A system undergoes three quasi-static processes sequentially as indicated in the given figure. is an isobaric process, is a polytrophic process with and is a process in which PV= constant. and . The heat transfer of the cycle is change in internal energy is and the work done is . for polytropic process constant. (Take n2 = 0.693). Then

JEE MainPhysicsToughThermodynamics

A system undergoes three quasi-static processes sequentially as
indicated in the given figure. $1 \rightarrow 2$ is an isobaric process,
$2 \rightarrow 3$ is a polytrophic process with
$\gamma = \frac{4}{3}$ and $3 \rightarrow 1$ is a process in which PV=
constant.
$P_{2} = P_{1} = 4 \times 10^{5}Nm^{- 2},P_{3} = 1 \times 10^{5}Nm^{- 2}$ and $\ V_{1} = 1m^{3}$ .
The heat transfer of the cycle is $\Delta Q,$ change in internal
energy is $\Delta U$ and the work done is $\Delta W$ . for polytropic
process $PV^{\gamma} =$ constant.
(Take $\mathcal{l}$ n2 = 0.693). Then

Show answer and solution

For the process

3 \rightarrow 1

P_{1}V_{1} = P_{3}V_{3} \Rightarrow V_{3} = \frac{P_{1}V_{1}}{P_{3}} = \frac{4 \times 10^{5} \times 1}{1 \times 10^{5}} = 4m^{3}

For the process

2 \rightarrow 3

P_{2}V_{2}^{\gamma} = P_{3}V_{3}^{\gamma} \Rightarrow V_{2} = \left( \frac{P_{3}}{P_{2}} \right)^{\frac{1}{\gamma}}V_{3} = 4\left( \frac{1}{4} \right)^{\frac{3}{4}} = 4^{\frac{1}{4}} = \sqrt{2}m^{3}

For the process

1 \rightarrow 2

W_{12} = P_{1}\left( V_{2} - V_{1} \right) = \left( \sqrt{2} - 1 \right)4 \times 10^{5}J

For the process

2 \rightarrow 3

W_{23} = \frac{\left( P_{2}V_{2} - P_{3}V_{3} \right)}{(\gamma - 1)} = 12(\sqrt{2} - 1) \times 10^{5}J

For the process

3 \rightarrow 1

w_{31} = - P_{1}V_{1}ln\frac{V_{3}}{V_{1}} - 4 \times 10^{5} \times 1 \times ln(4)

For cyclic process,

\Delta\ U = 0

\therefore\Delta Q = \Delta U + \Delta W = \Delta W = 4 \times 10^{5}(\sqrt{2} - 1)

1 \rightarrow 2 \rightarrow 3 \rightarrow 1

\Delta Q = + 12 \times 10^{5}(\sqrt{2} - 1) - 4 \times 10^{5} \times 2 \times 0.693

= 1.08 \times 10^{5}J

Source: Grass-G / P-08 Physics paper 23 October.docx

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