Heat, Work and Internal Energy

Internal energy ($U$) of a system is the total energy stored within it due to the random microscopic motion and mutual interaction of its constituent molecules: the kinetic energy of molecules moving, rotating and vibrating, together with the potential energy arising from the forces between them. It does not include any kinetic or potential energy the system possesses as a whole, due to its bulk motion or its position in an external field (a moving or elevated tank of gas has extra mechanical energy, but this is separate from, and additional to, its internal energy).

For an ideal gas in particular, molecules are assumed to exert no forces on one another except during instantaneous collisions, so there is no intermolecular potential energy at all; the internal energy of an ideal gas is therefore purely the kinetic energy of its molecules' random motion, and (since average molecular kinetic energy is set entirely by temperature) depends only on the gas's temperature $T$, and not at all on its pressure or volume individually. This is an important simplifying fact used repeatedly in this unit.

A crucial property of internal energy is that it is a state function: its value at any moment depends only on the current state of the system (for an ideal gas, only on $T$), never on the history of how the system arrived at that state. Consequently, the change in internal energy, $\Delta U = U_f - U_i$, between an initial state and a final state is completely fixed by those two states alone, and is exactly the same whichever path -- however long, twisted, or roundabout -- the system took to get from one to the other. …