Degrees of Freedom
Mono / diatomic / polyatomic — Cv, Cp, γ.
Key Notes
Degrees of freedom (DoF) f: independent ways a molecule can store energy.
Monatomic gas (He, Ar, Ne): 3 translational ⇒ f = 3.
Diatomic at room T (N₂, O₂): 3 translation + 2 rotation = 5. (Vibrations frozen out at room T.)
Diatomic at high T: vibrations contribute 2 more ⇒ f = 7.
Polyatomic non-linear (H₂O, NH₃): 3 translation + 3 rotation = 6 (room T). Vibrations add more at high T.
Equipartition theorem: each quadratic DoF gets ½k_BT per molecule.
Specific heat per mole: C_v = (f/2)R, C_p = (f/2 + 1)R. γ = C_p/C_v = 1 + 2/f.
Monatomic γ = 5/3. Diatomic (room T) γ = 7/5 = 1.4. Polyatomic γ = 4/3 (approximately).
Formulas
Equipartition (per molecule)
Each quadratic degree of freedom gets ½kT.
Internal energy
f-fold larger than monatomic for same T.
Specific heats
Per mole values.
Adiabatic index γ
Monatomic 5/3; diatomic 7/5; polyatomic 4/3.
Important Points
f counts INDEPENDENT QUADRATIC energy storage modes.
Translation always contributes 3 DoF.
Rotation: 2 for linear molecules (no rotation along symmetry axis), 3 for non-linear.
Vibration: each mode = 2 DoF (KE + PE), but FROZEN OUT at low T (quantum effect).
Adiabatic index γ decreases as f increases — diatomic γ = 1.4 vs monatomic 5/3.
At very high T, more vibrational modes activate ⇒ C_v approaches Dulong-Petit limit for solids and high-T behavior for gases.
Degrees of Freedom notes from sciphylab (also known as SciPhy, SciPhy Lab, SciPhy Labs, Physics Lab). Class 11 physics revision for JEE Mains, JEE Advanced, NEET UG, AP Physics 1/2/C, SAT, and CUET-UG.