Internal Energy

Internal energy — the sum of the randomly distributed kinetic and potential energies of all the particles in a body.

Both words matter. Random excludes any orderly motion: a cricket ball thrown across a field has kinetic energy, but that shared motion is not part of its internal energy. Sum means internal energy depends on how much material there is, so it is an extensive quantity, unlike temperature.

The kinetic contribution — depends on temperature alone. Raise the temperature and the particles move faster, so this part increases.

The potential contribution — depends on the separation of the particles, and therefore on the state. Pulling particles apart against their mutual attraction raises their potential energy.

Example: 1 kg of water at 100 °C and 1 kg of steam at 100 °C are at the same temperature, so the particles have the same mean kinetic energy. The steam has far more internal energy, because its particles have been separated and so have much greater potential energy.

Changing internal energy — there are exactly two ways to change the internal energy of a system: transferring energy by heating, or doing work on it.

Example: a bicycle pump warms up when you compress the air quickly. No heating has taken place — you have done work on the gas by pushing the piston in, and that work has raised the internal energy.

An ideal gas is a special case — in an ideal gas the particles are assumed to have no forces between them except during collisions, so there is no molecular potential energy to speak of. The internal energy of an ideal gas is therefore entirely kinetic, and depends only on its temperature. This is why heating a fixed mass of ideal gas and raising its temperature are the same statement.