For which of the following reactions, `DeltaH = DeltaU` ?

Options

A. `H_2(g) + l_2(g) rightarrow 2Hl(g)`
B. `N_2(g) + 3H_2(g) rightarrow 2NH_3(g)`
C. `C(s) + H_2O(g) rightarrow CO(g) + H_2(g)`
D. `CH_4(g) + H_2O(g) rightarrow CO(g) + 3H_2(g)`

Answer

To determine for which of the given reactions \( \Delta H = \Delta U \), we need to understand the relationship between the change in enthalpy (\( \Delta H \)) and the change in internal energy (\( \Delta U \)). The relationship can be expressed as:

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

where \( \Delta n_g \) is the change in the number of moles of gas (moles of gaseous products minus moles of gaseous reactants) and \( R \) is the universal gas constant.

From this equation, we can infer:
- If \( \Delta n_g = 0 \), then \( \Delta H = \Delta U \).
- If \( \Delta n_g < 0 \), then \( \Delta H < \Delta U \).
- If \( \Delta n_g > 0 \), then \( \Delta H > \Delta U \).

Now, let's analyze the reactions to find \( \Delta n_g \) for each:

1. **Reaction 1:** \( N_2 + 3H_2 \rightarrow 2NH_3 \)
- Reactants: 1 mole of \( N_2 \) (gas) + 3 moles of \( H_2 \) (gas) = 4 moles of gas
- Products: 2 moles of \( NH_3 \) (gas) = 2 moles of gas
- \( \Delta n_g = 2 - 4 = -2 \)

2. **Reaction 2:** \( C + 2H_2 + O_2 \rightarrow H_2O \)
- Reactants: 1 mole of \( C \) (solid) + 2 moles of \( H_2 \) (gas) + 1 mole of \( O_2 \) (gas) = 3 moles of gas
- Products: 1 mole of \( H_2O \) (gas) = 1 mole of gas
- \( \Delta n_g = 1 - 3 = -2 \)

3. **Reaction 3:** \( 1 + 3 \rightarrow 4 \)
- This reaction is not clearly defined, but assuming it has gaseous reactants and products, we can analyze it similarly.

4. **Reaction 4:** \( 1 + 3 \rightarrow 1 \)
- Again, without specific details, we cannot calculate \( \Delta n_g \).

From the analysis of the first two reactions, we see that both have \( \Delta n_g < 0 \), which implies \( \Delta H < \Delta U \).

However, we are looking for a reaction where \( \Delta n_g = 0 \).

If we analyze the first reaction:
- \( N_2 + 3H_2 \rightarrow 2NH_3 \) gives \( \Delta n_g = 0 \) (if we consider only gaseous components).

Thus, the answer is:

**The reaction for which \( \Delta H = \Delta U \) is:**
- **Option 1: \( N_2 + 3H_2 \rightarrow 2NH_3 \)**