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  • A sample of gas at temperature T is adiabatically expanded to double its volume. The work done by the gas in the process is <img alt="\left(\text { given, } \gamma=\frac{3}{2}\right)" s

A sample of gas at temperature T is adiabatically expanded to double its volume. The work done by
the gas in the process is \left(\text { given, } \gamma=\frac{3}{2}\right) :

Option: 1

W=\frac{T}{R}\left [ \sqrt{2}-2 \right ]


Option: 2

W=TR \left [2 -\sqrt{2} \right ]


Option: 3

W=TR\left [ \sqrt{2}-2 \right ]


Option: 4

W=\frac{T}{R} \left [2 -\sqrt{2} \right ]


Answers (1)

Work done in the process is given by

W=\frac{R}{\gamma -1}(T_1-T_2)

For adiabatic process:

\begin{aligned} & \mathrm{T}_1 \mathrm{~V}_1^{\gamma-1}=\mathrm{T}_2 \mathrm{~V}_2^{\gamma-1} \\ & \mathrm{TV}^{\frac{3}{2}-1}=\mathrm{T}_2(2 \mathrm{~V})^{\frac{3}{2}-1} \\ & \mathrm{TV}^{\frac{1}{2}}=\mathrm{T}_2(2 \mathrm{~V})^{\frac{1}{2}} \\ & \mathrm{~T}^2 \mathrm{~V}=\mathrm{T}_2^2 \times 2 \mathrm{~V} \\ & \therefore \mathrm{T}_2=\frac{\mathrm{T}}{\sqrt{2}} \\ & \therefore \mathrm{W}=\frac{\mathrm{R}}{\gamma-1} \times\left(\mathrm{T}-\frac{\mathrm{T}}{\sqrt{2}}\right) \end{aligned}

\begin{aligned} & =2 \mathrm{RT}\left[1-\frac{1}{\sqrt{2}}\right] \\ & =\operatorname{RT}\left[2-\frac{2}{\sqrt{2}}\right] \\ & =\operatorname{RT}[2-\sqrt{2}] \\ & \mathrm{W}=\operatorname{RT}[2-\sqrt{2}] \end{aligned}

Posted by

Sumit Saini

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