For an ideal gas, the heat of reaction at constant pressure and constant volume are related as

Options

A. `H + E = pV`
B. `E = H + pDeltaV`
C. `q_(p) = q_(v) + Delta_(g) nRT`
D. None of these

Explanation

To derive the relationship between the heat of reaction at constant pressure (ΔH) and constant volume (ΔU) for an ideal gas, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Definitions**:
- ΔH (enthalpy change) is the heat absorbed or released at constant pressure.
- ΔU (internal energy change) is the heat absorbed or released at constant volume.

2. **Starting with the Relationship**:
- The relationship between ΔH and ΔU can be expressed as:
\[ \Delta H = \Delta U + P \Delta V \]
- Here, \(P\) is the pressure and \(\Delta V\) is the change in volume.

3. **Considering the Volume Change**:
- For reactions involving solids and liquids, the volume change (\(P \Delta V\)) is negligible. Thus, we can simplify the equation for those cases.
- However, for gaseous reactions, the volume change cannot be neglected.

4. **Expressing ΔU in Terms of Volume**:
- The change in internal energy can be expressed as:
\[ \Delta U = P(V_2 - V_1) \]
- Where \(V_1\) is the volume of gas phase reactants and \(V_2\) is the volume of gas phase products.

5. **Using the Ideal Gas Law**:
- For ideal gases, we can use the ideal gas equation \(PV = nRT\).
- For 1 mole of gaseous reactants producing \(n_2\) moles of gaseous products:
\[ PV_1 = n_1RT \quad \text{and} \quad PV_2 = n_2RT \]

6. **Substituting into the Equation**:
- Substitute these into the equation for ΔH:
\[ \Delta H = \Delta U + (n_2RT - n_1RT) \]
- Simplifying gives:
\[ \Delta H = \Delta U + (n_2 - n_1)RT \]

7. **Defining Δn**:
- Let \(\Delta n = n_2 - n_1\), which represents the change in the number of moles of gaseous products minus gaseous reactants.
- Thus, we can write:
\[ \Delta H = \Delta U + \Delta n \cdot RT \]

8. **Final Relationship**:
- Therefore, the relationship between the heat of reaction at constant pressure and constant volume for an ideal gas is:
\[ \Delta H = \Delta U + \Delta n \cdot RT \]

### Conclusion:
The heat of reaction at constant pressure (ΔH) is related to the heat of reaction at constant volume (ΔU) by the equation:
\[ \Delta H = \Delta U + \Delta n \cdot RT \]