Equilibrium, State Variables, and the Zeroth Law

Statistical mechanics derives the laws of heat by counting microstates. Before the counting begins, the target of the derivation has to be stated in its own terms. Thermodynamics is that target: a closed, self-consistent description of macroscopic matter that predates the atomic picture and survives it unchanged. It represents a system of order $10^{23}$ particles by a few numbers — volume, pressure, temperature, energy — and asserts exact relations among them. The whole of the later course can be read as the project of computing those numbers and those relations from a partition function. This lesson sets up the objects the relations are about.

Systems, surroundings, and walls

A thermodynamic system is the portion of the universe under study; the surroundings are everything else that can exchange energy or matter with it. The two are separated by a wall (boundary), and the classification of walls by what they permit fixes what kind of system one has.

• Diathermal wall — permits energy transfer as heat. Two systems separated by a rigid, impermeable, diathermal wall can still change each other's state by exchanging thermal energy.
• Adiabatic wall — forbids heat transfer. A system bounded entirely by adiabatic walls is thermally isolated; only work crosses its boundary.
• Rigid wall — forbids volume change, so the system does no expansion work.
• Permeable / semipermeable wall — permits matter transfer, of all species or of selected ones.

A system that exchanges neither energy nor matter with its surroundings is isolated; one that exchanges energy but not matter is closed; one that exchanges both is open. A reservoir (or bath) is a system so large that finite exchanges with it leave its intensive properties unchanged — a heat reservoir holds its temperature fixed, a particle reservoir its chemical potential. Reservoirs are the fixed backdrops against which the canonical and grand-canonical ensembles are later defined.

State variables and process quantities

A system in equilibrium is specified by a small set of state variables. For a simple fluid, any two of $(P,V,T)$ fix the third and with the particle number $N$ determine every other equilibrium property. State variables split by how they scale when two identical copies of a system are combined:

• Extensive variables double: volume $V$, internal energy $U$, entropy $S$, particle number $N$, magnetization $M$.
• Intensive variables are unchanged: pressure $P$, temperature $T$, chemical potential $\mu$, mass density $\rho$.

A ratio of two extensive variables is intensive; this is the origin of molar and specific quantities. Each extensive variable is paired with an intensive conjugate through the energy: $P$ with $V$, $T$ with $S$, $\mu$ with $N$. Those pairings organize the entire formalism of thermodynamic potentials.

Internal energy, entropy, volume, and temperature are state functions. Heat $Q$ and work $W$ are not. They are process quantities: amounts of energy in transit, defined only for a process, with values that depend on the path taken between endpoints. Their infinitesimals are inexact, written $\delta Q$ and $\delta W$ to mark that they are not the differential of any state function.¹ The distinction is not pedantry; it is the mathematical content of the first and second laws, and the next two lessons turn on it.

Equilibrium and relaxation

A system is in thermodynamic equilibrium when its macroscopic state variables are uniform and unchanging in time, with no macroscopic flows of energy or matter. An isolated system prepared in an arbitrary state relaxes toward equilibrium over a characteristic relaxation time set by its internal dynamics, after which the state variables settle to constant values. Equilibrium thermodynamics describes only these end states, not the relaxation itself; the approach to equilibrium is the province of kinetic theory and the later fluctuation lessons.

Equilibrium is layered. Mechanical equilibrium means the pressure is balanced across every internal boundary; thermal equilibrium means the temperature is uniform; diffusive (or chemical) equilibrium means the chemical potential of each species is uniform. Full equilibrium requires all three. A system can sit in one and not the others: a gas at uniform pressure with a temperature gradient is in mechanical but not thermal equilibrium, and heat will flow until the gradient vanishes.

Quasi-static and reversible processes

A finite-rate process drives a system out of equilibrium: pushing a piston quickly launches pressure waves, and the gas has no single well-defined pressure while they cross it. To keep the state variables meaningful throughout a process, thermodynamics idealizes.

A quasi-static process is the limit of infinitely slow driving; a real process approximates it when the driving time is long compared with the relaxation time. Only for a quasi-static path is the work integral $W=-\int P\mathbf{d}V$ computable from the system's own pressure, because only then is $P$ defined at each step.

