log `K_p/K_c`+log RT=0 is a relationship for the reaction :

Options

A. (A) `PCI_(5)hArrPCI_(3)+CI_(2)`
B. (B) `2SO_(2)+O_(2)hArr2SO_(3)`
C. (C) `H_(2)+1_(2)hArr2HI`
D. (D) `N_(2)+3H_(2)hArr2NH_(3)`

Answer

To solve the question regarding the relationship \( \log \frac{K_p}{K_c} + \log RT = 0 \), we can follow these steps:

### Step 1: Rewrite the given equation
Starting with the equation:
\[ \log \frac{K_p}{K_c} + \log RT = 0 \]
We can combine the logarithms using the property \( \log a + \log b = \log(ab) \):
\[ \log \left( \frac{K_p}{K_c} \cdot RT \right) = 0 \]

### Step 2: Exponentiate both sides
To eliminate the logarithm, we exponentiate both sides:
\[ \frac{K_p}{K_c} \cdot RT = 10^0 \]
Since \( 10^0 = 1 \), we have:
\[ \frac{K_p}{K_c} \cdot RT = 1 \]

### Step 3: Rearrange the equation
Rearranging gives us:
\[ K_p = K_c \cdot RT \]

### Step 4: Relate to the change in moles
The relationship \( K_p = K_c (RT)^{\Delta n} \) can be compared to our derived equation. Here, \( \Delta n \) is defined as the change in the number of moles of gas, calculated as:
\[ \Delta n = \text{(moles of gaseous products)} - \text{(moles of gaseous reactants)} \]
From our equation \( K_p = K_c \cdot RT \), we can see that \( \Delta n = 0 \) because:
\[ K_p = K_c (RT)^0 \implies \Delta n = 0 \]

### Step 5: Identify the reaction
We need to find a reaction where the change in moles of gas (\( \Delta n \)) is equal to 0.

### Step 6: Evaluate the options
We will evaluate the given reactions to find one that has \( \Delta n = 0 \):
1. **Option A**: \( \text{PCl}_3 + \text{Cl}_2 \rightarrow \text{PCl}_5 \)
- Reactants: 2 moles (1 PCl3 + 1 Cl2)
- Products: 1 mole (1 PCl5)
- \( \Delta n = 1 - 2 = -1 \)

2. **Option B**: \( \text{N}_2 + 3\text{H}_2 \rightarrow 2\text{NH}_3 \)
- Reactants: 4 moles (1 N2 + 3 H2)
- Products: 2 moles (2 NH3)
- \( \Delta n = 2 - 4 = -2 \)

3. **Option C**: \( 2\text{CO} + \text{O}_2 \rightarrow 2\text{CO}_2 \)
- Reactants: 3 moles (2 CO + 1 O2)
- Products: 2 moles (2 CO2)
- \( \Delta n = 2 - 3 = -1 \)

4. **Option D**: \( \text{H}_2 + \text{I}_2 \rightarrow 2\text{HI} \)
- Reactants: 2 moles (1 H2 + 1 I2)
- Products: 2 moles (2 HI)
- \( \Delta n = 2 - 2 = 0 \)

### Conclusion
The reaction that validates the relationship \( \log \frac{K_p}{K_c} + \log RT = 0 \) is:
\[ \text{H}_2 + \text{I}_2 \rightarrow 2\text{HI} \]