An ideal gas undergoes a thermodynamic process in which internal energy $(U)$ of the gas depends on pressure $(P)$ of the gas as $= a P^{4}$ , where a is a positive constant. Assuming gas to be monoatomic, the molar heat capacity of the gas for given process will be

Option 1 - <math> <mfrac> <mrow> <mrow> <mn>3</mn> <mi>R</mi> </mrow> </mrow> <mrow> <mrow> <mn>4</mn> </mrow> </mrow> </mfrac> </math>

Option 2 - <math> <mfrac> <mrow> <mrow> <mn>2</mn> <mi>R</mi> </mrow> </mrow> <mrow> <mrow> <mn>3</mn> </mrow> </mrow> </mfrac> </math>

Option 3 - <math> <mfrac> <mrow> <mrow> <mn>9</mn> <mi>R</mi> </mrow> </mrow> <mrow> <mrow> <mn>4</mn> </mrow> </mrow> </mfrac> </math>

Option 4 - <math> <mfrac> <mrow> <mrow> <mn>4</mn> <mi>R</mi> </mrow> </mrow> <mrow> <mrow> <mn>9</mn> </mrow> </mrow> </mfrac> </math>

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Alok Kumar Singh | Contributor-Level 10

5 months ago

Correct Option - 3

Detailed Solution:

$U = a P^{4}$

$\frac{P V}{γ - 1} = a P^{4}$

$P^{- 3} V = a (γ - 1)$

$P V^{- 1 / 3} = c o n s t a n t$

$C o m p a r i n g w i t h P V^{x} = c o n s t a n t$

$x = - \frac{1}{3}$

$C = \frac{R}{γ - 1} + \frac{R}{1 - x} = \frac{R}{\frac{5}{3} - 1} + \frac{R}{1 + \frac{1}{3}} = \frac{3 R}{2} + \frac{3 R}{4}$

$C = \frac{9 R}{4}$

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Physics Thermodynamics 2025

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An ideal gas undergoes a thermodynamic process in which internal energy ( U ) of the gas depends on pressure ( P ) of the gas as = a P 4 , where a is a positive constant. Assuming gas to be monoatomic, the molar heat capacity of the gas for given process will be

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