10.1 Internal Energy and State Functions

Internal energy U is the microscopic energy stored inside the system. In general it includes random kinetic energy of particles plus potential-energy contributions associated with intermolecular interactions. For an ideal gas, intermolecular forces are neglected except during collisions, so internal energy is modeled as purely kinetic in origin.

That ideal-gas simplification gives a powerful result: internal energy depends only on temperature, not directly on pressure or volume. Pressure and volume still matter because they determine temperature through the equation of state, but for a fixed amount of ideal gas, any two states with equal temperature have equal internal energy.

Each expression highlights a different bridge. The NkBT form emphasizes microscopic particle counting. The nRT form connects to mole-based chemistry and lab quantities. The PV form is convenient in process problems when pressure and volume are directly measured from diagrams or instrument readouts.

This relation is conceptually important: process details do not appear in Delta U for this model. Whether heating was slow or fast, whether pressure stayed constant or not, and whether the path was curved on a PV diagram, the same initial and final temperatures give the same internal-energy change.

A state function depends only on the current thermodynamic state, not on history. Internal energy U, temperature T, pressure P, and volume V are state variables. Heat Q and work W are different: they describe energy transfer during change and therefore depend on path.

This distinction explains why a cyclic process can have nonzero net heat transfer and nonzero net work even though the system returns to its initial state. Since state functions return to original values on a full cycle, net Delta U over one cycle is zero, so first-law balance requires net Q to equal net W for that cycle under our sign convention.

Naming the boundary condition first dramatically reduces mistakes. If the wall is rigid, immediately set W = 0. If the process is adiabatic, set Q = 0 before algebra. These are not optional afterthoughts; they are primary physics constraints that should shape the equation path from line one.

Use the simulation for qualitative reasoning: hold phase fixed and raise temperature to observe kinetic-intensity growth. Then compare two states at equal temperature but different volume contexts to reinforce that, in the ideal model, U tracks temperature directly while pressure and volume influence U only through the state equation.