Statement I: The work done on an ideal gas in changing its volume from `V_(1)` to `V_(2)` under a polytropic porcess is given by the integral `int _(v_(1))^(v_(2)) P. Dv` taken along the process
Statement II: No work is done under an isochroci process of the gas.

Options

A. (a)Statement I: is true, Statement II is true and Statement II is the correct explanation for Statement I.
B. (b)Statement I: is true, Statement II is true and Statement II is NOT the correct explanation for Statement I.
C. (c)Statement I is true, Statement II is false.
D. (d)Statement I is false, Statement II is fase.

Explanation

To solve the question, we need to analyze both statements provided:

### Step 1: Analyze Statement I
Statement I claims that the work done on an ideal gas in changing its volume from \( V_1 \) to \( V_2 \) under a polytropic process is given by the integral:

\[ W = \int_{V_1}^{V_2} P \, dV \]

In a polytropic process, the pressure \( P \) can be expressed as a function of volume \( V \). Therefore, the integral correctly represents the work done during the volume change from \( V_1 \) to \( V_2 \).

**Conclusion for Statement I**: This statement is **True**.

### Step 2: Analyze Statement II
Statement II states that no work is done under an isochoric process of the gas.

In an isochoric process, the volume of the gas remains constant (\( V = \text{constant} \)). Since work done \( W \) is defined as:

\[ W = P \Delta V \]

where \( \Delta V \) is the change in volume. In an isochoric process, \( \Delta V = 0 \), hence:

\[ W = P \cdot 0 = 0 \]

**Conclusion for Statement II**: This statement is also **True**.

### Step 3: Determine the Relationship Between the Statements
While both statements are true, we need to determine if Statement II provides a correct explanation for Statement I.

Statement I discusses the work done during a polytropic process, while Statement II discusses an isochoric process where no work is done. Thus, Statement II does not explain the work done in a polytropic process.

**Final Conclusion**:
- Statement I is True.
- Statement II is True, but it does not explain Statement I.

### Final Answer
- Statement I: True
- Statement II: True, but not a correct explanation for Statement I.

---