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Question

An ideal mono atomic gas is expanded from state point A to state point B by the process AB. If W be the work done by the gas and $∆ U$ be the change in its internal energy, then $\frac{W}{∆ U}$ is equal to

Moderate

A

$\frac{3}{4}$
B

$\frac{2}{3}$
C

$\frac{1}{4}$
D

$\frac{1}{3}$

Solution

At A, $P_{0} V_{0} = {nRT}_{0}$ at B, $T_{B} = \frac{2 P_{0} . 2 V_{0}}{nR} = 4 T_{0}$
Now, $W = \frac{1}{2} (P_{0} + 2 P_{0}) (2 V_{0} - V_{0}) = \frac{3}{2} P_{0} V_{0}$
$∆ U = n . C_{v} (T_{B} - T_{A}) = n \frac{3}{2} R (4 T_{0} - T_{0}) = \frac{9}{2} {nRT}_{0} = \frac{9}{2} P_{0} V_{0}$
$∴ \frac{W}{∆ U} = \frac{\frac{3 P_{0} V_{0}}{2}}{\frac{9}{2} P_{0} V_{0}} = \frac{1}{3}$

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