Internal Energy of Ideal Gas
U = (f/2) nRT — depends only on T.
Key Notes
Internal energy U: total energy of all microscopic motions and interactions inside a system.
For an ideal gas, U depends ONLY on temperature: U = (f/2)·n·R·T, where f = degrees of freedom.
Monatomic gas (He, Ar): f = 3, U = (3/2)nRT. Diatomic gas (N₂, O₂): f = 5, U = (5/2)nRT (translation + rotation).
Polyatomic: more degrees of freedom (translation + rotation + vibration).
ΔU depends only on initial and final STATES — not on path (state function).
Real gases also have potential energy of intermolecular forces; U depends on V slightly.
First law of thermodynamics: ΔU = Q − W. U changes when heat enters or work is done by gas.
Equipartition theorem: each degree of freedom contributes ½k_BT per molecule (½RT per mole).
Formulas
Internal energy (ideal gas)
Depends only on T for ideal gas.
Change in internal energy
C_v = molar specific heat at constant volume = (f/2)R.
Equipartition (per molecule)
Each quadratic degree of freedom: ½k_BT.
Monatomic ideal gas
Only translational KE.
Diatomic (room T)
Translation + rotation (vibration frozen out at room T).
Important Points
U is a STATE FUNCTION — depends only on the state, not on how you got there.
Ideal gas: U = f(T) only. Real gases have small V-dependence due to intermolecular forces.
Heating ideal gas at constant V: all heat goes to ΔU = nC_vΔT. Heating at constant P: some heat is work.
Specific heat C_p > C_v by R (Mayer's relation): heat for V change becomes work.
Equipartition predicts U; quantum effects FREEZE OUT vibrational modes at low T.
Common pitfall: thinking heat = internal energy. NO: heat is energy in transit; U is energy stored.
Internal Energy of Ideal Gas 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.