Internal Energy
Internal energy (U) is the total energy of a thermodynamic system associated with its microscopic constituents: the kinetic energy of their random motion plus the potential energy of their interactions (and, in chemistry and nuclear physics, chemical and nuclear binding energy), excluding the kinetic and potential energy of the system as a whole. It is a state function; its changes are given by the first law, ΔU = Q − W.
Curious
A cup of hot tea on a moving train has kinetic energy because the train moves, but that is not part of its internal energy. Internal energy is the energy of the molecules' own random motion and the forces between them. Heating the tea or stirring it hard (doing work) both raise its internal energy (see First Law).
For a simple gas, internal energy depends only on temperature. Each way a molecule can move or rotate stores on average (1/2)kBT of energy (the equipartition theorem; see Temperature).
Exploring
| Gas type | Active degrees of freedom (room temp.) | U per mole | Cv per mole |
|---|---|---|---|
| Monatomic (He, Ar) | 3 translational | (3/2)RT | (3/2)R = 12.5 J/K |
| Diatomic (N2, O2) | 3 translational + 2 rotational | (5/2)RT | (5/2)R = 20.8 J/K |
| Diatomic, high T | + 2 vibrational | (7/2)RT | (7/2)R = 29.1 J/K |
Deep Dive
Fundamental relation. Combining the first and second laws gives dU = T dS − p dV + Σμi dNi, so U is naturally a function of S, V and particle numbers. The other potentials, enthalpy, Helmholtz and Gibbs free energies, are Legendre transforms of U suited to different constraints. Only differences of U are physically meaningful in classical thermodynamics; absolute values require a reference, though relativity assigns E = mc² to rest mass (see Mass-Energy Equivalence).
Statistical mechanics. U is the ensemble average of the energy, U = ⟨E⟩ = −∂ ln Z/∂β for the canonical partition function Z. Equipartition gives (1/2)kBT per quadratic degree of freedom only when level spacing is small compared with kBT; vibrational modes of N2 (level spacing about 0.29 eV) are frozen out at room temperature, a quantum effect that puzzled nineteenth-century physicists including Maxwell (see Quantum Harmonic Oscillator).
Joule's free expansion. Joule's 1845 experiment found that a gas expanding into a vacuum did not change temperature measurably, implying that U of an ideal gas depends on T only. More precise later measurements show small changes for real gases because intermolecular forces make U depend on volume as well.
Concept Relationship Map
Requires: first law, temperature
Enables: enthalpy, specific heat capacity, free energies
Commonly confused with: heat; temperature
Sources
- Clausius, R. (1850). Annalen der Physik, 79, 368–397, 500–524.
- Emden, R. (1938). “Why do we have winter heating?” Nature, 141, 908–909.
- Schroeder, D.V. (2000). An Introduction to Thermal Physics. Addison-Wesley.