A diatomic gas undergoes a process represented by `PV^(1.3)`= constant . Choose the incorrect statement

Options

A. The gas expands by absorbing heat from the surroundings
B. The gas cools down during expansion
C. The work done by surrounding during expansion of the gas is negative
D. None of these

Answer

To solve the problem, we need to analyze the statements related to the diatomic gas undergoing a polytropic process represented by \( PV^{1.3} = \text{constant} \). We will determine which statement is incorrect based on the properties of the gas and the nature of the process.

### Step-by-Step Solution:

1. **Identify the Process**:
The given process \( PV^{1.3} = \text{constant} \) indicates a polytropic process where \( n = 1.3 \).

2. **Determine the Value of \( \gamma \)**:
For a diatomic gas, the value of \( \gamma \) (the ratio of specific heats) is approximately \( 1.4 \).

3. **Calculate Molar Heat Capacity \( C \)**:
The molar heat capacity for a polytropic process can be calculated using the formula:
\[ C = \frac{n - \gamma}{n - 1} R \]
Substituting \( n = 1.3 \) and \( \gamma = 1.4 \):
\[ C = \frac{1.3 - 1.4}{1.3 - 1} R = \frac{-0.1}{0.3} R = -\frac{1}{3} R \]
This indicates that the molar heat capacity \( C \) is negative.

4. **Analyze Heat Transfer \( Q \)**:
The heat transfer \( Q \) during the process is given by:
\[ Q = nC\Delta T \]
Since \( C \) is negative and \( \Delta T \) (the change in temperature) is also negative during expansion (the gas cools), the product \( nC\Delta T \) will be positive. This means heat is absorbed from the surroundings.

5. **Determine Work Done**:
For the work done during the expansion, we can express it as:
\[ W = \int P \, dV \]
From the polytropic equation, we can express pressure \( P \) as:
\[ P = kV^{-1.3} \]
The work done by the gas during expansion will be positive, indicating that the gas does work on the surroundings.

6. **Evaluate Statements**:
- **Statement 1**: "Gas expands by absorbing heat from the surroundings." This is correct as we derived that \( Q > 0 \).
- **Statement 2**: "Gas cools down during expansion." This is also correct as the temperature decreases during expansion.
- **Statement 3**: "Work done by the surroundings during the expansion process is negative." This is correct since the work done on the gas is negative when the gas expands.

7. **Conclusion**:
Since all statements evaluated are correct, we conclude that the question likely implies that none of the statements are incorrect. Therefore, the answer is that none of these statements are incorrect.