If `DeltaH` is the change in enthylpy and `DeltaU`, the change in internal energy accompanying a gaseous reactant then

Options

A. `DeltaH` is always greater than `DeltaE`
B. `Delta H lt DeltH` only if the number of moles of the products is greater than the number of moles of the reactants
C. `DeltaH` is always less than `DeltaE`
D. `Delta H lt Delta E` only if the number of moles of products is less than the number of moles of the reactants

Answer

To solve the problem regarding the relationship between the change in enthalpy (ΔH) and the change in internal energy (ΔU) for a gaseous reactant, we can follow these steps:

### Step 1: Understand the relationship between ΔH and ΔU
The relationship between the change in enthalpy (ΔH) and the change in internal energy (ΔU) is given by the equation:
\[ \Delta H = \Delta U + \Delta n_g RT \]
where:
- Δn_g = change in the number of moles of gas (nP - nR)
- R = universal gas constant
- T = temperature in Kelvin

### Step 2: Analyze the three possible conditions for Δn_g
1. **Δn_g < 0**: This occurs when the number of moles of reactants (nR) is greater than the number of moles of products (nP). In this case, ΔH can be expressed as:
\[ \Delta H = \Delta U - |\Delta n_g| RT \]
Here, since we are subtracting a positive term from ΔU, it follows that:
\[ \Delta H < \Delta U \]

2. **Δn_g > 0**: This occurs when the number of moles of products (nP) is greater than the number of moles of reactants (nR). In this case, ΔH can be expressed as:
\[ \Delta H = \Delta U + \Delta n_g RT \]
Here, since we are adding a positive term to ΔU, it follows that:
\[ \Delta H > \Delta U \]

3. **Δn_g = 0**: This occurs when the number of moles of products (nP) is equal to the number of moles of reactants (nR). In this case, ΔH can be expressed as:
\[ \Delta H = \Delta U \]

### Step 3: Evaluate the options based on the analysis
- **Option 1**: "ΔH is always greater than ΔU" - This is incorrect because ΔH can be less than ΔU when Δn_g < 0.

- **Option 2**: "ΔH is less than ΔU only if the number of moles of product is greater than the number of moles of reactant" - This is incorrect since ΔH is greater than ΔU in this case.

- **Option 3**: "ΔH is always less than ΔU" - This is incorrect as ΔH can be equal to or greater than ΔU depending on the number of moles.

- **Option 4**: "ΔH is less than ΔU only if the number of moles of product is less than that of moles of reactant" - This is correct, as we derived that ΔH < ΔU when Δn_g < 0.

### Conclusion
The correct answer is **Option 4**: "ΔH is less than ΔU only if the number of moles of product is less than that of moles of reactant."

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