3.1 State, equilibrium, and the equation of state

Thermodynamics is the bookkeeping of energy, heat, and work in systems at or near equilibrium. It describes a macroscopic body not by the positions of its $10^{23}$ molecules but by a handful of state variables, and its first task is to say how many of those are independent and how they constrain one another.

State variables and equilibrium

A fluid in equilibrium is specified by a small set of state variables: pressure $p$, volume $V$ (or specific volume $v = V/m$, or density $\rho = 1/v$), temperature $T$, internal energy $U$, entropy $S$. A variable is a state variable if it depends only on the present equilibrium condition, not on the path by which the system reached it — so a change in it around any closed cycle is zero. Equilibrium itself means the variables are uniform and unchanging: no net flows of heat, matter, or momentum.

The state variables are not independent. For a simple fluid, fixing any two of them fixes all the rest. The relation that ties them together is the equation of state.

The ideal gas

For a dilute gas the equation of state takes its simplest form,

where $n = N/V$ is the number density, $k_B = 1.38\times10^{-23}\,\text{J/K}$ is Boltzmann’s constant, $R = N_A k_B = 8.314\,\text{J/(mol·K)}$ is the universal gas constant, and $M$ the molar mass. The ideal-gas law is the macroscopic shadow of the molecular picture: it follows directly from the kinetic-theory result $p = \tfrac13 n m\langle v^2\rangle$ once the mean molecular kinetic energy is identified with the temperature, $\tfrac12 m\langle v^2\rangle = \tfrac32 k_B T$ (developed in the kinetic theory chapter).

The simulation is a two-dimensional ideal gas at adjustable temperature, number, and volume. Slide $T$, $N$, and $V$ and watch the measured pressure track $pV = N k_B T$ — the gas law emerging as a running average over molecular impacts.

Real gases depart from this law when the molecules are dense enough that their finite size and mutual attraction matter; that correction is the van der Waals equation of state, developed with the intermolecular forces of the liquid state. For the rest of this chapter the ideal gas is the working example, simple enough that every thermodynamic relation can be carried through in closed form.