AIIMS 2006 Chemistry Enthalpy and Internal Energy MCQ Question

To solve the problem regarding the enthalpy change (ΔH) and the internal energy change (ΔU) for the reaction:

$$\text{N}_2 (g) + 3\text{H}_2 (g) \rightarrow 2\text{NH}_3 (g)$$

with $\Delta H = -92.38 \, \text{kJ}$ at $298 \, \text{K}$, we will use the following relationship between enthalpy change (ΔH) and internal energy change (ΔU):

$$\Delta H = \Delta U + \Delta n \cdot R \cdot T$$

Where:

• $\Delta n$ is the change in the number of moles of gas.
• $R$ is the universal gas constant, approximately $8.314 \, \text{J/(mol K)}$ or $0.008314 \, \text{kJ/(mol K)}$.
• $T$ is the temperature in Kelvin.

Step 1: Calculate Δn

In the given reaction:

• Reactants: $1$ mole of $\text{N}_2$ + $3$ moles of $\text{H}_2$ = $4$ moles of gas.
• Products: $2$ moles of $\text{NH}_3$ = $2$ moles of gas.

Thus, the change in the number of moles of gas ($\Delta n$) is:

$$\Delta n = \text{moles of products} - \text{moles of reactants} = 2 - 4 = -2$$

Step 2: Calculate ΔU

Now, we can substitute the values into the formula for ΔH:

1. Substitute $\Delta n = -2$, $R = 0.008314 \, \text{kJ/(mol K)}$, and $T = 298 \, \text{K}$ into the equation:

$$\Delta H = \Delta U + \Delta n \cdot R \cdot T$$

1. Calculate the term $\Delta n \cdot R \cdot T$:

$$\Delta n \cdot R \cdot T = -2 \cdot 0.008314 \, \text{kJ/(mol K)} \cdot 298 \, \text{K}$$

Calculating this gives:

$$\Delta n \cdot R \cdot T = -2 \cdot 0.008314 \cdot 298 \approx -4.97 \, \text{kJ}$$

1. Now, substituting this back into the ΔH equation:

$$-92.38 \, \text{kJ} = \Delta U - 4.97 \, \text{kJ}$$

1. Rearranging to solve for ΔU:

$$\Delta U = -92.38 \, \text{kJ} + 4.97 \, \text{kJ} = -87.41 \, \text{kJ}$$

Conclusion

The calculated internal energy change (ΔU) is approximately:

$$\Delta U \approx -87.41 \, \text{kJ}$$

The closest option provided in the question is B) -87.42 kJ, which is thus the correct answer.

Explanation of Other Options:

• A) -92.38 kJ: This represents ΔH, not ΔU; hence it is incorrect.
• C) -97.34 kJ: This value does not align with our calculations and is incorrect.
• D) -89.9 kJ: This value is also not close to our calculated ΔU and is incorrect.

In summary, the correct answer is B) -87.42 kJ, derived from the relationship between enthalpy and internal energy considering the change in the number of gas moles in the reaction.