Thermal Equilibrium and Thermodynamic States

What does thermodynamics describe?

Rubbing your palms makes them warmer, while a steam engine uses energy supplied through heating to move a piston. Thermodynamics studies thermal states and the conversion and transfer of energy through heat and work. Its starting point is the behaviour of a whole sample: a litre of gas, a metal block or water in a vessel. It does not require the position and velocity of every molecule. Pressure P, volume V, absolute temperature T, amount of substance n, and composition are enough to describe many equilibrium situations.

Mechanics asks how forces change a body’s motion. Thermodynamics asks how its internal state changes. A moving bullet has kinetic energy because its centre of mass moves; that fact alone does not make it hotter. On impact, some of that organised kinetic energy becomes internal energy of the bullet and wood, and their temperatures may rise. The two descriptions can cooperate, but centre-of-mass speed and temperature are different quantities.

A microscopic description would track the huge number of molecules and their motions. Kinetic theory instead uses statistical information about molecular motion. Thermodynamics works with directly measurable bulk variables. Molecules continue moving even when the bulk variables are steady: thermodynamic equilibrium does not mean every molecule is motionless or mechanically balanced.

Systems, surroundings and thermal contact

Choose the matter you want to study and call it the system; everything outside it is its surroundings. A boundary separates them. Here the usual system is a fixed amount of gas in a cylinder. A boundary may conduct energy as heat, insulate against heat transfer, or move so that work can cross it. Specifying the boundary is essential: “insulated” tells us about heat transfer, not automatically about work.

An adiabatic wall is a thermal insulator: it prevents heat transfer across it. A diathermic wall conducts energy as heat when there is a temperature difference. Two gas containers separated by a fixed insulating wall can remain at different temperatures. Replacing that wall by a conducting one allows energy to pass from the hotter sample to the colder one until thermal equilibrium is reached. At equilibrium there is no net heat transfer between the samples.

Thermal equilibrium means equal temperatures; it does not require equal pressures, volumes, masses or internal energies. Constraints decide which other variables can change. If both vessels are rigid, their volumes stay fixed while their pressures can change during heating and cooling. A movable conducting partition can also permit volume changes. A movable insulating partition can transmit work, so insulation alone does not guarantee that the states on either side remain unchanged.

The zeroth law gives temperature its meaning

Place A and B separately in thermal contact with a third system C, while keeping A and B thermally isolated from one another. Wait until each is in thermal equilibrium with C. Now thermally isolate C and connect A and B by a conducting boundary. No further net heat transfer occurs. This observation is the zeroth law: systems separately in thermal equilibrium with the same third system are in thermal equilibrium with one another.

A thermometer plays the role of C. A calibrated property, such as the length of a liquid column or the pressure of a fixed-volume gas, changes with temperature. Allowing the thermometer to reach thermal equilibrium with a body lets us assign the same temperature to both. A small thermometer limits disturbance of the body. Calibration supplies numerical values; the zeroth law explains why comparing those values is physically meaningful. Fowler named the law in 1931, after the first and second laws were already established, so it received the earlier number “zeroth”.

Equilibrium states and an equation of state

An equilibrium state has macroscopic variables that remain constant under the stated constraints. Full thermodynamic equilibrium also excludes unbalanced internal gradients that drive change. A steady flow maintained by external heating and cooling is not automatically equilibrium. Mechanical equilibrium of a body means zero net force and torque; thermal equilibrium is about the absence of a temperature-driven energy transfer. These are related conditions, not interchangeable definitions.

For a fixed amount and composition of a simple ideal gas, P, V and T are linked. Choosing any two independent variables fixes the third. Pressure is measured in pascals, volume in cubic metres, T in kelvin, and n in moles. The constant R is about 8.314 J mol⁻¹ K⁻¹. Some books use the symbol μ instead of n for the number of moles; both describe the same quantity here.

At fixed n, the equation PV = nRT defines a surface in P–V–T space. An equilibrium state corresponds to a point on it. An isotherm is the curve cut from that surface at a fixed T; on a P–V plot it satisfies P = nRT/V. During a sudden free expansion, pressure and temperature need not be uniform enough for one pair of values to describe the whole gas. The initial and final equilibrium states lie on the surface; the transient expansion is not a path of equilibrium points on it. An exploding gas mixture is another case where a single bulk P and T may fail during the change.

Intensive and extensive variables

Imagine dividing a uniform equilibrium sample into two equal parts without changing its local state. Each part has half the volume, mass, mole number and internal energy: these are extensive quantities. Each has the original pressure, temperature and density: these are intensive quantities. Density is intensive because both mass and volume halve together, leaving their ratio unchanged.

Scaling checks complement unit checks. In PV = nRT, P and T are intensive, while V and n scale with sample size, so both sides scale equally. Heat and work for corresponding processes also scale with amount, but that does not make them state variables: they describe energy transferred during a path. Internal energy U does belong to the state.

Thermodynamics also uses quantities such as entropy and enthalpy. They are macroscopic state functions, but they are not additional independent coordinates that every simple gas requires. Entropy has a precise thermodynamic definition beyond this chapter; “disorder” is only a limited intuition. Enthalpy is defined as H = U + PV, rather than a universal quantity of heat contained inside matter. The present chapter can be developed using P, V, T and U without adding a separate theory of these quantities.

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