Which of the following relationship are correct?

Options

A. `( DeltaH- DeltaE)/( DeltanxxT ) =`Constant
B. `DeltaG= -T DeltaS_("Total")`
C. `K= e^(- DeltaG^(@) //RT)`
D. `[(del(DeltaH))/(delT)]_(P)= DeltaC_(p)`

Explanation

To determine which relationships are correct, we will analyze each statement one by one based on thermodynamic principles.

### Step-by-Step Solution:

1. **Analyzing the first relationship:**
\[ \Delta H = -\Delta E \]
This statement is incorrect. The correct relationship is:
\[ \Delta H = \Delta E + \Delta nRT \]
where \(\Delta n\) is the change in the number of moles of gas. Thus, the first relationship is not correct.

2. **Analyzing the second relationship:**
\[ \Delta G = -T \Delta S \]
This relationship is indeed correct. It relates the change in Gibbs free energy (\(\Delta G\)) to the temperature (T) and the change in entropy (\(\Delta S\)). Therefore, the second relationship is correct.

3. **Analyzing the third relationship:**
\[ K = e^{-\Delta G^\circ / RT} \]
This relationship is also correct. It relates the equilibrium constant (K) to the standard Gibbs free energy change (\(\Delta G^\circ\)). Thus, the third relationship is correct.

4. **Analyzing the fourth relationship:**
\[ \mu = \frac{\Delta H}{\Delta T} \text{ at constant pressure} \]
This is the definition of the Joule-Thomson coefficient, which describes how the temperature of a gas changes when it is allowed to expand freely at constant enthalpy. Therefore, the fourth relationship is correct.

### Conclusion:
The correct relationships are:
- **Second relationship**: \(\Delta G = -T \Delta S\)
- **Third relationship**: \(K = e^{-\Delta G^\circ / RT}\)
- **Fourth relationship**: Joule-Thomson coefficient \(\mu = \frac{\Delta H}{\Delta T}\) at constant pressure.

### Summary of Correct Relationships:
- **Correct**: 2, 3, and 4.
- **Incorrect**: 1.