For which of the following reactions, `DeltaH` is equal to `DeltaU` ?

Options

A. `N_(2) (g) + 3 H_(2) (g) to 2 N H _(3) (g)`
B. `2 H I (g) to H_(2) (g) + I_(2) (g)`
C. `2 N O_(2) (g) to N_(2) O_(4) (g)`
D. `2 S O _(2) + O_(2) (g) to 2 S O _(3) (g)`

Explanation

To determine for which of the following reactions the change in enthalpy (ΔH) is equal to the change in internal energy (ΔU), we can use the relationship between these two thermodynamic quantities:

\[ \Delta U = \Delta H - \Delta N_G RT \]

Where:
- ΔU = change in internal energy
- ΔH = change in enthalpy
- ΔN_G = change in the number of moles of gas (moles of products - moles of reactants)
- R = universal gas constant
- T = temperature in Kelvin

For ΔH to equal ΔU, the term ΔN_G must be equal to zero. This means that the number of moles of gaseous products must equal the number of moles of gaseous reactants.

### Step-by-Step Solution:

1. **Identify the Reactions**: List the reactions provided in the options.

2. **Calculate ΔN_G for Each Reaction**:
- For each reaction, count the number of moles of gaseous products and subtract the number of moles of gaseous reactants.
- \( \Delta N_G = \text{(moles of gaseous products)} - \text{(moles of gaseous reactants)} \)

3. **Check Each Reaction**:
- If \( \Delta N_G = 0 \), then \( \Delta H = \Delta U \).
- If \( \Delta N_G \neq 0 \), then \( \Delta H \neq \Delta U \).

4. **Evaluate Each Option**:
- **Option 1**: Calculate \( \Delta N_G \).
- **Option 2**: Calculate \( \Delta N_G \).
- **Option 3**: Calculate \( \Delta N_G \).
- **Option 4**: Calculate \( \Delta N_G \).

5. **Determine the Correct Option**:
- Identify which option has \( \Delta N_G = 0 \).

### Example Calculation:
- **Option 1**: Assume it has 2 moles of products and 4 moles of reactants.
\[ \Delta N_G = 2 - 4 = -2 \quad (\Delta H \neq \Delta U) \]

- **Option 2**: Assume it has 2 moles of products and 2 moles of reactants.
\[ \Delta N_G = 2 - 2 = 0 \quad (\Delta H = \Delta U) \]

- **Option 3**: Assume it has 1 mole of product and 2 moles of reactants.
\[ \Delta N_G = 1 - 2 = -1 \quad (\Delta H \neq \Delta U) \]

- **Option 4**: Assume it has 2 moles of products and 3 moles of reactants.
\[ \Delta N_G = 2 - 3 = -1 \quad (\Delta H \neq \Delta U) \]

### Conclusion:
From the calculations, only **Option 2** has \( \Delta N_G = 0 \), which means for this reaction, \( \Delta H = \Delta U \).