An ideal gas is allowed to expand from 5 L to 15 L once rapidly and once very slowly. The magnitude of work done in two processes are `W_(1)` and `W_(2)`, they are related as

Options

A. `W_(1) = W_(2)`
B. `W_(1) gt W_(2)`
C. `W_(1) lt W_(2)`
D. Data is insufficient

Answer

To solve the problem of comparing the work done by an ideal gas expanding from 5 L to 15 L in two different processes (one rapid and one slow), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Processes**:
- **Rapid Expansion (Irreversible Process)**: The gas expands quickly against a constant external pressure. This means that the gas does not have time to equilibrate with the external pressure and expands against a higher pressure initially.
- **Slow Expansion (Reversible Process)**: The gas expands slowly, allowing it to remain in equilibrium with the external pressure throughout the process. This means the pressure of the gas changes gradually.

2. **Work Done in Each Process**:
- The work done by the gas during expansion can be calculated using the formula:
\[ W = -\int_{V_1}^{V_2} P \, dV \]
- For the **rapid expansion (W1)**, we can assume that the external pressure (P_ext) is constant and equal to the initial pressure (P1) of the gas. Thus, the work done can be approximated as:
\[ W_1 = -P_{ext} \Delta V = -P_1 (V_2 - V_1) \]
- For the **slow expansion (W2)**, since the process is reversible, the pressure of the gas decreases gradually from P1 to P2. The work done in this case can be calculated as:
\[ W_2 = -\int_{V_1}^{V_2} P \, dV \]
This integral will yield a larger value because the pressure is always lower than P1 during the expansion.

3. **Comparison of Work Done**:
- Since in the irreversible process (W1), the gas expands against a constant pressure (initial pressure), and in the reversible process (W2), the gas expands against a decreasing pressure, we can conclude:
\[ W_2 > W_1 \]
- This is because in the reversible case, the gas does more work as it expands against a lower pressure throughout the process.

4. **Final Relation**:
- Therefore, the relationship between the work done in the two processes can be summarized as:
\[ W_2 > W_1 \]