Solutions and Phase Equilibria Questions and Answers

Answer: D

The incident is accounted for by (II) and (III).

The 1986 disaster at Lake Nyos in Cameroon was a limnic eruption. Volcanic CO₂ seeping into the bottom of the deep, stratified lake dissolved in the cold, high-pressure bottom water over many years (II); by Henry's law, high pressure at depth allows large amounts of gas to stay dissolved. A disturbance, possibly wind, a landslide or cooling of surface water, overturned the layers (III). As deep water rose, the pressure fell, the dissolved CO₂ came out of solution explosively, and the dense gas cloud flowed down the valleys, killing about 1,700 people.

The gas accumulated in the bottom layers, not the upper ones, so (I) is wrong, and the relevant law is Henry's law for gas solubility, not Raoult's law for vapour pressure, so (IV) is wrong.

Answer: B

Sublimation corresponds to I → III.

In this pressure-temperature diagram region I, at low temperature beside the steep fusion line, is the solid; region II, above the vaporisation curve, is the liquid; and region III, at high temperature and low pressure, is the gas. The three lines meet at the triple point.

Sublimation is the direct change from solid to gas, crossing the short sublimation line below the triple point, so it is I → III. I → II is melting, II → III is vaporisation, and III → II is condensation.

Answer: C

The explanation is that toluene has a lower heat of vaporization than water.

The Clausius-Clapeyron equation links how fast vapour pressure rises with temperature to the enthalpy of vaporization:

\[ \ln P = -\frac{\Delta H_{vap}}{RT} + C \]

A small ΔH_vap means a shallow curve. Toluene starts with a higher vapour pressure than water at 25 °C, yet reaches 1 atm (its boiling point) at a higher temperature, 110.6 °C against 100 °C. Its vapour pressure must therefore rise more slowly with temperature, which requires the smaller ΔH_vap: about 33 kJ/mol for toluene against about 41 kJ/mol for hydrogen-bonded water.

The first option states the opposite of the usual trend (a higher vapour pressure normally goes with a lower boiling point). A higher molar mass would lower the vapour pressure, not raise it, and the density of the vapour does not govern evaporation.

Answer: B

The total pressure is 2P_A°P_B°/(P_A° + P_B°).

By Raoult's law, p_A = x_AP_A° and p_B = x_BP_B°. Setting them equal, with x_B = 1 − x_A:

\[ \begin{aligned} x_{A}P_{A}^{o} &= (1-x_{A})P_{B}^{o} \\ x_{A} &= \frac{P_{B}^{o}}{P_{A}^{o}+P_{B}^{o}} \\ P &= 2x_{A}P_{A}^{o} = \frac{2P_{A}^{o}P_{B}^{o}}{P_{A}^{o}+P_{B}^{o}} \end{aligned} \]

The first option is the partial pressure of one component, half the total. The last two are ratios of pressures, dimensionless numbers, so they cannot be a pressure at all.

Answer: A

The boiling point at 76 mm is close to 167 K.

Use the Clausius-Clapeyron equation between the normal boiling point (760 mm, 200 K) and the new pressure (76 mm, T₂):

\[ \begin{aligned} \ln\frac{p_{2}}{p_{1}} &= -\frac{\Delta H}{R} \\ &\quad \times \left(\frac{1}{T_{2}}-\frac{1}{T_{1}}\right) \end{aligned} \]

\[ \begin{aligned} \ln 0.1 &= -2.303 \\ -2.303 &= -2303\left(\frac{1}{T_{2}}-\frac{1}{200}\right) \\ \frac{1}{T_{2}} &= 0.005 + 0.001 = 0.006 \\ T_{2} &= 166.7\ \text{K} \end{aligned} \]

Lower pressure must lower the boiling point, so 200 K (unchanged) and 440 K (higher) are impossible, and 20 K is what you get by scaling the temperature in proportion to the pressure, which the equation does not allow.

Answer: B

The ratio is bc/ac (option B), by the lever rule.

On a temperature-composition diagram, when the overall composition c lies in the two-phase region, the mixture splits into the two layers whose compositions are a and b. The amounts of the two layers are in inverse ratio to the distances of their compositions from c:

\[ \frac{n_a}{n_b} = \frac{bc}{ac} \]

The nearer the point is to a, the more of layer a there is, so the quantity of layer a relative to layer b is the longer lever bc over the shorter ac. The stem words this as the composition of layer a relative to layer b, and the key takes it as this ratio of amounts. ac/bc is the inverse, and ab, the distance between the two layers, does not enter the ratio, so C and D are wrong. The diagram is hand drawn and marks a and b on the composition axis, so the distances are read as composition differences.