Which of the following relation is/are correct?

Options

A. `Delta G = Delta H - T Delta S`
B. `Delta G = Delta H + T[(delta(Delta G))/(delta T)]_(P)`
C. `Delta G = Delta H + T Delta S`
D. `Delta G = Delta H + Delta nRT`

Answer

To determine which of the given relations are correct, we will analyze each one step by step.

### Step 1: Analyze the first relation
The first relation given is:
\[ \Delta G = \Delta H - T \Delta S \]

This is known as the Gibbs free energy equation, which relates the change in Gibbs free energy (\(\Delta G\)), change in enthalpy (\(\Delta H\)), temperature (T), and change in entropy (\(\Delta S\)). This equation is fundamental in thermodynamics and is used to predict the spontaneity of a reaction.

**Conclusion**: The first relation is correct.

### Step 2: Analyze the second relation
The second relation given is:
\[ \Delta G = \Delta H + T \frac{\Delta G}{\Delta T} \]

This relation can be derived from the Gibbs-Helmholtz equation. At constant pressure, we can express the change in Gibbs free energy with respect to temperature. The correct form of this relation should be:
\[ \frac{\Delta G}{\Delta T} = -S \]
Thus, if we rearrange the terms, we can see that the second relation can also be valid under certain conditions.

**Conclusion**: The second relation is correct.

### Step 3: Analyze the third relation
The third relation given is:
\[ \Delta G + T \Delta S = \Delta H + \Delta N R T \]

This relation is not a standard form and does not hold true in general thermodynamic contexts. The correct relation involving \(\Delta G\) and \(\Delta H\) does not include the term \(\Delta N R T\) in this way. Therefore, this relation is incorrect.

**Conclusion**: The third relation is incorrect.

### Final Answer
Based on the analysis:
- The first relation is correct.
- The second relation is correct.
- The third relation is incorrect.

Thus, the correct relations are **A and B**.

---