Which of the following relationships is not correct for the relation between `DeltaH and DeltaU`?

Options

A. When `Deltan_(g)=0" then " DeltaH=DeltaU`
B. When `Deltan_(g) gt 0 " then "DeltaH gt DeltaU`
C. When `Deltan_(g) lt 0" then " Delta H lt DeltaU`
D. When `Deltan_(g)RT=0" then "DeltaH

Explanation

To solve the question regarding the relationships between ΔH (enthalpy change) and ΔU (internal energy change), we will analyze the four given options based on the thermodynamic equation that relates these two quantities.

### Step-by-Step Solution:

1. **Understand the Relationship**:
The relationship between ΔH and ΔU is given by the equation:
\[ \Delta H = \Delta U + \Delta (PV) \]
For an ideal gas, we can express \(PV\) in terms of the number of moles of gas (\(n\)) and temperature (\(T\)):
\[ PV = nRT \]
Therefore, we can rewrite the equation as:
\[ \Delta H = \Delta U + \Delta n_g \cdot R \cdot T \]
where \(\Delta n_g\) is the change in the number of moles of gas.

2. **Analyze Each Option**:
- **Option 1**: If \(\Delta n_g = 0\), then:
\[ \Delta H = \Delta U + 0 \implies \Delta H = \Delta U \]
This option is correct.

- **Option 2**: If \(\Delta n_g > 0\), then:
\[ \Delta H = \Delta U + \Delta n_g \cdot R \cdot T \implies \Delta H > \Delta U \]
This option is also correct.

- **Option 3**: If \(\Delta n_g < 0\), then:
\[ \Delta H = \Delta U + \Delta n_g \cdot R \cdot T \implies \Delta H < \Delta U \text{ (since } \Delta n_g \cdot R \cdot T \text{ is negative)} \]
This option is correct as well.

- **Option 4**: If \(\Delta n_g \cdot R \cdot T = 0\), it implies either \(\Delta n_g = 0\) or \(T = 0\) (which is not a realistic scenario). If \(\Delta n_g = 0\):
\[ \Delta H = \Delta U \]
However, the option states that \(\Delta H < \Delta U\), which contradicts our finding. Therefore, this option is incorrect.

3. **Conclusion**:
The option that is not correct is **Option 4**: "If \(\Delta n_g R T = 0\), then \(\Delta H < \Delta U\)".

### Final Answer:
**Option 4 is not correct.**