A sample of 1 mole gas at temperature T is adiabatically expanded to double its volume. If adiabatic constant for the gas is , then the work done by the gas in the process is:

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Solution

Q.
A sample of 1 mole gas at temperature T is adiabatically expanded to double its volume. If adiabatic constant for the gas is $γ = \frac{3}{2}$ , then the work done by the gas in the process is: [2024]

1 $R T [2 - \sqrt{2}]$

2 $\frac{R}{T} [2 - \sqrt{2}]$

3 $R T [2 + \sqrt{2}]$

4 $\frac{T}{R} [2 + \sqrt{2}]$

Ans.
(1)

WD by gas in adiabatic process

$W = \frac{P_{2} V_{2} - P_{1} V_{1}}{1 - γ} = \frac{n R [T_{2} - T_{1}]}{1 - γ}$

$T_{1} V_{1}^{γ - 1} = T_{2} V_{2}^{γ - 1}$

   $T \times {(V)}^{1 / 2} = T_{2} \times {(2 V)}^{1 / 2} \Rightarrow T_{2} = \frac{T}{\sqrt{2}}$

$W = \frac{R [\frac{T}{\sqrt{2}} - T]}{1 - \frac{3}{2}} = \frac{2 R}{\sqrt{2}} [\sqrt{2} T - T]$

   $\Rightarrow W = R T [2 - \sqrt{2}]$

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