Which of the following option is correct ?

Options

A. `[(del lnK_(p))/(delT)]=(DeltaH^(@))/(RT^(2))`
B. `(dellnK)/(delT)=(E_(a))/(RT^(2))`
C. `[(dellnK_(p))/(delT)]=(DeltaU)/(RT^(2))`
D. All of these

Explanation

To determine the correct option from the given statements, we will analyze each statement step by step.

### Step-by-Step Solution:

1. **Understanding the First Statement:**
- The first statement is:
\[ \frac{\partial \ln K_p}{\partial T} = \frac{\Delta H^0}{RT^2} \]
- This equation is known as the Clausius-Clapeyron equation, which relates the change in the natural logarithm of the equilibrium constant \( K_p \) to temperature \( T \) and the standard enthalpy change \( \Delta H^0 \).
- **Conclusion:** This statement is correct.

2. **Understanding the Second Statement:**
- The second statement is:
\[ \frac{\partial \ln K}{\partial T} = \frac{E_A}{RT^2} \]
- This equation is derived from the Arrhenius equation, which describes the temperature dependence of reaction rates. Here, \( E_A \) is the activation energy.
- By taking the natural logarithm of the Arrhenius equation and differentiating with respect to temperature, we arrive at this relation.
- **Conclusion:** This statement is also correct.

3. **Understanding the Third Statement:**
- The third statement is:
\[ \frac{\partial \ln K_p}{\partial T} = \frac{\Delta U}{RT^2} \]
- This statement is incorrect. The correct relation involves the enthalpy change \( \Delta H^0 \), not the internal energy change \( \Delta U \).
- If we express \( \Delta H^0 \) in terms of \( \Delta U \) and the work done (using the relation \( \Delta H = \Delta U + P \Delta V \)), we find that this does not simplify to the given statement.
- **Conclusion:** This statement is incorrect.

### Final Conclusion:
- The correct options are **A** and **B**.
- The incorrect option is **C**.