By Kumar Sir, Physics faculty at Pramana Institute (IIT Kanpur, 25 years teaching, ex-Allen)
Every year, JEE Main and NEET ask at least one question that needs Cv, Cp or γ for a gas. Most students memorise a table of values, then panic when the question mixes two gases or says "linear triatomic". If you know one number, the degrees of freedom f, you can write all three in your head.
Why one number is enough
By the law of equipartition, each degree of freedom carries ½kT of energy per molecule, or ½RT per mole. So the internal energy of n moles is fixed by f, and so is everything that follows from it.
U = (f/2) nRT Cv = (f/2) R Cp = Cv + R = (f/2 + 1) R γ = Cp/Cv = 1 + 2/f
The trick: find f first
Count 3 for translation, then add rotation. Ignore vibration unless the question asks for it.
| Gas type | Examples | f | Cv | Cp | γ |
|---|---|---|---|---|---|
| Monatomic | He, Ne, Ar | 3 | 3R/2 | 5R/2 | 5/3 ≈ 1.67 |
| Diatomic or linear (rigid) | O₂, N₂, CO₂ | 5 | 5R/2 | 7R/2 | 7/5 = 1.4 |
| Non-linear polyatomic (rigid) | H₂O, NH₃, CH₄ | 6 | 3R | 4R | 4/3 ≈ 1.33 |
The CO₂ trap: CO₂ has three atoms, but it is linear, so it rotates about only two axes. Treated as rigid, it has f = 5 and γ = 1.4, the same as a diatomic gas, not 4/3.
Worked example 1: internal energy
Find the internal energy of 2 mol of N₂ at 300 K (R = 8.314 J mol⁻¹ K⁻¹).
N₂ is diatomic, so f = 5. U = (5/2) × 2 × 8.314 × 300 = 12,471 J ≈ 12.5 kJ.
Worked example 2: γ of a mixture
For a mixture, add the Cv contributions of each gas. Since Cv = R/(γ − 1), this gives a one-line rule:
(n₁ + n₂)/(γmix − 1) = n₁/(γ₁ − 1) + n₂/(γ₂ − 1)
For 1 mol of He (γ = 5/3) and 1 mol of O₂ (γ = 7/5): 1/(2/3) + 1/(2/5) = 1.5 + 2.5 = 4. So 2/(γ − 1) = 4, giving γ = 1.5.
Check: Cv = (1.5R + 2.5R)/2 = 2R, Cp = 3R, and γ = 3R/2R = 1.5.
When the trick needs care
- "At high temperature" or "include vibration": each vibrational mode adds 2 to f (kinetic plus potential energy). A diatomic gas then has f = 7, Cv = 7R/2 and γ = 9/7.
- Real gases: these results are for ideal gases. JEE and NEET questions assume an ideal gas unless they say otherwise.
FAQs
What is γ for a diatomic gas? 7/5 = 1.4, if the molecule is treated as rigid with no vibration.
Why does CO₂ have 5 degrees of freedom? It is linear, so rotation about its own axis doesn't count. That leaves 3 translational and 2 rotational degrees.
Is Cp − Cv = R for every ideal gas? Yes. Mayer's relation holds for any ideal gas, per mole.
Practise questions like these in a batch of at most 40 at Pramana, Geeta Bhawan Square, Indore. See our IIT JEE and NEET programs, or book free counselling.
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