Two identical samples of a gas are allowed to expand (i) isothermally (ii) adiabatically. Work done is

Options

A. more in the isothermal process
B. more in the adiabatic process
C. neither of them
D. equal in both processes

Answer

To solve the problem of comparing the work done during isothermal and adiabatic expansion of identical samples of gas, we can follow these steps:

### Step 1: Understand the Processes
- **Isothermal Expansion**: This occurs at a constant temperature. The internal energy of the gas remains constant, and all the heat added to the system is converted into work done by the gas.
- **Adiabatic Expansion**: This occurs without heat exchange with the surroundings. The internal energy of the gas decreases as it does work on the surroundings.

### Step 2: Work Done in Isothermal Expansion
The work done (W) during isothermal expansion can be calculated using the formula:
\[ W_{\text{isothermal}} = nRT \ln \left( \frac{V_f}{V_i} \right) \]
where:
- \( n \) = number of moles of gas
- \( R \) = universal gas constant
- \( T \) = absolute temperature
- \( V_f \) = final volume
- \( V_i \) = initial volume

### Step 3: Work Done in Adiabatic Expansion
The work done during adiabatic expansion can be calculated using the formula:
\[ W_{\text{adiabatic}} = \frac{P_i V_i - P_f V_f}{\gamma - 1} \]
where:
- \( P_i \) = initial pressure
- \( P_f \) = final pressure
- \( \gamma \) = ratio of specific heats (Cp/Cv)

### Step 4: Compare the Work Done
To compare the two processes, we can analyze the graphs of the two processes on a PV diagram:
- The isothermal process is represented by a hyperbolic curve, while the adiabatic process is steeper than the isothermal curve.
- The area under the curve in the PV diagram represents the work done.

### Step 5: Conclusion
From the analysis of the graphs:
- The area under the isothermal curve (work done during isothermal expansion) is greater than the area under the adiabatic curve (work done during adiabatic expansion).
- Therefore, the work done in isothermal expansion is greater than that in adiabatic expansion.

### Final Answer
The work done is more in the isothermal process.

---