internal energy

Internal energy is the total energy stored inside a system in the ceaseless jiggling and arrangement of its molecules. Even when a cup of coffee is sitting perfectly still on the table, its molecules are zooming, spinning, and vibrating, and they tug on one another with electrical forces. Add up all that hidden microscopic kinetic and potential energy and you get the internal energy. It answers the question: how much energy does the stuff itself contain, quite apart from how fast the whole object happens to be moving or how high up it sits?

More precisely, the internal energy, written U, is the sum of the kinetic energies of all the particles (their random motion) plus the potential energies of all the forces between them (the bonds and attractions). It deliberately does NOT include the energy of the object as a whole — a moving, high-flying tank of gas has extra bulk kinetic and gravitational energy, but its internal energy is the same as an identical tank at rest on the ground. For an ideal gas, whose molecules do not attract one another, U depends only on temperature: raise T and you raise U.

Internal energy is a state function, meaning it depends only on the current state of the system (its P, V, T), not on the path taken to get there. You usually cannot measure U's absolute value, but you can always measure its change, ΔU, and that is what the first law of thermodynamics is built around. A common confusion: internal energy is not the same as heat. Heat is energy in transit across a boundary; internal energy is energy already parked inside the system.