Ncert Exercises & Solutions

The figure shows the \((P\text-V)\) diagram of an ideal gas undergoing a change of state from \(A\) to \(B.\) Four different paths \(\mathrm{I, II, III}\) and \(\mathrm{IV},\) as shown in the figure, may lead to the same change of state.

(a)	The change in internal energy is the same in cases \(\mathrm{IV}\) and \(\mathrm{III}\) but not in cases \(\mathrm{I}\) and \(\mathrm{II}.\)
(b)	The change in internal energy is the same in all four cases.
(c)	The work done is maximum in case \(\mathrm{I}.\)
(d)	The work done is minimum in case \(\mathrm{II}.\)

Which of the following options contains only correct statements?

1.	(b), (c), (d)	2.	(a), (d)
3.	(b), (c)	4.	(a), (c), (d)

Hint: The internal energy is a path-dependent variable.

Step 1: Find the change in internal energy.

The change in internal energy for the process \(A\) to \(B,\) is given by;

\({dU}_{{A} \rightarrow {~B}}={nC}_{{V}} {dT}={nC}_{{V}}\left({~T}_{{B}}-{T}_{{A}}\right)\)

which depends only on temperatures at \(A\) and \(B.\)

Step 2: Find the work done.

The work done for \(A\) to \(B,\) is given by;
\({dW}_{{A} \rightarrow {~B}}=\text { Area under the }{P}\text-{V} \text {curve which is maximum for the path } \mathrm{I} \text {. }\)
Hence, option (3) is the correct answer.

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