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n mole a perfect gas undergoes a cyclic ...

n mole 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 `V_1` to `V_2 = 2V_1` and pressure changes from `P_1` to `P_2`
`B to C` : Isobaric compression at pressure `P_2` to initial volume `V_1`
` C to A` : Isochoric change leading to change of pressure from `P_2` to` P_1` .
Total work done in the complete cycle ABCA is :

A

`nRT(ln2 -1/2)`

B

`nRT(ln2 +1/2)`

C

`nRTln2`

D

0

Text Solution

Verified by Experts

The correct Answer is:
A

`ArarrB` = isothermal process
`B rarrC` = isobaric process
`Crarr A =` isochoric process
Also, `V_2 = 2V_1`
Work done by gas in the complete cycle ABCA is
`implies W = W_(AB)+W_(BC) + W_(CA) " "....(i)`
` W_(CA) = 0`, as isochoric process
`implies W_(AB) =P_1V_1` ln `(v_2/v_1)=nRT ` ln(2)
`implies W_(BC) =P_(2) (V_1-V_2)=P_2 ((V_2)/2-V_2)=P_2V_2(1/2-1) =-(P_2V_2)/2=-(nRT)/2`
Now put the value of `W_(AB), W_(BC) and W_(CA)` in equation, we get,
`implies W=nRT ` ln (2) `-(nRT)/2+0`
`implies W=nRT ["ln (2)"-1/2]`
`implies W=nRT [ ln (2) -1/2]`
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