At constant temperature , the equilibrium constant `(K_P)` for the decomposition reaction
` N_2O_4 hArr 2NO_2, ` is expressed by ` K_p = ((4x^(2) p))/((1-x^(2)))`
where p= pressure ,x = extent of decomposition .which one of the following statements is true?

Options

A. ` K_p` increase with increase of p
B. `K_p` increases with increase of x
C. `K_p` increases with decrease of x
D. `K_p ` remains constant with change in P and X.

Answer

To solve the problem, we need to analyze the equilibrium constant \( K_p \) for the decomposition reaction of \( N_2O_4 \) to \( 2NO_2 \) given by the equation:

\[ K_p = \frac{4x^2 p}{1 - x^2} \]

where \( p \) is the pressure and \( x \) is the extent of decomposition.

### Step-by-Step Solution:

1. **Understanding the Reaction**:
The reaction is:
\[ N_2O_4 \rightleftharpoons 2NO_2 \]
This indicates that one mole of \( N_2O_4 \) decomposes into two moles of \( NO_2 \).

2. **Identifying Variables**:
- \( p \): pressure of the system.
- \( x \): extent of decomposition, which indicates how much of \( N_2O_4 \) has decomposed.

3. **Expression for \( K_p \)**:
The expression given for \( K_p \) is:
\[ K_p = \frac{4x^2 p}{1 - x^2} \]
This shows that \( K_p \) is a function of both \( p \) and \( x \).

4. **Effect of Pressure on \( K_p \)**:
- According to Le Chatelier's principle, if the pressure of the system is increased, the equilibrium will shift to the side with fewer moles of gas. In this case, since we have 1 mole of \( N_2O_4 \) producing 2 moles of \( NO_2 \), increasing pressure would favor the formation of \( N_2O_4 \).
- However, since \( K_p \) is defined at a constant temperature, it remains constant regardless of changes in pressure or extent of decomposition.

5. **Effect of Extent of Decomposition on \( K_p \)**:
- If \( x \) increases, it indicates more \( N_2O_4 \) is decomposing into \( NO_2 \). This would increase the numerator \( 4x^2 p \) and decrease the denominator \( 1 - x^2 \).
- The equilibrium will adjust to maintain the value of \( K_p \) constant.

6. **Conclusion**:
Since \( K_p \) is only dependent on temperature and remains constant with changes in \( p \) and \( x \), the correct statement is that \( K_p \) remains constant with changes in pressure and extent of decomposition.

### Final Answer:
The correct statement is: **\( K_p \) remains constant with the change in \( p \) and \( x \)**.