Which statement(s) is correct,

Options

A. `((delta H)/(delta T))_(P) - ((deltaU)/(deltaT))_(V) = R`
B. `((deltaH)/(deltaT))_(P) gt ((deltaU)/(deltaT))_(V)`
C. `((deltaH)/(deltaV))_(T)` for ideal gas is zero
D. All of these

Answer

To solve the question regarding which statements are correct, we will analyze each statement based on the principles of thermodynamics.

### Step-by-Step Solution:

1. **Understanding Enthalpy and Internal Energy**:
- We start with the definitions:
- Change in enthalpy (\( \Delta H \)) at constant pressure is given by:
\[ \Delta H = C_p \Delta T \]
- Change in internal energy (\( \Delta U \)) at constant volume is given by:
\[ \Delta U = C_v \Delta T \]

2. **Relating \( C_p \) and \( C_v \)**:
- The relationship between the heat capacities for an ideal gas is:
\[ C_p - C_v = R \]
- This shows that the difference between the heat capacities at constant pressure and constant volume equals the universal gas constant \( R \).

3. **Analyzing the First Statement**:
- The first statement claims:
\[ \frac{\Delta H}{\Delta T} \text{ at constant pressure} - \frac{\Delta U}{\Delta T} \text{ at constant volume} = C_p - C_v \]
- From our earlier relationship, we can conclude that this is correct because:
\[ C_p - C_v = R \]

4. **Analyzing the Second Statement**:
- The second statement claims:
\[ \frac{\Delta H}{\Delta T} \text{ at constant pressure} - \frac{\Delta U}{\Delta T} \text{ at constant volume} > 0 \]
- Since \( C_p - C_v = R \) and \( R > 0 \) for an ideal gas, this statement is also correct.

5. **Analyzing the Third Statement**:
- The third statement claims that:
\[ \frac{\Delta H}{\Delta V} \text{ at constant temperature} = 0 \]
- For an ideal gas, we know that:
\[ H = U + PV = U + RT \]
- Since internal energy \( U \) is a function of temperature only, at constant temperature, \( \Delta U \) does not change with volume. Thus:
\[ \frac{\Delta H}{\Delta V} = 0 + R \frac{\Delta T}{\Delta V} = 0 \]
- Therefore, this statement is also correct.

6. **Conclusion**:
- Since all three statements are correct, the final conclusion is that the correct option is "All of the above."