Assertion: `C_(P)-C_(V)=R` for an ideal gas.
Reason: `((delE)/(delV))_(T)=0` for an ideal gas.

Options

A. If both (`A`) and (`R`) are correct and (`R`) is the correct explanation of (`A`).
B. If both (`A`) and (`R`) are correct, but (`R`) is not the correct explanation of (`A`).
C. If (`A`) is correct, but (`R`) is incorrect.
D. If (`A`) is incorrect, but (`R`) is correct.

Explanation

To solve the given assertion and reason question, we will break it down step by step.

### Step 1: Understand the Assertion
The assertion states that \( C_P - C_V = R \) for an ideal gas.

- **Explanation**:
- \( C_P \) is the molar heat capacity at constant pressure.
- \( C_V \) is the molar heat capacity at constant volume.
- \( R \) is the universal gas constant.

### Step 2: Derive the Relationship
To derive the relationship \( C_P - C_V = R \):

1. Start with the definition of enthalpy (\( H \)):
\[ H = U + PV \]
where \( U \) is the internal energy.

2. The change in enthalpy at constant pressure is given by:
\[ dH = nC_P dT \]

3. The change in internal energy is given by:
\[ dU = nC_V dT \]

4. The differential form of enthalpy can also be expressed as:
\[ dH = dU + d(PV) \]

5. From the ideal gas equation, we know:
\[ PV = nRT \implies d(PV) = nRdT \]

6. Substitute \( dU \) and \( d(PV) \) into the enthalpy equation:
\[ dH = nC_V dT + nRdT \]

7. Equating the two expressions for \( dH \):
\[ nC_P dT = nC_V dT + nRdT \]

8. Dividing through by \( n dT \) (assuming \( dT \neq 0 \)):
\[ C_P = C_V + R \]

9. Rearranging gives:
\[ C_P - C_V = R \]

### Step 3: Understand the Reason
The reason states that \( \left( \frac{\partial E}{\partial V} \right)_T = 0 \) for an ideal gas.

- **Explanation**:
- This means that the change in internal energy \( E \) with respect to volume \( V \) at constant temperature \( T \) is zero.

### Step 4: Analyze the Reason
1. For an ideal gas, the internal energy \( U \) depends only on temperature, not on volume or pressure.
2. Therefore, at constant temperature, any change in volume does not affect the internal energy:
\[ \left( \frac{\partial U}{\partial V} \right)_T = 0 \]
3. This implies that the internal energy remains constant with changes in volume at constant temperature.

### Step 5: Conclusion
Both the assertion and reason are correct statements. However, they are not directly related to each other. The assertion discusses the relationship between heat capacities, while the reason discusses the behavior of internal energy with volume at constant temperature.

### Final Answer
The correct answer is **B**: Both Assertion and Reason are correct, but Reason is not the correct explanation for Assertion.

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