Temperature ↔ Kinetic Energy
⟨KE⟩ = (3/2) k_B T — linear in T.
Key Notes
Temperature is a measure of AVERAGE KINETIC ENERGY of molecules.
For an ideal gas (monatomic): ⟨KE⟩ = (3/2)k_BT per molecule.
Per mole: ⟨KE⟩ × N_A = (3/2)RT (using k_B·N_A = R).
Linear relation: T ∝ ⟨KE⟩. Absolute T = 0 ⇒ molecules at rest (classical limit).
For diatomic gas (5 DoF): U = (5/2)nRT — translation + rotation.
Polyatomic: more DoF, more energy at same T.
Equipartition theorem: each quadratic degree of freedom gets ½k_BT per molecule.
Temperature is a STATISTICAL concept — needs many molecules to be meaningful.
Formulas
Mean KE per molecule (3D)
Monatomic ideal gas — only translation.
Mean KE per mole
Per mole basis.
Equipartition theorem
Holds for each independent DoF.
Diatomic at room T
3 translational + 2 rotational DoF.
Important Points
T is THE measure of average molecular KE — fundamental definition.
Higher T ⇒ faster molecules ⇒ more KE ⇒ more pressure (at constant V).
Absolute zero (T = 0 K) ⇒ zero molecular motion (classical limit; quantum gives zero-point energy).
Different gases at same T have SAME mean KE per molecule (regardless of m). Heavier ones move slower.
Translation: 3 DoF. Rotation: 2 DoF (for linear molecules) or 3 (for non-linear). Vibration: 2 DoF per mode.
At low T, vibrational modes FREEZE OUT (quantum effect) — only translation remains.
Temperature ↔ Kinetic Energy 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.