Solutions — Practice Questions with Answers

Raoult's law states that the vapour pressure of a solvent above a solution is directly proportional to its mole fraction in the solution: $P_{solution} = x_{solvent}\times P^0_{solvent}$, where $P^0_{solvent}$ is the vapour pressure of the pure solvent.

Since adding a non-volatile solute reduces the solvent's mole fraction (which must be less than 1), the solution's vapour pressure is correspondingly lower than that of the pure solvent.

Colligative properties are properties of a solution that depend only on the number (concentration) of solute particles present, not on their specific chemical identity or nature.

The four main colligative properties are: (1) relative lowering of vapour pressure, (2) elevation of boiling point, (3) depression of freezing point, and (4) osmotic pressure.

Adding a non-volatile solute lowers the solvent's vapour pressure at any given temperature (per Raoult's law). Since boiling occurs when vapour pressure equals atmospheric pressure, this lowered vapour pressure means the solution must be heated to a higher temperature than the pure solvent for its vapour pressure to reach atmospheric pressure — this is precisely why boiling point elevation occurs for solutions containing non-volatile solutes.

Osmotic pressure is the minimum external pressure that must be applied to a solution to just prevent the net inward flow of solvent molecules through a semi-permeable membrane separating it from pure solvent (i.e., to stop osmosis).

Van't Hoff equation: $\pi = CRT$, where $\pi$ is osmotic pressure, $C$ is molar concentration, $R$ is the gas constant, and $T$ is absolute temperature — analogous in form to the ideal gas equation.

The van't Hoff factor ($i$) is the ratio of the actual (experimentally observed) colligative property value to the theoretically calculated value assuming no dissociation or association of the solute — $i = \dfrac{\text{observed value}}{\text{calculated value (no dissociation/association)}}$.

For solutes that dissociate (e.g., ionic compounds splitting into ions), $i > 1$, since more actual particles are present than the simple formula unit count would suggest. For solutes that associate (e.g., molecules pairing up), $i < 1$, since fewer independent particles are effectively present than the simple molecular count would suggest.

An ideal solution obeys Raoult's law across the entire concentration range, typically because the solute-solvent intermolecular interactions are essentially similar in strength to the solute-solute and solvent-solvent interactions — no significant enthalpy or volume change occurs on mixing.

A non-ideal solution deviates from Raoult's law (either positively or negatively), because solute-solvent interactions differ significantly (either weaker or stronger) from solute-solute and solvent-solvent interactions — typically accompanied by a non-zero enthalpy and/or volume change on mixing.

NaCl is an ionic compound that, upon dissolving in water, dissociates essentially completely into its constituent ions: $NaCl \rightarrow Na^+ + Cl^-$.

Since each formula unit of NaCl produces two independent particles (one $Na^+$ ion and one $Cl^-$ ion) in solution, the effective particle concentration is approximately double what the simple formula unit concentration would suggest — this is precisely why the van't Hoff factor for NaCl in dilute solution is close to 2, reflecting this near-complete dissociation into two ions per formula unit.

Isotonic solutions are two solutions that have the same osmotic pressure as each other — when separated by a semi-permeable membrane, no net movement of solvent occurs between them.

Medical importance: Intravenous fluids administered to patients must be isotonic with blood plasma/cells — if a fluid were hypotonic (lower osmotic pressure than cells), water would flow into cells, potentially causing them to swell and burst; if hypertonic (higher osmotic pressure), water would flow out of cells, causing them to shrink (crenate) — both outcomes can cause serious cellular damage, which is precisely why isotonic solutions are essential for safe intravenous fluid administration.

Colligative properties (like freezing point depression or boiling point elevation) are directly related to the molar concentration (and hence the number of moles) of solute present, via known constants ($K_f$ or $K_b$) specific to the solvent used.

By experimentally measuring a colligative property change for a solution prepared with a known mass of solute in a known mass/volume of solvent, the number of moles of solute present can be calculated (using the appropriate colligative property equation) — since moles = mass/molar mass, and both mass and moles are now known, the unknown molar mass of the solute can be readily determined by rearranging this relationship.

Positive deviation occurs when solute-solvent interactions are weaker than the solute-solute and solvent-solvent interactions present in the pure components — this weaker mutual attraction increases the escaping tendency of both components, resulting in a higher observed vapour pressure than Raoult's law would predict.

