For which of the reversible reaction `K_p=K_c`

Options

A. `N_(2)(g)+3H_(2)(g) Leftrightarrow 2NH_(3) (g)`
B. `2SO_(2)(g)+O_(2) (g) Leftrightarrow 2SO_(3) (g)`
C. `PCl_(3) (g) +Cl_(2) (g) Leftrightarrow PCl_(3) (g)`
D. `N_(2(g))+O_(2(g)) Leftrightarrow 2NO(g)`

Answer

To determine for which reversible reaction \( K_p = K_c \), we need to analyze the relationship between \( K_p \) and \( K_c \) using the formula:

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

where:
- \( R \) is the universal gas constant,
- \( T \) is the temperature in Kelvin,
- \( \Delta N_g \) 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 have four reactions to analyze:
- **Reaction A:** \( N_2(g) + 3H_2(g) \rightleftharpoons 2NH_3(g) \)
- **Reaction B:** \( 2SO_2(g) + O_2(g) \rightleftharpoons 2SO_3(g) \)
- **Reaction C:** \( PCl_3(g) + Cl_2(g) \rightleftharpoons PCl_5(g) \)
- **Reaction D:** \( N_2(g) + O_2(g) \rightleftharpoons 2NO(g) \)

2. **Calculate \( \Delta N_g \) for Each Reaction:**

- **For Reaction A:**
- Products: 2 moles of \( NH_3 \)
- Reactants: 1 mole of \( N_2 \) + 3 moles of \( H_2 \) = 4 moles
- \( \Delta N_g = 2 - 4 = -2 \)

- **For Reaction B:**
- Products: 2 moles of \( SO_3 \)
- Reactants: 2 moles of \( SO_2 \) + 1 mole of \( O_2 \) = 3 moles
- \( \Delta N_g = 2 - 3 = -1 \)

- **For Reaction C:**
- Products: 1 mole of \( PCl_5 \)
- Reactants: 1 mole of \( PCl_3 \) + 1 mole of \( Cl_2 \) = 2 moles
- \( \Delta N_g = 1 - 2 = -1 \)

- **For Reaction D:**
- Products: 2 moles of \( NO \)
- Reactants: 1 mole of \( N_2 \) + 1 mole of \( O_2 \) = 2 moles
- \( \Delta N_g = 2 - 2 = 0 \)

3. **Determine \( K_p \) and \( K_c \) Relationship:**

- **For Reaction A:**
\[ K_p = K_c \cdot (RT)^{-2} \quad \Rightarrow \quad K_p \neq K_c \]

- **For Reaction B:**
\[ K_p = K_c \cdot (RT)^{-1} \quad \Rightarrow \quad K_p \neq K_c \]

- **For Reaction C:**
\[ K_p = K_c \cdot (RT)^{-1} \quad \Rightarrow \quad K_p \neq K_c \]

- **For Reaction D:**
\[ K_p = K_c \cdot (RT)^{0} \quad \Rightarrow \quad K_p = K_c \]

4. **Conclusion:**
The only reaction for which \( K_p = K_c \) is **Reaction D**:
\[ N_2(g) + O_2(g) \rightleftharpoons 2NO(g) \]

### Final Answer:
The correct option is **D**.