Classical Thermodynamics
Thermodynamics is the physics of heat, work and temperature, stated for macroscopic bodies and without any hypothesis about the constitution of matter. Its raw material is a handful of state variables; its whole content is four laws; and its working machinery is the family of thermodynamic potentials generated from the internal energy by Legendre transforms—the same construction that carries the Lagrangian to the Hamiltonian in Hamiltonian Mechanics. The subject was forced into existence by the steam engine: Carnot's memoir on the motive power of fire [Carnot:1824] predates any correct concept of energy, and the theory built on it has survived every later revolution of physics untouched. The canonical modern exposition is [Callen:1985].
This chapter develops the phenomenological theory: equilibrium and temperature, the equations of state, the first and second laws with their processes and cycles, entropy, the potentials and the Maxwell relations, phase equilibria, and the third law. The molecular reading of the same quantities begins in Kinetic Theory of Gases and is completed in Statistical Mechanics; the experiment that grounds the first law has its own chapter, Experiment: The Mechanical Equivalent of Heat.
Classical Thermodynamics: all derivations of this chapter are pending.
Systems, equilibrium and temperature
State variables and equilibrium
[Reserved: thermodynamic systems, walls and reservoirs; extensive and intensive variables; the equilibrium postulate that a finite set of state variables \((U,V,N,\ldots)\) suffices to describe a macroscopic body at rest, in the axiomatic form of [Callen:1985]; quasi-static and reversible processes; exact and inexact differentials, with heat and work as path functions and the state variables as point functions.]
The zeroth law and empirical temperature
[Reserved: thermal contact and thermal equilibrium; the zeroth law—transitivity of thermal equilibrium—as stated by Maxwell in his textbook of the subject [Maxwell:1871] and later named the zeroth law by Fowler and Guggenheim [Fowler:1939]; the resulting existence of an empirical temperature as a label on isotherms; thermometric substances and the gas scale.]
Let three bodies \(A\), \(B\), \(C\) be brought into thermal contact in pairs. Without exception it is found that if \(A\) is in thermal equilibrium with \(C\), and \(B\) is in thermal equilibrium with \(C\), then \(A\) and \(B\) are already in thermal equilibrium when placed in contact: no further change of any of their state variables occurs. This is the fact that makes a thermometer possible—\(C\) being a small standard instrument read in place of the bodies themselves. It was made the ground of thermometry by Maxwell [Maxwell:1871] and named the zeroth law by Fowler and Guggenheim [Fowler:1939].
Derivation. Write \(A\sim B\) for “\(A\) and \(B\) are in thermal equilibrium”. The relation is reflexive and symmetric by construction, and the observed regularity is precisely transitivity; hence \(\sim\) is an equivalence relation on the set of equilibrium states, and its classes partition that set. For a simple fluid, whose equilibrium states are labelled by the pair \((P,V)\), equilibrium with one fixed reference body is a single condition on two variables, so each class is a curve in the \((P,V)\) plane—an isotherm. Choose a thermometric body together with a coordinate that varies monotonically along its own states, and assign to every class the value \(\theta\) that coordinate takes when the reference body is equilibrated with a member of the class. By construction
Any second assignment \(\theta'\) with the same property labels the same classes, so \(\theta'=g(\theta)\) with \(g\) strictly monotone: empirical temperature is fixed only up to a monotone relabelling. Removing that residual freedom is the work of Section 54.4.3.
∎Thermometry and the kelvin
[Reserved: William Thomson's absolute thermometric scale founded on the Carnot cycle [Thomson:1848], anticipating Section 54.4.3; the constant-volume gas thermometer as its classical realisation; the modern kelvin, defined since 2019 by fixing the Boltzmann constant at \(1.380649\times 10^{-23}\,\mathrm{J}/\mathrm{K}\) [BIPM:2019], and its relation to the measurement conventions of Measurement, SI Units, and the Theory of Errors.]
Equations of state
The gas laws
[Reserved: Boyle's law \(PV=\text{const}\) at fixed temperature, from the pressure–volume data of the 1662 defence of the spring of the air [Boyle:1662]; Gay-Lussac's law of thermal expansion, with the equal expansivity of all gases [GayLussac:1802] and Gay-Lussac's own attribution of earlier unpublished work to Charles; Dalton's law of partial pressures [Dalton:1802]; Avogadro's hypothesis that equal volumes at equal temperature and pressure hold equal numbers of molecules [Avogadro:1811].]