Example: A mixture of ethanol and acetone shows positive deviation, since the ethanol-acetone interactions (via disrupted hydrogen bonding compared to pure ethanol) are weaker than the interactions present in the pure components alone, resulting in greater-than-expected vapour pressure for the mixture.

Moles of glucose (solute) $= 18/180 = 0.1\text{ mol}$; Moles of water (solvent) $= 90/18 = 5\text{ mol}$

Mole fraction of solute: $x_{solute} = \dfrac{0.1}{0.1+5} = \dfrac{0.1}{5.1} \approx 0.0196$

Relative lowering of vapour pressure $= x_{solute}$ (by Raoult's law, for dilute solutions)

Relative lowering $\approx \mathbf{0.0196}$ (or 1.96%)

Molality $= \dfrac{\text{moles of solute}}{\text{mass of solvent in kg}} = \dfrac{6/60}{0.2} = \dfrac{0.1}{0.2} = 0.5\text{ mol/kg}$

$\Delta T_f = K_f\times m = 1.86\times0.5$

$\Delta T_f = \mathbf{0.93\text{ K}}$

Moles of sucrose $= 34.2/342 = 0.1\text{ mol}$

Molar concentration $C = \dfrac{0.1\text{ mol}}{1\text{ L}} = 0.1\text{ mol/L}$

$\pi = CRT = 0.1\times0.0821\times300$

$\pi = \mathbf{2.463\text{ atm}}$

Molality $m = 0.1\text{ mol/kg}$ (since 0.1 mol dissolved in 1 kg water)

Theoretical (calculated, no dissociation) $\Delta T_f = K_f\times m = 1.86\times0.1 = 0.186\text{ K}$

Van't Hoff factor: $i = \dfrac{\text{observed }\Delta T_f}{\text{calculated }\Delta T_f} = \dfrac{0.348}{0.186}$

$i \approx \mathbf{1.87}$ (close to the ideal value of 2 for complete dissociation into $Na^+$ and $Cl^-$, with some deviation due to ionic interactions in real solution)

Molality $= \dfrac{12/60}{0.2} = \dfrac{0.2}{0.2} = 1\text{ mol/kg}$

$\Delta T_b = K_b\times m = 0.52\times1$

$\Delta T_b = \mathbf{0.52\text{ K}}$

Calculated (undissociated) $\Delta T_f = K_f\times m = 1.86\times0.5 = 0.93\text{ K}$

Van't Hoff factor: $i = \dfrac{1.033}{0.93} \approx 1.111$

For dissociation into $n=2$ ions with degree of dissociation $\alpha$: $i = 1+\alpha(n-1) = 1+\alpha$

$1.111 = 1+\alpha \Rightarrow \alpha = 0.111$

Degree of dissociation $\alpha \approx 0.111$ (11.1%)

Molality: $\Delta T_f = K_f\times m \Rightarrow m = \dfrac{\Delta T_f}{K_f} = \dfrac{0.372}{1.86} = 0.2\text{ mol/kg}$

Moles of solute $= m\times(\text{mass of solvent in kg}) = 0.2\times0.1 = 0.02\text{ mol}$

Molar mass $= \dfrac{\text{mass}}{\text{moles}} = \dfrac{2.5}{0.02}$

Molar mass $= \mathbf{125\text{ g/mol}}$

Since $\pi = CRT$, at the same temperature, $\pi \propto C$

$\dfrac{\pi_1}{\pi_2} = \dfrac{C_1}{C_2} = \dfrac{0.1}{0.2} = \mathbf{\dfrac{1}{2}}$

The solution with 0.2 M concentration has twice the osmotic pressure of the 0.1 M solution.

Total moles $= 5+45=50$

Mole fraction of solute: $x_{solute} = 5/50 = 0.1$

Mole fraction of solvent: $x_{solvent} = 45/50 = 0.9$

Relative lowering of vapour pressure $= x_{solute} = \mathbf{0.1}$ (10%)

For an electrolyte solution, the van't Hoff equation is modified: $\pi = iCRT$

$\pi = 3\times0.01\times0.0821\times300$

$\pi = 3\times0.2463$

$\pi = \mathbf{0.7389\text{ atm}}$