The work done, W, during an isothermal process in which the gas expands from an intial volume `V_(1)`, to a final volume `V_(2)` is given by (R : gas constant, T : temperature )

Options

A. `R (V_(2)-V_(1)) log_(e _ ((T_(1))/(T_(2))`
B. `R (T_(2)-T_(1)) log_(e ) ((V_(1))/(V_(2)))`
C. `RT log_(e ) [(V_(2))/(V_(1))]`
D. `2RT log_(e ) [(V_(1))/(V_(2))]`

Answer

To solve the question regarding the work done, W, during an isothermal process where a gas expands from an initial volume \( V_1 \) to a final volume \( V_2 \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Work Done in an Isothermal Process**:
The work done \( W \) on or by a gas during an isothermal process can be calculated using the formula:
\[ W = \int_{V_1}^{V_2} P \, dV \]
where \( P \) is the pressure and \( dV \) is the change in volume.

2. **Use the Ideal Gas Law**:
According to the ideal gas law, we have:
\[ PV = nRT \]
For an isothermal process, the temperature \( T \) remains constant. Therefore, we can express pressure \( P \) in terms of volume \( V \):
\[ P = \frac{nRT}{V} \]

3. **Substitute Pressure in the Work Done Formula**:
Substitute \( P \) into the work done equation:
\[ W = \int_{V_1}^{V_2} \frac{nRT}{V} \, dV \]

4. **Factor Out Constants**:
Since \( nRT \) is constant during the isothermal process, we can factor it out of the integral:
\[ W = nRT \int_{V_1}^{V_2} \frac{1}{V} \, dV \]

5. **Integrate**:
The integral of \( \frac{1}{V} \) is:
\[ \int \frac{1}{V} \, dV = \ln V \]
Therefore, we have:
\[ W = nRT \left[ \ln V \right]_{V_1}^{V_2} \]

6. **Evaluate the Integral**:
Evaluating the definite integral gives:
\[ W = nRT \left( \ln V_2 - \ln V_1 \right) \]

7. **Use Logarithmic Properties**:
Using the property of logarithms that states \( \ln a - \ln b = \ln \left( \frac{a}{b} \right) \), we can rewrite the expression:
\[ W = nRT \ln \left( \frac{V_2}{V_1} \right) \]

### Final Result:
Thus, the work done during the isothermal expansion of the gas is given by:
\[ W = nRT \ln \left( \frac{V_2}{V_1} \right) \]