The ideal-gas equation of state
[Reserved: the combination of the gas laws into the single equation \(PV=nRT\) by Clapeyron [Clapeyron:1834]; the gas constant, exact since 2019 as the product of the Boltzmann and Avogadro constants, \(8.314462618\,\mathrm{J}/\mathrm{mol}/\mathrm{K}\) [BIPM:2019] [Mohr:2025]; Regnault's precision measurements of the small departures of real gases from the law [Regnault:1847], the experimental seed of Section 54.6.3; the ideal gas as the universal dilute limit.]
Every gas, whatever its chemical species, obeys one and the same equation of state in the limit of low density. Three separate observations enter it: at fixed temperature the product \(PV\) of a fixed quantity of gas is constant [Boyle:1662]; at fixed pressure the volume grows with temperature by a coefficient of thermal expansion that is the same for all gases [GayLussac:1802]; and at fixed pressure and temperature equal volumes of different gases contain equal numbers of molecules [Avogadro:1811]. Together they say that
with one constant \(R\) common to every substance. Departures from Equation (54.2) set in at finite density and were mapped systematically by Regnault [Regnault:1847]; they are the subject of Section 54.6.3. Since the redefinition of the SI in 2019 the gas constant is exact, \(R=8.314462618\,\mathrm{J}/\mathrm{mol}/\mathrm{K}\) [BIPM:2019] [Mohr:2025].
Derivation. Fix the amount of gas. Boyle's observation says that on each isotherm the product \(PV\) depends on the empirical temperature alone, \(PV=f(\theta)\), so it is a label on the classes of Equation (54.1). Gay-Lussac's says that at fixed pressure \(V=V_{0}(1+\alpha\theta_{\mathrm{C}})\), with \(\theta_{\mathrm{C}}\) the Celsius reading and \(\alpha\) the same number for every gas; defining
turns it into \(V\propto T\) at fixed \(P\). The two together force \(f\) to be proportional to \(T\), so \(PV=CT\) with \(C\) depending only on how much gas is present. Avogadro's observation fixes that dependence: at given \(P\) and \(T\) the volume is proportional to the number of molecules, hence \(C=nR\) with \(R\) independent of the species, which is Equation (54.2). The scale Equation (54.3) so constructed is the ideal-gas temperature; that it coincides with the substance-independent scale of thermodynamics is shown in Section 54.4.3, and only then is \(T\) more than a convention.
∎The first law
Work, heat and internal energy
[Reserved: the first law \(\dd U=\delta Q-\delta W\): internal energy as a state function whose change splits into heat and work, neither of which is one by itself; Mayer's inference of the equivalence of heat and work from the two heat capacities of air [Mayer:1842]; Joule's paddle-wheel measurement of the mechanical equivalent [Joule:1850], treated in full in Experiment: The Mechanical Equivalent of Heat; Helmholtz's statement of the conservation of energy across all of physics [Helmholtz:1847].]
Carry a thermally insulated body from one equilibrium state to another by doing work on it. The work required is found to depend only on the two states and not at all on how it is delivered: a paddle wheel churning water, the same wheel churning mercury, cast iron rubbing on cast iron, and an electric current driven by a falling weight all produce the same change of state for the same work [Joule:1850] [Joule:1843]. Equivalently, heat and work are interconvertible at one fixed rate—the mechanical equivalent of heat—first inferred from the two heat capacities of air by Mayer [Mayer:1842] and then measured directly by Joule [Joule:1850]. The measurement is treated in full in Experiment: The Mechanical Equivalent of Heat.
Derivation. Let \(W_{\mathrm{ad}}(A\to B)\) denote the work done by an insulated system in passing from state \(A\) to state \(B\); the observation is that this number depends on \(A\) and \(B\) alone. Fix a reference state \(O\) and define
which is well posed precisely because of the observed path independence, and which fixes a state function \(U\) up to the additive constant \(U(O)\). Join now the same two states by a process that is not insulated; the work \(W\) done by the system differs from \(W_{\mathrm{ad}}\), and the deficit
defines the heat absorbed. It is an amount of energy, measured in joules, and owns no independent unit; in differential form Equation (54.5) is the first law \(\dd U=\delta Q-\delta W\). The converse shows why the observation is not a triviality: if \(W_{\mathrm{ad}}\) depended on the path, one could carry the system from \(A\) to \(B\) along the costlier route and back along the cheaper one, returning it to \(A\) while extracting net work from nothing—perpetual motion of the first kind, which no mechanism has ever achieved.
