JEE MAIN - Physics (2021 - 24th February Morning Shift - No. 5)
n mole of a perfect gas undergoes a cyclic process ABCA (see figure) consisting of the following processes.
A $$ \to $$ B : Isothermal expansion at temperature T so that the volume is doubled from V1 to V2 = 2V1 and pressure charges from P1 to P2
B $$ \to $$ C : Isobaric compression at pressure P2 to initial volume V1.
C $$ \to $$ A : Isochoric change leading to change of pressure from P2 to P1.
Total workdone in the complete cycle ABCA is :
_24th_February_Morning_Shift_en_5_1.png)
A $$ \to $$ B : Isothermal expansion at temperature T so that the volume is doubled from V1 to V2 = 2V1 and pressure charges from P1 to P2
B $$ \to $$ C : Isobaric compression at pressure P2 to initial volume V1.
C $$ \to $$ A : Isochoric change leading to change of pressure from P2 to P1.
Total workdone in the complete cycle ABCA is :
nRTln 2
0
$$nRT\left( {\ln 2 - {1 \over 2}} \right)$$
$$nRT\left( {\ln 2 + {1 \over 2}} \right)$$
Explanation
_24th_February_Morning_Shift_en_5_2.png)
A $$ \to $$ B = isothermal process
B $$ \to $$ C = isobaric process
C $$ \to $$ A = isochoric process
also, V2 = 2V1
work done by gas in the complete cycle ABCA is -
$$ \Rightarrow $$ w = wAB + wBC + wCA .....(1)
$$ \Rightarrow $$ wCA = 0, as isochoric process
$$ \Rightarrow $$ wAB = 2P1V1 ln$$\left( {{{{v_2}} \over {{v_1}}}} \right)$$ = 2 nRT ln(2)
$$ \Rightarrow $$ wBC = P2(V1 $$-$$ V2) = P2 (V1 $$-$$ 2V1) = $$-$$P2V1 = $$-$$nRT
$$ \Rightarrow $$ Now put the value of wAB, wBC and wCA in equation, we get
$$ \Rightarrow $$ w = 2nRT ln(2) $$-$$ nRT + 0
$$ \Rightarrow $$ w = nRT [2ln (2) $$-$$ 1]
$$ \Rightarrow $$ w = nRT [ln (2) $$-$$ $${{1 \over 2}}$$]
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