JEE Mains Physics · Kinetic Theory

Degrees of Freedom and Specific Heats

Each degree of freedom of a molecule holds ½kT of energy on average, so the count f fixes the molar heat capacities, Cv = fR/2 and Cp = Cv + R, and their ratio γ = 1 + 2/f.

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Why this matters

Eighteen PYQs, one of them asking for a number, and one from 2026. Five count degrees of freedom or state the equipartition law, and thirteen work with Cv, Cp and γ: a formula in terms of f, a value for a given molecule, or a comparison between two gases. Count each vibrational mode as two, and γ follows straight from 1 + 2/f.

Concept 1 of 2: Counting degrees of freedom

A degree of freedom is an independent way for a molecule to hold energy. Every molecule can move in three directions. A linear molecule can also spin about two axes; spin about its own axis stores almost nothing. A non-linear molecule can spin about three. Each mode of vibration stores both kinetic and potential energy, so it counts twice. Equipartition then gives ½kT to every degree of freedom.

Definition

Equipartition: average energy $\tfrac{1}{2}kT$ per degree of freedom per molecule, $\tfrac{1}{2}RT$ per mole.
Translation: 3 for every molecule. A monatomic gas has no rotational degrees of freedom.
Rotation: 2 for a linear molecule (every diatomic, and CO₂), 3 for a non-linear one (H₂O, NH₃, CH₄).
Vibration: each mode adds 2, one kinetic and one potential. At room temperature most diatomics are rigid, with no vibration.
Mean energy per molecule $= \dfrac{f}{2}kT$ . For a rigid diatomic, rotation carries $kT$ and translation $\tfrac{3}{2}kT$ .
In JEE wording, "triatomic" without more detail usually means non-linear.

Gas	f (trans + rot + vib)	Cv	Cp	γ
Monatomic (He, Ne, Ar)	3 (3 + 0 + 0)	3R/2	5R/2	5/3 ≈ 1.67
Rigid diatomic (N₂, O₂ near room temperature)	5 (3 + 2 + 0)	5R/2	7R/2	7/5 = 1.40
Diatomic with one vibrational mode	7 (3 + 2 + 2)	7R/2	9R/2	9/7 ≈ 1.29
Rigid linear triatomic (CO₂)	5 (3 + 2 + 0)	5R/2	7R/2	7/5 = 1.40
Rigid non-linear (H₂O, NH₃, CH₄)	6 (3 + 3 + 0)	3R	4R	4/3 ≈ 1.33
Non-linear with v vibrational modes	6 + 2v	(3 + v)R	(4 + v)R	(4 + v)/(3 + v) Each vibrational mode adds 2 to f, so it adds R to both Cv and Cp.

\gamma = 1 + 2/f

: more degrees of freedom always means a smaller

\gamma

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2024 · 4 April 2024 · Q91Moderate

Example 1 · Kinetic Theory · Degrees of Freedom and Specific Heats

The translational degrees of freedom (f) and rotational degrees of freedom (

f_{r}

) of

CH_{4}

molecule are :

Counting a vibrational mode once

A vibration stores kinetic and potential energy, so one mode adds 2 to f. Counting it as 1 gives the wrong Cv and γ.

Linear and non-linear triatomics

CO₂ is a straight line and has 5 degrees of freedom when rigid; H₂O is bent and has 6. The shape, not the number of atoms, decides the rotations.

Drill 4 more on counting degrees of freedom

Concept 2 of 2: Cv, Cp and γ from the degrees of freedom

One mole of an ideal gas has internal energy (f/2)RT, so warming it by 1 K at constant volume takes (f/2)R: that is Cv. At constant pressure the gas also expands and does work R for every kelvin, so Cp = Cv + R. Their ratio γ = 1 + 2/f falls as f grows, because the extra R becomes a smaller share of a bigger Cv.

Definition

$C_v = \dfrac{f}{2}R$ , $C_p = \left(\dfrac{f}{2} + 1\right)R$ , $C_p - C_v = R$ (Mayer's relation, per mole of ideal gas).
$\gamma = \dfrac{C_p}{C_v} = 1 + \dfrac{2}{f}$ ; $\dfrac{C_v}{C_p} = \dfrac{f}{f + 2}$ ; $f = \dfrac{2}{\gamma - 1}$ .
In terms of $\gamma$ : $C_v = \dfrac{R}{\gamma - 1}$ , $C_p = \dfrac{\gamma R}{\gamma - 1}$ .
$\Delta U = nC_v\Delta T = \dfrac{nR\,\Delta T}{\gamma - 1}$ for any process between two temperatures.
In the classical theory $\gamma$ does not depend on temperature, as long as no new mode of motion switches on.
To compare two gases, find each $\gamma$ from its own f, then divide.

Heat capacities of an ideal gas

C_v = \frac{f}{2}R \qquad C_p = C_v + R \qquad \gamma = 1 + \frac{2}{f}

Worked example

Find Cv, Cp and γ for a rigid non-linear triatomic gas, and the ratio of the γ of a rigid diatomic gas to this one.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2025 · 2 Apr 2025 · Q24Moderate

Example 2 · Kinetic Theory · Degrees of Freedom and Specific Heats

\gamma_{A}

is the specific heat ratio of monoatomic gas A having 3 translational degrees of freedom.

\gamma_{B}

is the specific heat ratio of polyatomic gas

B

having 3 translational, 3 rotational degrees of freedom and 1 vibrational mode. If

\frac{\gamma_{A}}{\gamma_{B}} = \left( 1 + \frac{1}{n} \right)

, then the value of

n

is .

Expecting γ to grow with f

γ = 1 + 2/f, so more degrees of freedom give a SMALLER γ. A gas with vibration has a lower γ than the same gas held rigid.

Cp − Cv = R is per mole

For specific heats per kilogram the difference is R/M. Check the units the question uses before subtracting.

Drill 12 more on cv, cp and γ from the degrees of freedom

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

Cv, Cp and γ from the degrees of freedom
Heat capacities of an ideal gas
$C_v = \frac{f}{2}R \qquad C_p = C_v + R \qquad \gamma = 1 + \frac{2}{f}$

Reference tables (1)

Counting degrees of freedom6 rows

Gas	f (trans + rot + vib)	Cv	Cp	γ
Monatomic (He, Ne, Ar)	3 (3 + 0 + 0)	3R/2	5R/2	5/3 ≈ 1.67
Rigid diatomic (N₂, O₂ near room temperature)	5 (3 + 2 + 0)	5R/2	7R/2	7/5 = 1.40
Diatomic with one vibrational mode	7 (3 + 2 + 2)	7R/2	9R/2	9/7 ≈ 1.29
Rigid linear triatomic (CO₂)	5 (3 + 2 + 0)	5R/2	7R/2	7/5 = 1.40
Rigid non-linear (H₂O, NH₃, CH₄)	6 (3 + 3 + 0)	3R	4R	4/3 ≈ 1.33
Non-linear with v vibrational modes	6 + 2v	(3 + v)R	(4 + v)R	(4 + v)/(3 + v) Each vibrational mode adds 2 to f, so it adds R to both Cv and Cp.

\gamma = 1 + 2/f

: more degrees of freedom always means a smaller

\gamma

Watch out for (4)

Counting a vibrational mode once
→ Counting degrees of freedom
Linear and non-linear triatomics
→ Counting degrees of freedom
Expecting γ to grow with f
→ Cv, Cp and γ from the degrees of freedom
Cp − Cv = R is per mole
→ Cv, Cp and γ from the degrees of freedom

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