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# One mole of an ideal gas at initial temperature T, undergoes a quasi -static process, during which the volume V is doubled. During the process, the internal energy U varies with volume as U=aV3, where a is a constant. The work done during this process is pRT where p is

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Solution

## Given Internal energyU=aV3⇒fnRT2=aV3 nRT=2aV3f..........(1) where f number of degrees of freedom. Ideal gas equation PV=nRT⇒PV=(2af)V3⇒P=(2af)V2 Now, work done when volume is doubled isW=V2∫V1PdV=2V∫V(2af)V2dV=(2af)(8−1)3V3=(2af)7V33=7nRT3=7RT3 (as n=1 & from (1)) OR Given Internal energy U∝V3 and U∝T ⇒T∝V3...........(1) If V is doubled, T increases 8 times. So, increase in temperature ΔT=8T−T=7T From ideal gas equation T∝PV From (1) ⇒PV∝V3 ⇒P∝V2 ⇒PV−2=Constant Its a Polytropic process with polytropic coefficient x=−2 So Work done in polytropic process W=nRΔT1−x W=(1)R(7T)1−(−2) W=7RT3

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