What is the difference between heat of reaction of constant pressure and that of constant volume for the reaction given below at $27^{\circ}C$ ? $2C_{6}H_{6}(l)+15O_{2}(g)\rightarrow12CO_{2}(g) +6H_{2}O(l)$

Question

Accepted Answer

To determine the difference between the heat of reaction at constant pressure ($\Delta H$) and constant volume ($\Delta U$) for the given reaction at $27^{\circ}C$, we can use the relationship between these two thermodynamic quantities:

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

where:
- $\Delta n$ is the change in the number of moles of gas,
- $R$ is the ideal gas constant ($8.314 \, \text{J/mol K}$),
- $T$ is the temperature in Kelvin.

**Step-by-step Explanation:**

1. **Calculate $\Delta n$:**
   - In the given reaction: 
     $$
     2C_{6}H_{6}(l) + 15O_{2}(g) \rightarrow 12CO_{2}(g) + 6H_{2}O(l)
     $$
   - Initial moles of gas = 15 (from $O_2$)
   - Final moles of gas = 12 (from $CO_2$)
   - $\Delta n = \text{moles of gaseous products} - \text{moles of gaseous reactants} = 12 - 15 = -3$

2. **Convert temperature to Kelvin:**
   - $T = 27^{\circ}C = 27 + 273.15 = 300.15 \, \text{K}$

3. **Calculate the difference $\Delta H - \Delta U$:**
   $$
   \Delta H - \Delta U = \Delta n \cdot R \cdot T = (-3) \cdot 8.314 \, \text{J/mol K} \cdot 300.15 \, \text{K}
   $$
   $$
   \Delta H - \Delta U = -3 \cdot 8.314 \cdot 300.15 = -7481.5 \, \text{J/mol} \approx -7.48 \, \text{kJ/mol}
   $$

**Conclusion:**

- The calculated difference $\Delta H - \Delta U$ is approximately $-7.48 \, \text{kJ/mol}$, which matches Option 1. Therefore, Option 1 is correct.

**Explanation for Incorrect Options:**

- **Option 2 (7.48 kJ):** This option suggests a positive difference, which is incorrect because $\Delta n$ is negative, leading to a negative value for $\Delta H - \Delta U$.

- **Option 3 (0.673 kJ):** This value is significantly smaller than the calculated difference and does not match the magnitude of the change in moles of gas.

- **Option 4 (-0.673 kJ):** This value is incorrect as it does not match the calculated difference of $-7.48 \, \text{kJ/mol}$ and is too small in magnitude.

Thus, the correct answer is indeed Option 1, $-7.48 \, \text{kJ/mol}$.