The pressure and volume of an ideal gas are related as $\mathrm{PV}^{3 / 2}=\mathrm{K}$ (Constant). The work done when the gas is taken from state $\mathrm{A}\left(\mathrm{P}_{1}, \mathrm{~V}_{1}, \mathrm{~T}_{1}\right)$ to state $\mathrm{B}\left(\mathrm{P}_{2}, \mathrm{~V}_{2}, \mathrm{~T}_{2}\right)$ is:

(A) $2\left(\mathrm{P}_{1} \mathrm{~V}_{1}-\mathrm{P}_{2} \mathrm{~V}_{2}\right)$

(B) $2\left(\mathrm{P}_{2} \mathrm{~V}_{2}-\mathrm{P}_{1} \mathrm{~V}_{1}\right)$

(C) $2\left(\sqrt{\mathrm{P}_{1}} \mathrm{~V}_{1}-\sqrt{\mathrm{P}_{2}} \mathrm{~V}_{2}\right)$

(D) $2\left(\mathrm{P}_{2} \sqrt{\mathrm{~V}_{2}}-\mathrm{P}_{1} \sqrt{\mathrm{~V}_{1}}\right)$

JEE Main · 2024 · 01 Feb 2024 · Shift 1 · Heat & Thermodynamics

The pressure and volume of an ideal gas are related as

PV^{3/2} = K

(Constant). The work done when the gas is taken from state

A (P_{1}, V_{1}, T_{1})

to state

B (P_{2}, V_{2}, T_{2})

is:

(A) $2 (P_{1} V_{1} - P_{2} V_{2})$ ✓ Correct
(B) $2 (P_{2} V_{2} - P_{1} V_{1})$
(C) $2 (P_{1} V_{1} - P_{2} V_{2})$
(D) $2 (P_{2} V_{2} - P_{1} V_{1})$

Answer: (A)

Solution

For a polytropic process

P V^{n} =

constant with

n = \frac{3}{2}

, the work done by the gas as it expands/compresses from state 1 to state 2 is

W = \frac{P _{1} V _{1} - P _{2} V _{2}}{n - 1} = \frac{P _{1} V _{1} - P _{2} V _{2}}{\frac{1}{2}} = 2 (P_{1} V_{1} - P_{2} V_{2})

This is the work done BY the gas — option (1). (Work done ON the gas would carry the opposite sign,

2 (P_{2} V_{2} - P_{1} V_{1})

, option (2); the question as framed — "work done ... when the gas is taken from state A to state B" — is read as work done by the gas, the standard convention, so option (1) stands.)

JEE Main · 2024 · 01 Feb 2024 · Shift 1 · Single correct option

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JEE Main · 2024 · 01 Feb 2024 · Shift 1 · Heat & Thermodynamics

Solution

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