For the reaction
`N_(2)(g) + 3H_(2)(g) rarr 2NH_(2)(g)`
Which of the following is correct?

Options

A. `DeltaH = DeltaU`
B. `DeltaH gt DeltaU`
C. `DeltaH lt DeltaU`
D. `DeltaH = 2 DeltaU`

Explanation

To solve the problem, we need to determine the relationship between the change in enthalpy (ΔH) and the change in internal energy (ΔU) for the given reaction:

\[ N_2(g) + 3H_2(g) \rightarrow 2NH_2(g) \]

### Step-by-Step Solution:

1. **Identify the Reaction**:
The reaction involves nitrogen gas (N₂) and hydrogen gas (H₂) reacting to form ammonia (NH₂).

2. **Apply the First Law of Thermodynamics**:
According to the first law of thermodynamics, the relationship between enthalpy (ΔH) and internal energy (ΔU) is given by:
\[ ΔH = ΔU + PΔV \]

3. **Calculate ΔV (Change in Volume)**:
For ideal gases, the change in volume (ΔV) can be related to the change in the number of moles (Δn) of gas:
\[ PΔV = ΔnRT \]
where R is the universal gas constant and T is the temperature.

4. **Determine Δn (Change in Number of Moles)**:
Calculate Δn using the formula:
\[ Δn = \text{(moles of products)} - \text{(moles of reactants)} \]
- Moles of products (NH₂): 2
- Moles of reactants (N₂ + 3H₂): 1 + 3 = 4
\[ Δn = 2 - 4 = -2 \]

5. **Substitute Δn into the Equation**:
Substitute Δn into the equation for ΔH:
\[ ΔH = ΔU + (-2)RT \]
This simplifies to:
\[ ΔH = ΔU - 2RT \]

6. **Rearrange the Equation**:
Rearranging gives us:
\[ ΔU = ΔH + 2RT \]

7. **Conclusion**:
From the final equation, we can conclude that:
\[ ΔU > ΔH \]
Therefore, the correct relationship between ΔH and ΔU for the given reaction is that ΔU is greater than ΔH.

### Final Answer:
The correct relation is \( ΔU = ΔH + 2RT \), which implies \( ΔU > ΔH \).

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