Internal energy, heat and work

Temperature marks how hot a body is and fixes the direction of heat flow: from higher to lower temperature, stopping when the temperatures are equal.

Internal energy U adds up the kinetic and the potential energies of all a system's molecules, measured in the frame in which the system's centre of mass is at rest. It counts only the disordered energy of random molecular motion.

If a whole box of gas moves with some velocity, the kinetic energy of the box as a whole is not part of U. In a gas, the molecular kinetic energy includes translational, rotational and vibrational motion.

For a gas, neglecting the small intermolecular forces, U is just the sum of the kinetic energies of the molecules' random motions.

U is a state variable: its value depends only on the present state (given P, V, T), not on the path by which that state was reached.

Two ways to change the internal energy of a gas in a cylinder with a piston: put it in contact with a hotter body so heat flows in, or push the piston in so work is done on it. Both can also run the other way: heat flows out to cooler surroundings, or the gas pushes the piston out and does work.

Heat is energy transferred because of a temperature difference between the system and its surroundings. Work is energy transferred by other means, such as moving a piston by raising or lowering a weight attached to it, with no temperature difference involved.

Heat is energy in transit. 'This gas holds so much heat' makes no more sense than 'this gas holds so much work'; 'this gas has so much internal energy' is meaningful, as is 'so much heat was supplied' or 'so much work was done'.

Heat and work are not state variables: they are two ways of moving energy that change U, which is a state variable. Everyday speech often mixes up heat and internal energy; thermodynamics keeps them apart.