Work done by a gas during isothermal expansion is ______________ than that done during adiabatic expansion.

Answer

To determine the relationship between the work done by a gas during isothermal expansion and adiabatic expansion, we can analyze both processes step by step.

### Step 1: Understand the Definitions
- **Isothermal Process**: This is a thermodynamic process in which the temperature of the system remains constant. For an ideal gas, the equation governing this process is \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the gas constant, and \( T \) is the absolute temperature.
- **Adiabatic Process**: In this process, there is no heat exchange with the surroundings (\( Q = 0 \)). The relationship between pressure and volume is given by \( PV^\gamma = \text{constant} \), where \( \gamma \) is the heat capacity ratio (\( C_p/C_v \)).

### Step 2: Work Done in Isothermal Expansion
For an isothermal process, the work done \( W \) can be calculated using the formula:
\[ W_{\text{isothermal}} = nRT \ln\left(\frac{V_f}{V_i}\right) \]
where \( V_f \) is the final volume and \( V_i \) is the initial volume.

### Step 3: Work Done in Adiabatic Expansion
For an adiabatic process, the work done can be expressed as:
\[ W_{\text{adiabatic}} = \frac{P_1V_1 - P_2V_2}{\gamma - 1} \]
Alternatively, using the temperatures:
\[ W_{\text{adiabatic}} = \frac{nR(T_1 - T_2)}{\gamma - 1} \]
where \( T_1 \) and \( T_2 \) are the initial and final temperatures respectively.

### Step 4: Comparing Work Done
To compare the two, we need to consider the nature of both processes:
- In an isothermal expansion, the gas absorbs heat from the surroundings to maintain constant temperature, allowing it to do more work.
- In an adiabatic expansion, the gas does work on the surroundings without heat exchange, leading to a drop in temperature.

### Step 5: Conclusion
From the analysis, we can conclude:
\[ W_{\text{isothermal}} > W_{\text{adiabatic}} \]
Thus, the work done by a gas during isothermal expansion is greater than that done during adiabatic expansion.