Which of the following is/are correct?

Options

A. `DeltaH=DeltaU+Delta(PV)` when P and V both changes
B. `DeltaH=DeltaU+PDeltaV` when pressure is constant
C. `DeltaH=DeltaU+VDeltaP` when volume is constant
D. `DeltaH=DeltaU+PDeltaV+VDeltaP` when P and V both changes

Explanation

To determine which of the statements regarding thermodynamics are correct, we will analyze the relationships between enthalpy (ΔH), internal energy (ΔU), pressure (P), and volume (V).

### Step-by-Step Solution:

1. **Understanding the Relationship Between ΔH and ΔU**:
The first relationship we need to consider is:
\[ \Delta H = \Delta U + \Delta (PV) \]
This equation is valid when both pressure (P) and volume (V) change during a process.

2. **Constant Pressure Scenario**:
If the pressure (P) is constant, we can simplify the equation:
\[ \Delta U = \Delta H - P \Delta V \]
Here, since P is constant, we can factor it out, leading to:
\[ \Delta U = P \Delta V \]
This shows that the change in internal energy (ΔU) is directly related to the work done by the system at constant pressure.

3. **Constant Volume Scenario**:
If the volume (V) is constant, the change in entropy (ds) can be expressed as:
\[ ds = V dP \]
In this case, since V is constant, it can be factored out of the differential, indicating that the change in entropy is dependent on the change in pressure.

4. **General Case for Δ(PV)**:
When both pressure and volume change, the change in the product PV can be expressed as:
\[ \Delta(PV) = P \Delta V + V \Delta P \]
This equation shows how the changes in both pressure and volume contribute to the overall change in the product PV.

5. **Conclusion**:
All the relationships derived above are correct under their respective conditions. Therefore, the conclusion is that all the statements provided in the question are correct.

### Summary of Correct Statements:
- ΔH = ΔU + Δ(PV) when both P and V change.
- ΔU = PΔV when P is constant.
- ds = VdP when V is constant.
- Δ(PV) = PΔV + VΔP when both P and V change.