Question

What is the difference between $\Delta H$ and $\Delta U$ for reaction given below at 298 K?
(R = 8.314 $\text{J}\ \text{K}^{-1}\ \text{mol}^{-1}$)
$2\text{C}_6\text{H}_{6(\ell)} + 15\text{O}_{2(g)} \rightarrow 12\text{CO}_{2(g)} + 6\text{H}_2\text{O}_{(\ell)}$

• A. $-2.72\ \text{kJ}$
• B. $-7.43\ \text{kJ}$
• C. $-7.8\ \text{kJ}$
• D. $-3.72\ \text{kJ}$

Hint:

Always double-check the state symbols! A common trap is including the 6 moles of liquid water or the 2 moles of liquid benzene in your $\Delta n_g$ calculation, which will lead to an incorrect answer. Only count components marked with a $(g)$.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The problem requires calculating the mathematical difference between enthalpy change ($\Delta H$) and internal energy change ($\Delta U$) for the liquid benzene combustion reaction at a specified temperature.

Step 2: Key Formula or Approach:
The fundamental relationship connecting enthalpy and internal energy for a chemical reaction is given by the equation: $$\Delta H = \Delta U + \Delta n_g RT \implies \Delta H - \Delta U = \Delta n_g RT$$ where $\Delta n_g$ is the net change in the number of moles of gaseous components: $$\Delta n_g = \sum n_{\text{gaseous products}} - \sum n_{\text{gaseous reactants}}$$

Step 3: Detailed Explanation:
Let's first determine $\Delta n_g$ from the balanced equation. Remember to look closely at the physical state subscripts ($g$, $\ell$): Gaseous products: $12\ \text{mol}$ of $\text{CO}_{2(g)}$. (Note: $\text{H}_2\text{O}$ is liquid, so its coefficient is ignored). Gaseous reactants: $15\ \text{mol}$ of $\text{O}_{2(g)}$. (Note: $\text{C}_6\text{H}_6$ is liquid, so its coefficient is ignored). $$\Delta n_g = 12 - 15 = -3\ \text{mol}$$ Now, substitute the values of $\Delta n_g$, $R$, and $T$ into our formula: $$\Delta H - \Delta U = (-3\ \text{mol}) \times (8.314\ \text{J}\ \text{K}^{-1}\ \text{mol}^{-1}) \times (298\ \text{K})$$ $$\Delta H - \Delta U = -3 \times 8.314 \times 298 = -7432.716\ \text{J}$$ Convert Joules to kilojoules (kJ) by dividing by 1000: $$\Delta H - \Delta U = \frac{-7432.716}{1000} \approx -7.43\ \text{kJ}$$

Step 4: Final Answer:
The difference between $\Delta H$ and $\Delta U$ is $-7.43\ \text{kJ}$, corresponding to option (B).