For which of the following gaseous reactions, the relationship log `K_C/K_P+"log RT = 0"` will bne correct ?

Options

A. `PCI_5 hArr PCI_3+CI_2`
B. `SO_2 + O_2 hArr 2SO_3`
C. `2NH_3 hArr N_2 +3H_2`
D. `N_2 +O_2 hArr 2NO`

Answer

To solve the problem, we need to determine for which of the given gaseous reactions the relationship \( \log \frac{K_C}{K_P} + \log RT = 0 \) holds true.

### Step-by-Step Solution:

1. **Understand the Relationship**:
The given relationship can be rewritten using logarithmic properties:
\[ \log \frac{K_C}{K_P} + \log RT = 0 \implies \log \left( \frac{K_C}{K_P} \cdot RT \right) = 0 \]
This implies:
\[ \frac{K_C}{K_P} \cdot RT = 1 \implies \frac{K_C}{K_P} = \frac{1}{RT} \]

2. **Use the Relationship Between \( K_C \) and \( K_P \)**:
We know the relationship between \( K_C \) and \( K_P \):
\[ K_P = K_C (RT)^{\Delta n} \]
where \( \Delta n \) is the change in the number of moles of gas (moles of products - moles of reactants).

3. **Substitute into the Equation**:
Substitute \( K_P \) in the equation:
\[ \frac{K_C}{K_P} = \frac{K_C}{K_C (RT)^{\Delta n}} = \frac{1}{(RT)^{\Delta n}} \]
Therefore, we have:
\[ \frac{1}{(RT)^{\Delta n}} = \frac{1}{RT} \]

4. **Equate the Two Expressions**:
From the above, we can equate:
\[ (RT)^{\Delta n} = RT \]
This simplifies to:
\[ \Delta n = 1 \]

5. **Determine \( \Delta n \) for Each Reaction**:
Now we need to find the reactions provided in the question and calculate \( \Delta n \) for each:
- **Reaction A**: \( \text{Cl}_2(g) + \text{PCl}_5(g) \rightarrow 2 \text{Cl}_3(g) \)
- \( \Delta n = 2 - 1 = 1 \)
- **Reaction B**: \( 2 \text{NO}(g) \rightarrow \text{N}_2(g) + \text{O}_2(g) \)
- \( \Delta n = 3 - 2 = 1 \)
- **Reaction C**: \( 2 \text{H}_2(g) + \text{O}_2(g) \rightarrow 2 \text{H}_2\text{O}(g) \)
- \( \Delta n = 2 - 3 = -1 \)
- **Reaction D**: \( \text{N}_2(g) + 3 \text{H}_2(g) \rightarrow 2 \text{NH}_3(g) \)
- \( \Delta n = 2 - 4 = -2 \)

6. **Identify the Correct Reaction**:
The only reactions where \( \Delta n = 1 \) are:
- Reaction A: \( \Delta n = 1 \)
- Reaction B: \( \Delta n = 1 \)

Therefore, both reactions A and B satisfy the condition.

### Conclusion:
The relationship \( \log \frac{K_C}{K_P} + \log RT = 0 \) is correct for reactions A and B.