In which of the following `Delta H = Delta U `?

Options

A. `N_(2)(g) + 3 H_(2)(g) rightarrow 2NH_(3)(g)`
B. `C(s) + O_(2)(g) rightarrow CO_(2)(g)`
C. `PCI_(5)(g) rightarrow PCI_(3)(g) + CI_(2)(g)`
D. `CaCO_(3)(s) rightarrow CaO(s) + CO_(2)(g)`

Answer

To determine in which of the given reactions \( \Delta H = \Delta U \), we will analyze each reaction using the relationship between enthalpy change (\( \Delta H \)) and internal energy change (\( \Delta U \)):

\[ \Delta H = \Delta U + n_gRT \]

where \( n_g \) is the change in the number of moles of gas (moles of products - moles of reactants), \( R \) is the universal gas constant, and \( T \) is the temperature in Kelvin.

### Step-by-Step Solution:

1. **First Reaction:**
\[ \text{N}_2(g) + 3\text{H}_2(g) \rightarrow 2\text{NH}_3(g) \]
- **Calculate \( n_p \) (moles of products):** 2 moles of \( \text{NH}_3 \) (gaseous).
- **Calculate \( n_r \) (moles of reactants):** 4 moles (1 mole of \( \text{N}_2 \) + 3 moles of \( \text{H}_2 \)).
- **Calculate \( n_g \):**
\[ n_g = n_p - n_r = 2 - 4 = -2 \]
- **Substitute into the equation:**
\[ \Delta H = \Delta U + (-2)RT \Rightarrow \Delta H = \Delta U - 2RT \]
- **Conclusion:** \( \Delta H \neq \Delta U \).

2. **Second Reaction:**
\[ \text{C}(s) + \text{O}_2(g) \rightarrow \text{CO}_2(g) \]
- **Calculate \( n_p \):** 1 mole of \( \text{CO}_2 \) (gaseous).
- **Calculate \( n_r \):** 1 mole of \( \text{O}_2 \) (gaseous) + 1 mole of \( \text{C} \) (solid) = 1.
- **Calculate \( n_g \):**
\[ n_g = n_p - n_r = 1 - 1 = 0 \]
- **Substitute into the equation:**
\[ \Delta H = \Delta U + 0 \Rightarrow \Delta H = \Delta U \]
- **Conclusion:** \( \Delta H = \Delta U \).

3. **Third Reaction:**
\[ \text{PCl}_5(g) \rightarrow \text{PCl}_3(g) + \text{Cl}_2(g) \]
- **Calculate \( n_p \):** 2 moles (1 mole of \( \text{PCl}_3 \) + 1 mole of \( \text{Cl}_2 \)).
- **Calculate \( n_r \):** 1 mole of \( \text{PCl}_5 \) (gaseous).
- **Calculate \( n_g \):**
\[ n_g = n_p - n_r = 2 - 1 = 1 \]
- **Substitute into the equation:**
\[ \Delta H = \Delta U + 1RT \]
- **Conclusion:** \( \Delta H \neq \Delta U \).

4. **Fourth Reaction:**
\[ \text{CaCO}_3(s) \rightarrow \text{CaO}(s) + \text{CO}_2(g) \]
- **Calculate \( n_p \):** 1 mole of \( \text{CO}_2 \) (gaseous).
- **Calculate \( n_r \):** 1 mole of \( \text{CaCO}_3 \) (solid).
- **Calculate \( n_g \):**
\[ n_g = n_p - n_r = 1 - 0 = 1 \]
- **Substitute into the equation:**
\[ \Delta H = \Delta U + 1RT \]
- **Conclusion:** \( \Delta H \neq \Delta U \).

### Final Conclusion:
The only reaction in which \( \Delta H = \Delta U \) is the **second reaction**:

\[ \text{C}(s) + \text{O}_2(g) \rightarrow \text{CO}_2(g) \]