Two moles of a monoatomic ideal gas undergo a cyclic process ABCDA as shown in figure. BCD is a semicircle. The efficiency of the cycle in $5 x$ %. Then the value of $X$ is:

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Solution

$Δ W_{A B} = 0$ $Δ U_{A B} = \frac{P_{o} V_{o}}{2}$ $Δ Q_{A B} = \frac{P_{o} V_{o}}{2}$

$Δ W_{B C} = \frac{π P_{o} V_{o}}{8} + \frac{P_{o} V_{o}}{2}$ $Δ U_{A B} = 2 P_{o} V_{o}$ $Δ Q_{A B} = \frac{π P_{o} V_{o}}{8} + \frac{P_{o} V_{o}}{2} + 2 P_{o} V_{o}$

$Δ W_{C D} = \frac{π P_{o} V_{o}}{8} + \frac{P_{o} V_{o}}{2}$ $Δ U_{C D} = - P_{o} V_{o}$ $Δ Q_{C D} = \frac{π P_{o} V_{o}}{8} + \frac{P_{o} V_{o}}{2} - P_{o} V_{o}$

$Δ W_{D A} = \frac{- 3 P_{o} V_{o}}{4}$ $Δ U_{D A} = \frac{- 3 P_{o} V_{o}}{2}$ $Δ Q_{D A} = \frac{- 9 P_{o} V_{o}}{4}$

$e f f i c i e n c y = \frac{W_{n e t}}{Q_{i n}} = \frac{\frac{π + 1}{4}}{\frac{π}{8} + 3} = 30$

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