A sample of gas at temperature T is adiabatically expanded to double its volume. Adiabatic constant for the gas is The work done by the gas in the process is (n = 1 mole)

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Solution

Q.
A sample of gas at temperature T is adiabatically expanded to double its volume. Adiabatic constant for the gas is $γ = 3 / 2 .$ The work done by the gas in the process is (n = 1 mole) [2024]

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

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

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

4 $R T [2 - \sqrt{2}]$

Ans.
(4)

$P V = n R T \Rightarrow P = \frac{n R T}{V}$

Also, $P V^{γ} =$ constant

$\frac{n R T}{V} V^{γ} =$ constant $\Rightarrow T V^{γ - 1} =$ constant

   $T V^{γ - 1} = T_{2} \cdot {(2 V)}^{γ - 1}$

$T_{2} = T \cdot {(\frac{1}{2})}^{1 / 2} = \frac{T}{\sqrt{2}}$

   $W = \frac{n R (T_{1} - T_{2})}{γ - 1} = 2 n R (T - \frac{T}{\sqrt{2}}) = R T (2 - \sqrt{2})$

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