JEE/NEET Physics · Kinetic Theory series · Part 3 of 4 · All parts →
- Degrees of freedom (f) = independent ways a molecule can store energy
- Monatomic: f = 3 (translation only) · diatomic: f = 5 (+2 rotation)
- Equipartition: each degree gets (1/2)kT per molecule — energy democracy
- U = (f/2)nRT and C_v = (f/2)R, C_p = C_v + R
- γ = 1 + 2/f: monatomic 5/3, diatomic 7/5 — the shapes pay differently
A helium atom stores energy one way — moving. An oxygen molecule stores it five ways — moving AND tumbling. The same heat divides among more piggy banks, and that single difference explains γ, C_p, C_v, and the speed of sound in every gas. Part 3 of the Kinetic Theory series.
- Counting the piggy banks
- Equipartition: energy democracy
- U = (f/2)nRT and the heat capacities
- γ = 1 + 2/f
- Solved examples
- Common mistakes
- This physics in your daily life
- Practice set
- Recap
Counting the Piggy Banks
A degree of freedom is an independent way to move or store energy. A single atom: 3 translational (x, y, z). A dumbbell molecule: add 2 rotations (about the two axes perpendicular to the bond — spinning about its own bond barely counts, too little inertia). At high temperature vibrations open 2 more, but room-temperature physics uses 3 or 5.
Equipartition: Energy Democracy
U = (f/2)nRT and the Heat Capacities
| Letter | What it means (plain words) | Value / unit |
|---|---|---|
| f | degrees of freedom | mono 3, dia 5 |
| C_v | molar heat capacity, constant volume | (f/2)R |
| C_p | molar heat capacity, constant pressure | C_v + R (the piston’s tip) |
| γ | = C_p/C_v | 1 + 2/f |
γ = 1 + 2/f
Monatomic: γ = 5/3 ≈ 1.67. Diatomic: 7/5 = 1.4 (air!). Molecules with more banks store heat at lower ‘temperature profit’, so their γ is smaller — and γ feeds directly into adiabats (PV^γ), sound speed (√(γP/ρ)), and the previous series’ engine formulas.
Solved Examples
He: monatomic, f = 3. O₂: diatomic, f = 5.
✔
Answer: 3 and 5
C_v = (5/2)R = 20.8 J/(mol·K); C_p = 29.1; γ = 7/5.
✔
Answer: C_v ≈ 20.8 J/mol·K
Translational: (3/2)RT = 3.74 kJ; rotational: (2/2)RT = RT = 2.49 kJ.
Total U = (5/2)RT = 6.23 kJ, of which 40% is tumbling.
✔
Answer: 3.74 kJ translational + 2.49 kJ rotational
- Counting the bond-axis rotation. A diatomic molecule has only TWO meaningful rotations — spinning about its own axis stores ~nothing (negligible moment of inertia).
- f = 3 for everything. Atoms 3, room-temp diatomics 5: the shape decides, not the gas.
- C_p − C_v = R confusion. This holds per MOLE for ideal gases; per kg it’s R/M.
- γ inverted. γ = C_p/C_v > 1 always: more banks (higher f) means SMALLER γ, not larger.
This Physics in Your Daily Life
- Sound travels faster in helium (and your voice goes squeaky) — v_sound = √(γP/ρ): helium’s tiny density wins despite similar γ: party trick by equipartition.
- Divers’ voices underwater sound cartoonish — sound speed changes with the medium’s γ and density: the muffled Donald Duck effect.
- Why air’s γ is 1.4 in every formula — because air is mostly diatomic N₂/O₂ with f = 5: engineering tables are molecule-shape tables.
- Microwave ovens exploit rotational freedom — polar water molecules tumble to align with the oscillating field: rotational degrees of freedom absorbing radiation as heat.
- Infrared spectroscopy reads vibrations — molecules’ extra degrees of freedom absorb specific colours of light: chemistry identifying substances by their piggy banks.
Give ₹100 to a traveller who only walks (helium atom): all of it goes to walking speed. Give ₹100 to a traveller who walks AND tumbles like a gymnast (oxygen molecule): the money spreads across five accounts, and each looks less impressive. Same energy, more banks, smaller temperature rise per rupee — that’s why diatomic gases have bigger C_v.
Helium: C_v = (3/2)R = 12.5 J/mol·K. Oxygen: (5/2)R = 20.8. Add the same 100 J to 1 mol of each: helium warms by 8 K, oxygen only by 4.8 K — the gymnast hid 40% of it in tumbling. γ follows: 1.67 vs 1.40.
Draw the atom as a dot with 3 arrows (x, y, z). Draw the diatomic as a dumbbell: same 3 arrows plus 2 curved rotation arrows about its waist. Every arrow is a piggy bank of size (1/2)kT. The picture IS the count.
Practice set (answers hidden — try first)
(NEET-level) f for a monatomic gas:
(JEE Main-level) C_v for N₂ (dia, R=8.31):
(NEET-level) γ for a diatomic gas:
(Concept) C_p − C_v =
(JEE Main-level) U of 1 mol monatomic gas at 300 K:
- f = independent energy banks
- mono 3, dia 5 (vibrations dormant at room T)
- each bank: (1/2)kT per molecule
- U = (f/2)nRT; C_v = (f/2)R; C_p = C_v + R
- γ = 1 + 2/f
- 🔁 degrees of freedom counting
- 🔁 equipartition rule
- 🔁 U and C formulas by f
- 🧠 Chant: ‘dots have three, dumbbells five’.
- 🧠 γ chain: ‘more banks, smaller gamma’.
- 🏠 Daily: helium squeak = sound speed in γ and ρ.
- 🏠 Daily: microwaves spin water’s rotational banks.
Quick revision
- Degrees of freedom (f) = independent ways a molecule can store energy
- Monatomic: f = 3 (translation only) · diatomic: f = 5 (+2 rotation)
- Equipartition: each degree gets (1/2)kT per molecule — energy democracy
- U = (f/2)nRT and C_v = (f/2)R, C_p = C_v + R
- γ = 1 + 2/f: monatomic 5/3, diatomic 7/5 — the shapes pay differently
- Equipartition: energy democracy
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