For which of the following reaction, `(K_(p))/(K_(c))` ratio is maximum ?

Options

A. `CO(g)+(1)/(2) O_(2)(g) leftrightarrowCO_(2)(g)`
B. `H_(2)(g)+I_(2)(g)leftrightarrow2HI(g)`
C. `PCl_(5)(g)leftrightarrowPCl_(3)(g)+Cl_(2)(g)`
D. `7H_(2)(g)+2NO_(2)(g)leftrightarrow2NH_(2)(g)+4H_(2)O(g)`

Explanation

To determine for which of the given reactions the ratio \( \frac{K_p}{K_c} \) is maximum, we need to analyze each reaction using the relationship between \( K_p \) and \( K_c \):

\[ K_p = K_c (RT)^{\Delta N_g} \]

Where:
- \( R \) is the gas constant,
- \( T \) is the temperature,
- \( \Delta N_g \) is the change in the number of moles of gas, calculated as the moles of gaseous products minus the moles of gaseous reactants.

### Step-by-Step Solution:

1. **Identify the Reactions**: We will analyze each reaction provided in the question.

2. **Calculate \( \Delta N_g \) for Each Reaction**:
- **Reaction A**:
\[ \text{CO} + \frac{1}{2} \text{O}_2 \rightarrow \text{CO}_2 \]
- Moles of products = 1 (CO₂)
- Moles of reactants = 1 + 0.5 = 1.5
- \( \Delta N_g = 1 - 1.5 = -0.5 \)

- **Reaction B**:
\[ \text{2 HI} \rightarrow \text{H}_2 + \text{I}_2 \]
- Moles of products = 2 (H₂ + I₂)
- Moles of reactants = 2 (2 HI)
- \( \Delta N_g = 2 - 2 = 0 \)

- **Reaction C**:
\[ \text{PCl}_5 \rightarrow \text{PCl}_3 + \text{Cl}_2 \]
- Moles of products = 2 (PCl₃ + Cl₂)
- Moles of reactants = 1 (PCl₅)
- \( \Delta N_g = 2 - 1 = 1 \)

- **Reaction D**:
\[ \text{A} \rightarrow \text{B} + \text{C} + \text{D} \]
- Assume total moles of products = 6
- Assume total moles of reactants = 9
- \( \Delta N_g = 6 - 9 = -3 \)

3. **Calculate \( K_p \) in Terms of \( K_c \)**:
- For **Reaction A**:
\[ K_p = K_c (RT)^{-0.5} \]
- For **Reaction B**:
\[ K_p = K_c (RT)^{0} \quad \Rightarrow \quad K_p = K_c \]
- For **Reaction C**:
\[ K_p = K_c (RT)^{1} \quad \Rightarrow \quad K_p = K_c (RT) \]
- For **Reaction D**:
\[ K_p = K_c (RT)^{-3} \]

4. **Determine the Ratios \( \frac{K_p}{K_c} \)**:
- For **Reaction A**:
\[ \frac{K_p}{K_c} = (RT)^{-0.5} \]
- For **Reaction B**:
\[ \frac{K_p}{K_c} = 1 \]
- For **Reaction C**:
\[ \frac{K_p}{K_c} = RT \]
- For **Reaction D**:
\[ \frac{K_p}{K_c} = (RT)^{-3} \]

5. **Compare the Ratios**:
- The maximum ratio \( \frac{K_p}{K_c} \) occurs for **Reaction C** since it is proportional to \( RT \), which is greater than 1, while the other reactions yield ratios less than or equal to 1.

### Conclusion:
The reaction for which the ratio \( \frac{K_p}{K_c} \) is maximum is **Reaction C**.