For the expression: dG = Vdp - SdT, which of the following is correct?

Options

A. `((delG)/(delT))_(p)=V`
B. `((delG)/(delp))_(T)=V`
C. `((delG)/(delT))_(S)=V`
D. `((delG)/(delp))_(T)=-S`

Answer

To solve the expression \( dG = Vdp - SdT \) and determine which statements are correct, we will analyze the expression under different conditions.

### Step 1: Understand the expression
The expression \( dG = Vdp - SdT \) represents the change in Gibbs free energy (\( G \)) in terms of changes in pressure (\( p \)) and temperature (\( T \)). Here, \( V \) is the volume and \( S \) is the entropy.

### Step 2: Analyze at constant pressure
If we assume constant pressure, then \( dp = 0 \). Substituting this into the expression gives:

\[ dG = V(0) - SdT \implies dG = -SdT \]

From this, we can derive:

\[ \left( \frac{dG}{dT} \right)_{p} = -S \]

### Step 3: Analyze at constant temperature
Now, if we consider constant temperature, then \( dT = 0 \). Substituting this into the expression gives:

\[ dG = Vdp - S(0) \implies dG = Vdp \]

From this, we can derive:

\[ \left( \frac{dG}{dp} \right)_{T} = V \]

### Step 4: Analyze at constant entropy
If we assume constant entropy, then \( dS = 0 \). In this case, we have:

\[ dG = Vdp - S(0) \implies dG = Vdp \]

Again, we find:

\[ \left( \frac{dG}{dp} \right)_{S} = V \]

### Step 5: Summary of results
1. At constant pressure: \( \left( \frac{dG}{dT} \right)_{p} = -S \)
2. At constant temperature: \( \left( \frac{dG}{dp} \right)_{T} = V \)
3. At constant entropy: \( \left( \frac{dG}{dp} \right)_{S} = V \)

### Conclusion
Based on the analysis:
- The first option is incorrect because \( \left( \frac{dG}{dT} \right)_{p} = -S \) and not equal to \( V \).
- The second option is correct because \( \left( \frac{dG}{dp} \right)_{T} = V \).
- The third option is incorrect because \( \left( \frac{dG}{dT} \right)_{S} \) does not equal \( V \).
- The fourth option is correct as it states \( \left( \frac{dG}{dp} \right)_{T} = V \).

Thus, the correct answer is option 2.