An ideal gas is carried through a thermodynamic cycle consisting of two isobaric and two isothermal processes, as shown in figure. Shown that the net work done in the entire cycle is given by the equation.
$W_{n e t} = P_{1} (V_{2} - V 1) I n \frac{P_{2}}{P_{1}}$

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Solution

As CD is an isothermal process : $P_{C} V_{C} = P_{D} V_{D}$
$∴$ $P_{2} V_{C} = P_{1} V_{2}$ $⟹ V_{C} = \frac{P_{1} V_{2}}{P_{2}}$

$∴$ $W_{C D} = n R T$ $l n \frac{P_{C}}{P_{D}} = P_{1} V_{2}$ $l n \frac{P_{2}}{P_{1}}$ $(∵ P_{1} V_{2} = n R T)$

Also AB is an isothermal process : $P_{A} V_{A} = P_{B} V_{B}$
$∴$ $P_{1} V_{1} = P_{2} V_{B}$ $⟹ V_{B} = \frac{P_{1} V_{1}}{P_{2}}$

$∴$ $W_{A B} = n R T^{'}$ $l n \frac{P_{A}}{P_{B}} = - P_{1} V_{1}$ $l n \frac{P_{2}}{P_{1}}$ $(∵ P_{1} V_{1} = n R T^{'})$

Process BC :
$W_{B C} = P_{2} (V_{C} - V_{B}) = P_{2} (\frac{P_{1} V_{2}}{P_{2}} - \frac{P_{1} V_{1}}{P_{2}}) = P_{1} V_{2} - P_{1} V_{1}$

Process DA :
$W_{D A} = P_{1} (V_{A} - V_{D}) = P_{1} V_{1} - P_{1} V_{2}$

$⟹$ Net work done $W_{n e t} = W_{A B} + W_{B C} + W_{C D} + W_{D A}$
$∴$ $W_{n e t} =$ $- P_{1} V_{1}$ $l n \frac{P_{2}}{P_{1}} +$ $P_{1} V_{2} - P_{1} V_{1} +$ $P_{1} V_{2}$ $l n \frac{P_{2}}{P_{1}} +$ $P_{1} V_{1} - P_{1} V_{2}$

$⟹$ $W_{n e t} = P_{1} (V_{2} - V_{1})$ $l n \frac{P_{2}}{P_{1}}$

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An ideal gas is carried through a thermodynamic cycle consisting of two isobaric and two isothermal processes, as shown in figure. Shown that the net work done in the entire cycle is given by the equation.Wnet=P1(V2−V1)InP2P1

An ideal gas is carried through a thermodynamic cycle consisting of two isobaric and two isothermal processes, as shown in figure. Shown that the net work done in the entire cycle is given by the equation.
$W_{n e t} = P_{1} (V_{2} - V 1) I n \frac{P_{2}}{P_{1}}$