If one mole of an ideal gas at $$P1$$, $$V1$$ is allowed to expand reversibly and isothermally (A$$ to $$B) its pressure is reduced to $$one-half$$ of the original pressure (see$$ fig.$$) This is followed by a constant volume cooling till its pressure is reduced to $$one-fourth$$ of the initial value B→C. Then is restored to its initial state by a reversible adiabatic compression (C$$ to $$A). The net work done by the gas is equal to:

Question

If one mole of an ideal gas at  $$P1$$,  $$V1$$ is allowed to expand reversibly and isothermally (A$$ to $$B) its pressure is reduced to $$one-half$$ of the original pressure (see$$ fig.$$) This is followed by a constant volume cooling till its pressure is reduced to $$one-fourth$$ of the initial value B→C. Then is restored to its initial state by a reversible adiabatic compression (C$$ to $$A). The net work done by the gas is equal to:

Infinity Learn · Accepted Answer

At A, let say, temperature is T∴$$P1V1=(1)RT$$ ⇒$$TA=T=P1V1R$$−−−−−(1)Work$$ done in isothermal process $$(AB)WAB=(1)RTln2V1V1=RTln(2)----(2)Work$$ done in isochoric process $$(BC)WBC=0----(3)Work$$ done in adiabatic process $$(CA)WCA=$$−△$$U=$$−nCvΔ$$T=$$−$$(1)R$$γ−$$1TA$$−TC−−−−$$(4) here, $$TC=P142V1nR=P1V12R=T2$$−−−−(5) from equation (1) From equations (1),(4) and (5) we get ⇒$$WCA=$$−Rγ−$$1T$$−$$T2=$$−$$RT2($$γ−$$1)$$ Thus, net work ⇒Wnet $$=WAB+WBC+WCA=RTln$$⁡(2)+$$−$$RT2($$γ−$$1)=RTln$$⁡$$2$$−$$12($$γ−$$1)

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