Reversibility is a stronger condition. A process is reversible if it can be run backward through the same sequence of states with no net change in either the system or its surroundings. Every reversible process is quasi-static, but not conversely: quasi-static compression against friction is quasi-static yet dissipative, and reversing it does not restore the surroundings. Reversibility additionally forbids dissipation and requires that exchanges occur across vanishing gradients — heat crossing between bodies differing in temperature by an infinitesimal $\mathbf{d}T$, work done against a pressure differing by an infinitesimal $\mathbf{d}P$. The idealization matters because the Carnot bound of the third lesson is saturated only by reversible processes.

The zeroth law and empirical temperature

Temperature enters thermodynamics not as a primitive but as a consequence of an empirical regularity about thermal equilibrium. Write $A\sim B$ for $A$ and $B$ are in thermal equilibrium when placed in diathermal contact — meaning no net heat flows and their state variables cease to change.

The relation $\sim$ is trivially reflexive and symmetric; the zeroth law asserts that it is also transitive, hence an equivalence relation. That is the entire content of the law, and it is not a logical necessity — one can imagine a world where it fails — but an experimental fact.

Because $\sim$ is an equivalence relation, it partitions all systems into disjoint classes, each class comprising the systems that are mutually in thermal equilibrium. Temperature is the label attached to a class: two systems have the same temperature iff they belong to the same class, iff they would not exchange net heat in diathermal contact. Any monotone relabeling of the classes is an equally valid empirical temperature; the zeroth law guarantees that some consistent labeling exists but does not single one out. Fixing a definite scale requires a thermometer.

The ideal-gas thermometer and the equation of state

A thermometer is a system with one easily read state variable — a mercury column's length, a resistor's resistance, a gas's pressure — that is brought to thermal equilibrium with the body of interest and calibrated to report temperature. Different thermometric substances agree only if their thermometric variables happen to track one another, which in general they do not, so the reading depends on the choice of substance.

The dilute gas escapes this ambiguity. Every gas, in the limit of vanishing density, obeys the same relation between pressure, volume, and temperature. Held at fixed volume, the pressure of a dilute gas is linear in the empirical temperature, and extrapolating the pressure to zero defines a substance- independent zero. Fixing one reference point — the triple point of water, assigned $273.16\mathrm{K}$ — sets the ideal-gas (absolute) temperature

with $P$ and $P_{\mathrm{tp}}$ the fixed-volume pressures at the measured temperature and at the triple point.² The limit removes the residual dependence on which gas fills the bulb. This absolute scale later coincides exactly with the thermodynamic temperature defined by the Carnot efficiency, a coincidence proved in the second-law lesson.

For $N$ molecules the dilute-gas relations combine into the ideal-gas equation of state,

where $n=N/N_A$ is the mole number, $R=N_A{k}_B=8.314\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1}$ the gas constant, and $N_A$ Avogadro's number. This is the archetype of an equation of state: a relation $f(P,V,T,N)=0$ constraining the state variables of an equilibrium system. It reduces the independent variables of a simple fluid from three to two and defines a two-dimensional equation-of-state surface in $(P,V,T)$ space. Every equilibrium state of the fluid is a point on that surface; a quasi-static process is a curve drawn on it.

An equation of state is only part of the thermodynamic description. It fixes the mechanical relation among $P$, $V$, and $T$ but says nothing on its own about the energy content or entropy; those require a second relation, the caloric equation of state $U(T,V)$, developed with the first law next. The two together, or equivalently a single fundamental relation such as $S(U,V,N)$, contain the complete thermodynamics of the system — the object the ensembles are built to compute.

Summary

• A thermodynamic system is delimited by walls classified by what they pass: diathermal (heat), adiabatic (none), rigid (no work), permeable (matter). A reservoir is a system large enough to hold its intensive variables fixed under exchange.
• State variables are fixed by the current equilibrium state; they are extensive or intensive and pair into conjugates $(P,V)$, $(T,S)$, $(\mu ,N)$. State functions have exact differentials; heat and work are path-dependent process quantities with inexact $\delta Q$, $\delta W$.
• A quasi-static process passes through equilibrium states so that $P$ and $T$ stay defined; a reversible process is quasi-static and additionally free of dissipation, returnable with no net change in the surroundings.
• The zeroth law makes thermal equilibrium transitive, hence an equivalence relation whose classes are labeled by temperature. The dilute-gas limit fixes a substance-independent absolute scale, and the ideal-gas law $PV=N{k}_{B}T$ is the archetypal equation of state, a surface in $(P,V,T)$ space.