Q.A solution is prepared by dissolving 394 g of a nonvolatile solute in 622 g of water. The vapour pressure of solution is found to be 30.74 mm Hg at 30 ⁰C. If vapour pressure of water at 30 ⁰C is 31.8 mm Hg, what is the molar mass of solute?

Relative lowering of vapour pressure depends on the number of dissolved particles; Na3PO4 dissociates into 4 ions, the most among the options.

Lowering of vapour pressure is a colligative property, proportional to the total particle concentration (i x molarity). At 0.1 M:

• Urea: i = 1 → 0.1
• BaCl2: i = 3 → 0.3
• Na2SO4: i = 3 → 0.3
• Na3PO4 → 3Na+ + PO4³-, i = 4 → 0.4

Na3PO4 gives the highest effective particle concentration, hence the lowest vapour pressure.

Both solutions contain the same number of moles of solute (0.2 mol) in the same amount of water, so their mole fractions of solute — and hence their relative lowering of vapour pressure — are equal. Correct option: (iv).

Concept. Relative lowering of vapour pressure (RLVP) is a colligative property: it depends on the number of solute particles, not their nature. By Raoult's law for a non-volatile solute,

$\frac{p^{0}-p_s}{p^{0}} = x_{\text{solute}} = \frac{n_{\text{solute}}}{n_{\text{solute}}+n_{\text{solvent}}}$

Why compare moles. Since both solutions use the same solvent quantity (1 L water), the one with more solute moles has the larger RLVP. So we just compare solute moles.

Steps.

• Urea: $n = \dfrac{12}{60} = 0.2\ \text{mol}$ (molar mass of urea, $\text{NH}_2\text{CONH}_2 = 60\ \text{g mol}^{-1}$).
• Sucrose: $n = \dfrac{68.4}{342} = 0.2\ \text{mol}$ (molar mass of sucrose, $\text{C}_{12}\text{H}_{22}\text{O}_{11} = 342\ \text{g mol}^{-1}$).
• Equal solute moles in equal solvent $\Rightarrow$ equal mole fraction $\Rightarrow$ equal RLVP.

Both urea and sucrose are non-electrolytes (van't Hoff factor $i = 1$), so no dissociation/association changes the count.

Vapour pressure is not a colligative property; the colligative one is the RELATIVE LOWERING of vapour pressure.

A colligative property depends only on the number of solute particles present, not on their nature. The four colligative properties are:

• Relative lowering of vapour pressure
• Depression in freezing point
• Elevation in boiling point
• Osmotic pressure

Options (b), (c) and (d) are all colligative. "Vapour pressure" on its own is a bulk property of the solution and is not colligative — only its RELATIVE LOWERING (a fractional change) qualifies.

Adding a non-volatile solute to a solvent reduces the fraction of solvent molecules at the surface that can escape into vapour, so the vapour pressure of the solution is lower than that of the pure solvent.

According to Raoult's law, for a solution of a non-volatile solute in a volatile solvent, the vapour pressure of the solution (p) is p = p° × x_solvent, where p° is the vapour pressure of pure solvent and x_solvent < 1 is the mole fraction of solvent (since some of the moles are now solute). Because x_solvent is less than 1, p < p°, i.e. the vapour pressure is lowered on dissolving the solute.

This lowering of vapour pressure is the root cause of the other colligative properties: it raises the boiling point (option d is therefore wrong — boiling point rises, it doesn't fall) and it lowers the freezing point (option c is wrong — freezing point falls, not rises).

Colligative properties depend only on the NUMBER of solute particles, not their identity: relative lowering of vapour pressure, elevation in boiling point, depression in freezing point, and osmotic pressure. Boiling point never falls when a non-volatile solute is added, so 'depression in boiling point' is not a real colligative property.

When a non-volatile solute is dissolved in a solvent, the vapour pressure of the solution is lowered (Raoult's law). A lower vapour pressure means the solution needs to be heated to a HIGHER temperature to reach atmospheric pressure and boil, so the boiling point RISES (elevation), never falls.

1. relative lowering of vapour pressure — genuine colligative property.
2. elevation in boiling point — genuine colligative property.
3. depression in freezing point — genuine colligative property.
4. depression in boiling point — this does not occur for a non-volatile solute; it is not a colligative property.

The relative lowering of vapour pressure equals the mole fraction of the solute in the solution. Given the lowering is 0.0225, the mole fraction of the non-volatile solute is directly 0.0225, which corresponds to option (D).

Raoult's law states directly that the relative lowering of vapour pressure equals the solute's mole fraction.

According to Raoult's law, for a solution of a non-volatile solute in a volatile solvent, the relative lowering of vapour pressure is given by:

$\frac{p°_{solvent} - p_{solution}}{p°_{solvent}} = x_{solute}$

This quantity equals the mole fraction of the solute, not the molarity, molality, or mole fraction of the solvent.

Raoult's law directly equates the relative lowering of vapour pressure to the solute's mole fraction.

For a solution of a non-volatile solute in a volatile solvent, Raoult's law states:

(P0 - Ps)/P0 = x(solute)

where P0 is the vapour pressure of the pure solvent, Ps is the vapour pressure of the solution, and x(solute) is the mole fraction of the solute. This relation is independent of the nature of the solute (only depends on the number of solute particles), which is what makes vapour pressure lowering a colligative property.

A non-volatile solute reduces the fraction of solvent molecules at the surface, lowering the rate of escape into vapour and hence the vapour pressure.

Vapour pressure arises from solvent molecules escaping from the liquid surface into the vapour phase. When a non-volatile solute is dissolved, some solute particles occupy positions at (or near) the surface, effectively reducing the fraction/number of solvent molecules present per unit area of the surface. This lowers the rate at which solvent molecules can escape into the vapour phase, so a smaller vapour pressure is needed to establish liquid–vapour equilibrium — i.e., the vapour pressure of the solution is lower than that of the pure solvent.