Internal Energy

The internal energy $U$ of a thermodynamic system is the sum of the kinetic and potential energies of all its molecules, measured relative to the system's own centre of mass: $U=E_K+E_P$, where $E_K$ (internal kinetic energy) comes from translational, rotational and vibrational molecular motion, and $E_P$ (internal potential energy, e.g. bond energy) comes from molecular interactions. For an ideal gas, molecules are assumed non-interacting, so $U$ is purely kinetic, depending only on temperature and particle number, never on volume (this fails for real gases like van der Waals gases). Internal energy is a state variable -- raising water from 30°C to 40°C by heating or by stirring gives exactly the same final $U$, regardless of path. Crucially, only random molecular motion and molecular interaction energy count toward $U$: the bulk kinetic energy of a moving container of gas, or its bulk gravitational potential energy, are never part of internal energy (see Ex …