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Physics-

General

Easy

Question

One mole of an ideal gas having initial volume V, pressure 2p and temperature T undergoes a cyclic process ABCDA as shown below.

The net work done in the complete cycle is

Zero
$\frac{1}{2} R T I n 2$
RT In 2
$\frac{3}{2} R T I n 2$

The correct answer is: RT In 2

Work done $increment W equals p increment V$
At constant pressure
$increment W equals p open parentheses V subscript f end subscript minus V subscript i end subscript close parentheses equals n R left parenthesis T subscript f end subscript minus T subscript i end subscript right parenthesis$
At constant temperature
$increment W equals n R T I n open parentheses fraction numerator V subscript f end subscript over denominator V subscript i end subscript end fraction close parentheses equals n R T I n open parentheses fraction numerator P subscript i end subscript over denominator P subscript f end subscript end fraction close parentheses$
$therefore blank increment W subscript A B end subscript equals 1 cross times R cross times open parentheses 2 T minus T close parentheses equals R T$
$increment W subscript B C end subscript equals 1 cross times R cross times 2 T I n fraction numerator 2 p over denominator P end fraction equals 2 R T I n blank 2$
$increment W subscript C D end subscript equals 1 cross times R cross times open parentheses T minus 2 T close parentheses equals negative R T$
$increment W subscript D A end subscript equals 1 cross times R cross times T I n open parentheses fraction numerator p over denominator 2 p end fraction close parentheses equals R T I n blank left parenthesis fraction numerator 1 over denominator 2 end fraction right parenthesis$
Net work done in the complete cycle is
$increment W equals increment W subscript A B end subscript plus increment W subscript B C end subscript plus increment W subscript C D end subscript plus increment W subscript D A end subscript$
$equals R T plus 2 R T I n blank 2 minus R T plus R T I n open parentheses fraction numerator 1 over denominator 2 end fraction close parentheses$
$equals 2 R T I n blank 2 plus R T I n blank 1 minus R T I n blank 2$
$equals 2 R T I n blank 2 minus R T I n blank 2$
$equals R T I n blank 2$

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