Chapter: Physics· JEE

A polyatomic ideal gas has 24 vibrational modes. What is the value of gamma ?

A polyatomic ideal gas has 24 vibrational modes. What is the value of \(\gamma\) ?

(1) 10.3 (2) 1.03 (3) 1.30 (4) 1.37

Doubtly Asked 15 Apr 2021 Updated 23 Oct 2026 Student doubt jeemainmarch2021 Save WhatsApp Report

2 answers

Best answer

(2) 1.03 is the correct option

Explaination ::

f= 3+3 +24 (vibrational modes) =30

\(\gamma = 1+{2\over30}={16\over15}\)

\(\gamma=1.067\)

Doubtly · 15 Apr 2021 Report

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Answer: (2) 1.03

For an ideal gas with \(f\) degrees of freedom, \(\gamma = 1 + \dfrac{2}{f}\).

Count the degrees of freedom of a polyatomic molecule:

Translational: 3
Rotational (non-linear molecule): 3
Vibrational: each vibrational mode has both kinetic and potential energy, so it contributes 2: \(2 \times 24 = 48\)

\[f = 3 + 3 + 48 = 54\]

\[\gamma = 1 + \frac{2}{54} = 1 + \frac{1}{27} \approx 1.037\]

This matches option (2), 1.03. (Counting each vibrational mode as only 1 would give \(f = 30\) and \(\gamma \approx 1.067\), which is not an option.)

Doubtly AI · 23 Oct 2026 AI-verified · Oct 2026 Report

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