Choose the correct statement with respect to the vapour pressure of a liquid among the following:

The vapour pressure of a liquid is the pressure exerted by its vapour when it is in equilibrium with the liquid phase at a given temperature. To understand how vapour pressure changes with temperature, we use the Clausius-Clapeyron equation. This equation describes the relationship between vapour pressure and temperature.

The Clausius-Clapeyron equation is:

$$\ln \left( \frac{P_2}{P_1} \right) = -\frac{\Delta H_{\text{vap}}}{R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right)$$

where $$P_1$$ and $$P_2$$ are the vapour pressures at temperatures $$T_1$$ and $$T_2$$ (in Kelvin), $$\Delta H_{\text{vap}}$$ is the enthalpy of vaporization, and $$R$$ is the gas constant.

We can rearrange this equation to express vapour pressure as a function of temperature. Let $$P$$ be the vapour pressure at temperature $$T$$. The equation becomes:

$$\ln P = -\frac{\Delta H_{\text{vap}}}{R} \cdot \frac{1}{T} + C$$

where $$C$$ is a constant. This shows that $$\ln P$$ is linearly related to $$\frac{1}{T}$$.

To find how $$P$$ depends on $$T$$, we solve for $$P$$ by exponentiating both sides:

$$\ln P = -\frac{A}{T} + B$$

where $$A = \frac{\Delta H_{\text{vap}}}{R}$$ and $$B = C$$ are constants. Exponentiating:

$$P = e^{B} \cdot e^{-\frac{A}{T}}$$

This simplifies to:

$$P = k \cdot e^{-\frac{A}{T}}$$

where $$k = e^B$$ is a constant.

This equation reveals that vapour pressure $$P$$ is an exponential function of $$\frac{1}{T}$$. As temperature $$T$$ increases, $$\frac{1}{T}$$ decreases. Since $$A$$ is positive (as $$\Delta H_{\text{vap}} > 0$$ and $$R > 0$$), the exponent $$-\frac{A}{T}$$ becomes less negative (i.e., increases) as $$T$$ increases. Therefore, $$e^{-\frac{A}{T}}$$ increases, causing $$P$$ to increase.

However, the increase is not linear. The term $$e^{-\frac{A}{T}}$$ depends on $$T$$ in the denominator of the exponent, resulting in a non-linear relationship. Specifically, $$P$$ increases rapidly with increasing $$T$$, which is characteristic of exponential growth. For example, at lower temperatures, the increase in vapour pressure is slower, but it accelerates as temperature rises.

Now, evaluating the options:

• Option A states that vapour pressure increases non-linearly with increasing temperature, which matches our analysis.
• Option B states it decreases non-linearly, which is incorrect because vapour pressure increases.
• Option C states it decreases linearly, which is also incorrect.
• Option D states it increases linearly, but our derivation shows it is exponential, not linear.

Hence, the correct answer is Option A.