∎Heat capacities
[Reserved: heat capacities at constant volume and constant pressure; enthalpy; the Mayer relation \(C_{P}-C_{V}=nR\) for the ideal gas, the relation from which Mayer extracted his equivalent of heat [Mayer:1842]; Regnault's calorimetric data [Regnault:1847]; the ratio \(\gamma=C_{P}/C_{V}\) and its appearance in the adiabatic law and in Laplace's correction of Newton's speed of sound [Laplace:1816].]
Processes and cycles
[Reserved: isothermal, isobaric, isochoric and adiabatic processes of the ideal gas; the adiabatic law \(PV^{\gamma}=\text{const}\); work as the area of the indicator diagram, the representation Clapeyron introduced [Clapeyron:1834]; free expansion into vacuum as the standard irreversible contrast, tested by Gay-Lussac [GayLussac:1807] and by Joule, and refined into the throttling experiment of Section 54.6.4.]
The second law
Carnot's theorem
[Reserved: the Carnot cycle of two isotherms and two adiabats; Carnot's theorem that no engine working between two reservoirs outperforms a reversible one, and that all reversible engines between the same reservoirs share one efficiency [Carnot:1824]—an argument that survived the caloric theory it was written in; Clapeyron's analytic and graphical restatement [Clapeyron:1834].]
The Clausius and Kelvin statements
[Reserved: Clausius's 1850 reconciliation of Carnot's theorem with the first law, and his statement that heat does not pass by itself from a colder to a hotter body [Clausius:1850]; Thomson's statement that no cyclic process converts heat wholly into work [Thomson:1853]; the proof that the two statements are equivalent; perpetual motion of the second kind.]
Two negative statements about cyclic processes have never been contradicted by any device built or proposed.
-
Clausius. No process has as its sole result the transfer of heat from a colder to a hotter body [Clausius:1850].
-
Kelvin. No cyclic process has as its sole result the complete conversion into work of heat drawn from a single reservoir at one temperature [Thomson:1853].
Neither is a consequence of the first law, which permits both: an engine converting the thermal energy of the ocean entirely into work would conserve energy exactly. They are an independent fact about Nature, and the modern form of the impossibility Carnot had already taken for granted [Carnot:1824].
Derivation. The two statements are equivalent, so the observation is a single one. Suppose Kelvin's statement false: some cyclic device draws heat \(Q>0\) from the cold reservoir and delivers work \(W=Q\) with no other effect. Dissipate that work as heat in the hot reservoir by friction. The compound operation is cyclic and its only result is the passage of \(Q\) from cold to hot, so Clausius's statement fails. Conversely, suppose Clausius's statement false: some device passes heat \(Q_{\mathrm{c}}\) from the cold reservoir to the hot one with no other effect. Beside it run an ordinary engine that takes \(Q_{\mathrm{h}}\) from the hot reservoir, delivers work \(W=Q_{\mathrm{h}}-Q_{\mathrm{c}}\) and rejects exactly \(Q_{\mathrm{c}}\) to the cold one. In the compound cycle the cold reservoir is left unchanged, and the sole result is the conversion into work of \(Q_{\mathrm{h}}-Q_{\mathrm{c}}\) drawn from the hot reservoir alone, so Kelvin's statement fails. Each denial implies the other, hence the two statements stand or fall together.
∎Thermodynamic temperature and the Carnot efficiency
[Reserved: the universality of the Carnot efficiency as the definition of a temperature independent of every thermometric substance, with \(\eta=1-T_{2}/T_{1}\) [Thomson:1848] [Thomson:1853]; the proof that this scale coincides with the ideal-gas scale of Section 54.2.2.]
A device working in a cycle between two reservoirs held at temperatures \(T_{\mathrm{h}}>T_{\mathrm{c}}\) never converts into work more than the fraction
of the heat it draws from the hot reservoir; real engines fall below it, and every engine that is reversible attains it exactly, whatever its working substance and whatever the mechanism [Carnot:1824] [Thomson:1853]. It is the independence from the working substance that lets Equation (54.6) be read backwards, as the definition of a temperature scale owing nothing to any thermometric material [Thomson:1848].
