`{:("column"-1,"Column"-2),((A)(DeltaG_("system"))_(T.P) = 0,(p) "Process is in equilibrium"),((B)DeltaS_("system")+DeltaS_("surrounding") gt 0, (q)"Process is nonspontaneous"),((C)DeltaS_("system") +DeltaS_("surroumding") lt 0 , (r) "Process is spontaneous"),((D)(DeltaG_("system"))_(T.P) gt 0,(s)"System is unable to do useful work"):}`

Explanation

To solve the matching question, we need to analyze the statements in Column 1 and find the corresponding statements in Column 2 based on thermodynamic principles.

### Step-by-Step Solution:

1. **Analyze Statement A**:
- **Statement**: ΔG(system) at constant temperature and pressure = 0.
- **Interpretation**: This condition indicates that the system is in equilibrium. At equilibrium, no net change occurs, and the system is not able to do useful work.
- **Match**: This corresponds to (p) "Process is in equilibrium".

2. **Analyze Statement B**:
- **Statement**: ΔS(system) + ΔS(surrounding) > 0.
- **Interpretation**: A positive change in the total entropy (system + surroundings) indicates that the process is spontaneous. A spontaneous process can do useful work.
- **Match**: This corresponds to (r) "Process is spontaneous".

3. **Analyze Statement C**:
- **Statement**: ΔS(system) + ΔS(surrounding) < 0.
- **Interpretation**: A negative change in the total entropy indicates that the process is non-spontaneous. A non-spontaneous process cannot do useful work.
- **Match**: This corresponds to (q) "Process is nonspontaneous".

4. **Analyze Statement D**:
- **Statement**: ΔG(system) at constant temperature and pressure > 0.
- **Interpretation**: A positive Gibbs free energy change indicates that the process is non-spontaneous. A non-spontaneous process cannot do useful work.
- **Match**: This corresponds to (s) "System is unable to do useful work".

### Final Matches:
- A → P ("Process is in equilibrium")
- B → R ("Process is spontaneous")
- C → Q ("Process is nonspontaneous")
- D → S ("System is unable to do useful work")

### Summary of Matches:
- A (ΔG = 0) → P
- B (ΔS(system) + ΔS(surrounding) > 0) → R
- C (ΔS(system) + ΔS(surrounding) < 0) → Q
- D (ΔG > 0) → S