Two moles of an ideal monoatomic gas at `27^(@)C`occupies a volume of V . If the gas is expanded adiabatically to the volume , 2V then the work done by the gas will be `[lambda=5//3,R=8.31j//molK]`

Options

A. `-2767.23j`
B. `2767.23j`
C. `2500j`
D. `-2500j`

Answer

To solve the problem of finding the work done by an ideal monoatomic gas during an adiabatic expansion from volume \( V \) to \( 2V \), we can follow these steps:

### Step 1: Identify the Given Information
- Number of moles, \( n = 2 \)
- Initial temperature, \( T_1 = 27^\circ C = 300 \, K \) (after converting to Kelvin)
- Initial volume, \( V_1 = V \)
- Final volume, \( V_2 = 2V \)
- Heat capacity ratio, \( \gamma = \frac{5}{3} \)
- Ideal gas constant, \( R = 8.31 \, J/(mol \cdot K) \)

### Step 2: Use the Adiabatic Condition
For an adiabatic process, we have the relation:
\[ T V^{\gamma - 1} = \text{constant} \]
This implies:
\[ \frac{T_1}{T_2} = \left(\frac{V_2}{V_1}\right)^{\gamma - 1} \]
Substituting the values:
\[ \frac{T_1}{T_2} = \left(\frac{2V}{V}\right)^{\frac{5}{3} - 1} = 2^{\frac{2}{3}} \]

### Step 3: Solve for \( T_2 \)
Rearranging gives:
\[ T_2 = T_1 \cdot 2^{-\frac{2}{3}} \]
Substituting \( T_1 = 300 \, K \):
\[ T_2 = 300 \cdot 2^{-\frac{2}{3}} \approx 300 \cdot 0.63 \approx 189 \, K \]

### Step 4: Calculate Change in Internal Energy
The change in internal energy \( \Delta U \) for a monoatomic ideal gas is given by:
\[ \Delta U = n C_v \Delta T \]
Where \( C_v = \frac{3}{2} R \). Thus:
\[ \Delta U = n \cdot \frac{3}{2} R (T_2 - T_1) \]
Substituting the values:
\[ \Delta U = 2 \cdot \frac{3}{2} \cdot 8.31 \cdot (189 - 300) \]
Calculating \( (189 - 300) = -111 \):
\[ \Delta U = 2 \cdot \frac{3}{2} \cdot 8.31 \cdot (-111) \]
\[ \Delta U = -2 \cdot 3 \cdot 8.31 \cdot 111 \approx -5533.26 \, J \]

### Step 5: Calculate Work Done by the Gas
For an adiabatic process, the work done \( W \) is related to the change in internal energy:
\[ W = -\Delta U \]
Thus:
\[ W = 5533.26 \, J \]

### Final Answer
The work done by the gas during the adiabatic expansion is approximately:
\[ W \approx 5533.26 \, J \]

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