005525 \times 17.535$

$17.535 - P_1 = 0.09688$

$P_1 = 17.535 - 0.09688$

$P_1 = 17.43812 \, \text{mm Hg}$

Visual Representation of Vapour Pressure Depression

Concentration of Glucose Vapour Pressure P° = 17.535 P = 17.438

Final Solution: The vapour pressure of the glucose solution at 293 K is 17.438 mm Hg.


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