A sample of gas at temperature `T` is adiabatically expanded to double its volume.The work done by the gas in the process is (given `gamma=(3)/(2)):`

Options

A. `W=TR[sqrt(2)-2]`
B. `W=(R)/(T)[2-sqrt(2)]`
C. `W=RT[2-sqrt(2)]`
D. `W=(T)/(R)[sqrt(2)-2]`

Answer

To solve the problem of work done by a gas during an adiabatic expansion to double its volume, we can follow these steps:

### Step 1: Understand the Adiabatic Process
In an adiabatic process, there is no heat exchange with the surroundings. The relationship between pressure, volume, and temperature for an ideal gas undergoing an adiabatic process is given by:
\[ T_1 V_1^{\gamma - 1} = T_2 V_2^{\gamma - 1} \]
where \( \gamma \) is the heat capacity ratio (given as \( \frac{3}{2} \)).

### Step 2: Define Initial and Final Conditions
Let:
- Initial volume \( V_1 = V \)
- Final volume \( V_2 = 2V \)
- Initial temperature \( T_1 = T \)
- Final temperature \( T_2 \) (to be determined)

### Step 3: Apply the Adiabatic Condition
Using the adiabatic condition:
\[ T_1 V_1^{\gamma - 1} = T_2 V_2^{\gamma - 1} \]
Substituting the values:
\[ T (V)^{\frac{3}{2} - 1} = T_2 (2V)^{\frac{3}{2} - 1} \]
This simplifies to:
\[ T V^{\frac{1}{2}} = T_2 (2^{\frac{1}{2}} V^{\frac{1}{2}}) \]
\[ T = T_2 \sqrt{2} \]
Thus, we find:
\[ T_2 = \frac{T}{\sqrt{2}} \]

### Step 4: Calculate Work Done
The work done \( W \) during an adiabatic process can be expressed as:
\[ W = \frac{P_1 V_1 - P_2 V_2}{\gamma - 1} \]
We can express pressure in terms of temperature and volume using the ideal gas law:
\[ P = \frac{nRT}{V} \]

Substituting for \( P_1 \) and \( P_2 \):
\[ P_1 = \frac{nRT_1}{V} = \frac{nRT}{V} \]
\[ P_2 = \frac{nRT_2}{2V} = \frac{nR(T/\sqrt{2})}{2V} = \frac{nRT}{2\sqrt{2}V} \]

Now substituting these into the work done equation:
\[ W = \frac{\left( \frac{nRT}{V} \cdot V \right) - \left( \frac{nRT}{2\sqrt{2}V} \cdot 2V \right)}{\frac{3}{2} - 1} \]
\[ W = \frac{nRT - \frac{nRT}{\sqrt{2}}}{\frac{1}{2}} \]
\[ W = 2nRT \left( 1 - \frac{1}{\sqrt{2}} \right) \]
\[ W = 2nRT \left( \frac{\sqrt{2} - 1}{\sqrt{2}} \right) \]

### Step 5: Final Expression
Assuming \( n = 1 \) (for simplicity):
\[ W = 2RT \left( \frac{\sqrt{2} - 1}{\sqrt{2}} \right) \]

### Step 6: Conclusion
Thus, the work done by the gas during the adiabatic expansion to double its volume is:
\[ W = RT(2(\sqrt{2} - 1)) \]