For the reaction `4NO_(2)(g) +O_(2)(g)hArr 2N_(2)O_(5)(g)`, which of the following facts holds good ?

Options

A. `K_(p)=K_(c)`
B. `K_(p)gt K_(c)`
C. `K_(p)ltK_(c)`
D. `K_(p)` and `K_(c)` cannot unless pressure of the system is provided.

Explanation

To solve the question regarding the reaction \( 4NO_2(g) + O_2(g) \rightleftharpoons 2N_2O_5(g) \), we need to analyze the equilibrium constants \( K_p \) and \( K_c \) and their relationship.

### Step-by-Step Solution:

1. **Identify the Reaction**:
The given reaction is:
\[ 4NO_2(g) + O_2(g) \rightleftharpoons 2N_2O_5(g) \]

2. **Determine the Change in Moles of Gas (\( \Delta N_g \))**:
- Count the moles of gaseous products and reactants.
- **Products**: There are 2 moles of \( N_2O_5 \).
- **Reactants**: There are 4 moles of \( NO_2 \) and 1 mole of \( O_2 \), totaling 5 moles.
- Calculate \( \Delta N_g \):
\[ \Delta N_g = \text{Moles of products} - \text{Moles of reactants} = 2 - 5 = -3 \]

3. **Relate \( K_p \) and \( K_c \)**:
The relationship between \( K_p \) and \( K_c \) is given by:
\[ K_p = K_c (RT)^{\Delta N_g} \]
where \( R \) is the gas constant and \( T \) is the temperature in Kelvin.

4. **Substitute \( \Delta N_g \)**:
Substitute \( \Delta N_g = -3 \) into the equation:
\[ K_p = K_c (RT)^{-3} = \frac{K_c}{(RT)^3} \]

5. **Analyze the Relationship**:
Since \( (RT)^3 \) is a positive quantity, it follows that:
\[ K_p < K_c \]
This indicates that \( K_p \) is smaller than \( K_c \).

6. **Conclusion**:
Therefore, the correct conclusion is that \( K_p \) is less than \( K_c \).

### Final Answer:
The correct fact that holds good for the reaction \( 4NO_2(g) + O_2(g) \rightleftharpoons 2N_2O_5(g) \) is that \( K_p < K_c \).

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