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 following reactions \( \Delta H = \Delta U \), we need to analyze the relationship between the change in enthalpy (\( \Delta H \)) and the change in internal energy (\( \Delta U \)). The relationship is given by the equation:

\[ \Delta H = \Delta U + \Delta n_g RT \]

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

For \( \Delta H \) to equal \( \Delta U \), the term \( \Delta n_g RT \) must be zero, which implies that \( \Delta n_g = 0 \). This means there is no change in the number of moles of gas during the reaction.

### Step-by-Step Solution:

1. **Identify the Reactions**: We need to analyze each reaction provided to determine the change in the number of moles of gas (\( \Delta n_g \)).

2. **Calculate \( \Delta n_g \) for Each Reaction**:
- **Reaction 1**: \( N_2 + 3H_2 \rightarrow 2NH_3 \)
- Reactants: 1 mole of \( N_2 \) + 3 moles of \( H_2 \) = 4 moles of gas
- Products: 2 moles of \( NH_3 \)
- \( \Delta n_g = 2 - 4 = -2 \) (not equal to 0)

- **Reaction 2**: \( C + O_2 \rightarrow CO_2 \)
- Reactants: 1 mole of \( O_2 \) (carbon is solid and does not count)
- Products: 1 mole of \( CO_2 \)
- \( \Delta n_g = 1 - 1 = 0 \) (equal to 0)

- **Reaction 3**: \( PCl_5 \rightarrow PCl_3 + Cl_2 \)
- Reactants: 1 mole of \( PCl_5 \)
- Products: 1 mole of \( PCl_3 \) + 1 mole of \( Cl_2 \) = 2 moles of gas
- \( \Delta n_g = 2 - 1 = 1 \) (not equal to 0)

- **Reaction 4**: \( CaCO_3 \rightarrow CaO + CO_2 \)
- Reactants: 1 mole of \( CaCO_3 \) (solid, does not count)
- Products: 1 mole of \( CO_2 \) (gas)
- \( \Delta n_g = 1 - 0 = 1 \) (not equal to 0)

3. **Conclusion**: The only reaction where \( \Delta n_g = 0 \) is the second reaction \( C + O_2 \rightarrow CO_2 \). Therefore, in this case, \( \Delta H = \Delta U \).

### Final Answer:
The reaction in which \( \Delta H = \Delta U \) is the second reaction: \( C + O_2 \rightarrow CO_2 \).