AIIMS 2005 Chemistry Equilibrium and Thermodynamics MCQ Question

To determine the standard enthalpy of formation ($\Delta H_f^\circ$) for the equilibrium reaction

$$\text{CaCO}_3(s) \rightleftharpoons \text{CaO}(s) + \text{CO}_2(g),$$

we can use the principles of chemical thermodynamics, specifically the relationship between the equilibrium constant and temperature.

Explanation of the Correct Answer (A)

The correct answer to the question is A) logₑ PCO₂ vs 1/T. This can be understood through the van 't Hoff equation, which relates the change in the equilibrium constant $K$ with temperature $T$:

$$\ln K = -\frac{\Delta H^\circ}{R} \cdot \frac{1}{T} + C,$$

where:

• $K$ is the equilibrium constant,
• $\Delta H^\circ$ is the standard enthalpy change,
• $R$ is the universal gas constant,
• $C$ is a constant.

For the given reaction, the equilibrium constant $K$ at a given temperature can be expressed in terms of the partial pressure of $\text{CO}_2$:

$$K = P_{CO_2}$$

Thus, we can rewrite the equation as:

$$\ln P_{CO_2} = -\frac{\Delta H^\circ}{R} \cdot \frac{1}{T} + C.$$

If we take the natural log of $P_{CO_2}$ and plot it against $\frac{1}{T}$, the slope of this line will be $-\frac{\Delta H^\circ}{R}$. Therefore, we can determine $\Delta H^\circ$ from the slope of the plot.

Why Other Options are Incorrect

B) logₑ PCO₂ vs T: This option would imply a direct relationship between the logarithm of the partial pressure and temperature, which does not correspond to the van 't Hoff equation. The equation directly relates $\ln K$ to $\frac{1}{T}$, not $T$. Therefore, this plot would not yield information regarding $\Delta H^\circ$.

C) ln PCO₂ vs logₑ T: In this case, the plot involves the natural logarithm of $P_{CO_2}$ versus the logarithm of temperature. This does not fit the form of the van 't Hoff equation and would not provide a linear relationship that expresses $\Delta H^\circ$ in a straightforward manner.

D) PCO₂ vs 1/T: This plot directly relates the partial pressure of $\text{CO}_2$ to $\frac{1}{T}$. However, this relationship is not linear. The expected nature of the relationship between $P_{CO_2}$ and $\frac{1}{T}$ does not allow for the direct extraction of $\Delta H^\circ$ as a slope, which is why this option is also incorrect.

Summary

To summarize, plotting $\log_e P_{CO_2}$ against $\frac{1}{T}$ (Option A) allows for the determination of the standard enthalpy of formation $\Delta H_f^\circ$ due to the linear relationship defined by the van 't Hoff equation. In contrast, the other options do not yield a usable linear form for determining $\Delta H^\circ$.