For the following gases equilibrium `N_(2) O_(4(g)) hArr 2NO_(2(g)), K_(p)` is found to be equal to `K_(c)` This is attained when temperature is

Options

A. `0^(@)` c
B. 273 K
C. 1 K
D. 12.19 K

Answer

To solve the question regarding the equilibrium of the reaction \( N_2O_4(g) \rightleftharpoons 2NO_2(g) \) and the relationship between \( K_p \) and \( K_c \), we need to analyze the conditions under which \( K_p = K_c \).

### Step-by-Step Solution:

1. **Understanding the Equilibrium Constants**:
- The equilibrium constant \( K_p \) is related to the partial pressures of the gases involved in the reaction.
- The equilibrium constant \( K_c \) is related to the concentrations of the gases.
- The relationship between \( K_p \) and \( K_c \) is given by the equation:
\[ K_p = K_c (RT)^{\Delta n} \]
where \( R \) is the universal gas constant, \( T \) is the temperature in Kelvin, and \( \Delta n \) is the change in the number of moles of gas during the reaction.

2. **Calculating \( \Delta n \)**:
- For the reaction \( N_2O_4(g) \rightleftharpoons 2NO_2(g) \):
- The number of moles of gaseous products = 2 (from \( 2NO_2 \))
- The number of moles of gaseous reactants = 1 (from \( N_2O_4 \))
- Therefore, \( \Delta n = 2 - 1 = 1 \).

3. **Setting the Condition for \( K_p = K_c \)**:
- From the equation \( K_p = K_c (RT)^{\Delta n} \), we can set \( K_p = K_c \):
\[ K_c = K_c (RT)^{1} \]
- This simplifies to:
\[ 1 = RT \]
- Thus, we can express the condition as:
\[ RT = 1 \]

4. **Finding the Temperature**:
- Rearranging the equation \( RT = 1 \) gives us:
\[ T = \frac{1}{R} \]
- The value of \( R \) is typically \( 0.0821 \, \text{L atm K}^{-1} \text{mol}^{-1} \) (when using atm for pressure).
- Substituting this value into the equation gives:
\[ T = \frac{1}{0.0821} \approx 12.2 \, \text{K} \]

5. **Conclusion**:
- The temperature at which \( K_p \) equals \( K_c \) for the given reaction is approximately \( 12.2 \, \text{K} \).

### Final Answer:
The temperature at which \( K_p = K_c \) for the reaction \( N_2O_4(g) \rightleftharpoons 2NO_2(g) \) is approximately \( 12.2 \, \text{K} \).