Internal Energy of a System

Internal energy is the running total of every molecule's kinetic energy, plus any potential energy from forces between them. Stir a cup of water long enough and your mechanical work shoves the molecules around faster; that extra jostling is extra internal energy, and you feel it as a slightly warmer cup. No heat was added through a hot surface, yet the temperature still rose, because work did the same job heat normally does: it increased the molecules' energy.

For an ideal gas, molecules are treated as points that never attract or repel each other, so there is no potential energy to track, only kinetic energy. Kinetic theory shows that average molecular kinetic energy is set entirely by temperature. Add up that kinetic energy over all n moles of a monatomic gas and you get U = 3/2 nRT, with R = 8.314 J/(mol K). Nothing about pressure, volume, or density enters directly; they only matter insofar as they affect temperature. Change the temperature and you change U, full stop, which is why problems on this topic always boil down to tracking a temperature change.

This sets up a clean distinction used constantly in the rest of thermodynamics: internal energy is a stored quantity the gas has, while heat and work are the two ways energy moves across the gas's boundary. A gas has internal energy at every instant; it never has heat sitting inside it. That distinction is exactly what makes the first law of thermodynamics, delta U = Q - W, meaningful: heat in and work done add or remove energy from the store called internal energy.