Derivation. Universality. Let \(E\) be reversible and \(X\) arbitrary, both cycling between the same two reservoirs, and choose the number of cycles of each so that they deliver the same total work \(W\). Run \(X\) forward and \(E\) backward. The compound device is cyclic and delivers no net work, so by the first law the heat it takes from the cold reservoir equals the heat it gives to the hot one, namely \(Q^{E}_{\mathrm{h}}-Q^{X}_{\mathrm{h}}\). Now \(Q^{X}_{\mathrm{h}}=W/\eta_{X}\) and \(Q^{E}_{\mathrm{h}}=W/\eta_{E}\), so if \(\eta_{X}>\eta_{E}\) that quantity is positive and the sole result of the compound cycle is a passage of heat from the colder reservoir to the hotter, contradicting Phenomenon 54.4. Hence \(\eta_{X}\leq\eta_{E}\); taking \(X\) reversible as well and exchanging the roles gives equality. All reversible engines between the same pair of reservoirs therefore share one efficiency, a function of the reservoirs alone.
Evaluation. That efficiency may then be computed with any convenient substance. Take one mole of ideal gas, with \(C_{V}\) constant for simplicity, around the cycle \(1\to2\) isothermal at \(T_{\mathrm{h}}\), \(2\to3\) adiabatic, \(3\to4\) isothermal at \(T_{\mathrm{c}}\), \(4\to1\) adiabatic. The internal energy of an ideal gas depends on \(T\) alone (Experiment: The Mechanical Equivalent of Heat), so on an isotherm the heat absorbed equals the work done and
\(Q_{\mathrm{c}}\) being the heat rejected. On an adiabat \(C_{V}\dd T=-P\dd V=-RT\,\dd V/V\), which integrates to \(TV^{R/C_{V}}=\text{const}\); the two adiabats give \(T_{\mathrm{h}}V_{2}^{R/C_{V}}=T_{\mathrm{c}}V_{3}^{R/C_{V}}\) and \(T_{\mathrm{h}}V_{1}^{R/C_{V}}=T_{\mathrm{c}}V_{4}^{R/C_{V}}\), and dividing one by the other leaves \(V_{2}/V_{1}=V_{3}/V_{4}\). The logarithms in Equation (54.7) therefore cancel in the ratio,
and \(\eta=1-Q_{\mathrm{c}}/Q_{\mathrm{h}}\) is Equation (54.6). Since the left-hand side of Equation (54.8) was just shown to be independent of the working substance, while the \(T\) on its right is the ideal-gas temperature Equation (54.3), the thermodynamic and ideal-gas scales agree wherever both are defined.
∎Entropy and the Clausius inequality
[Reserved: the Clausius equality, that \(\delta Q/T\) integrates to zero around any reversible cycle [Clausius:1854]; entropy as the resulting state function, named by Clausius in 1865 together with the summary that the energy of the world is constant and its entropy tends to a maximum [Clausius:1865]; the Clausius inequality and the increase of entropy in isolated systems; the entropy of the ideal gas.]
Carathéodory's axiomatics
[Reserved: the axiomatic route of Carathéodory, in which the second law is the existence, near every state, of states inaccessible by adiabatic processes, and entropy arises as an integrating factor of \(\delta Q\) [Caratheodory:1909]; its equivalence with the engine statements; what the axiomatisation clarifies and what it hides.]
Thermodynamic potentials
Legendre transforms and the four potentials
[Reserved: internal energy, enthalpy, the Helmholtz free energy introduced as the “free energy” of chemical processes [Helmholtz:1882], and the Gibbs free energy, all as Legendre transforms of one fundamental relation, in the systematic form Gibbs gave the subject [Gibbs:1876]; the natural variables of each potential, and the parallel with the Legendre transform of Hamiltonian Mechanics.]
Maxwell relations
[Reserved: the four Maxwell relations as equalities of mixed second derivatives of the potentials, in the form given in Maxwell's Theory of Heat [Maxwell:1871]; their use to reduce any measured response to derivatives of state variables; worked identities for the ideal and the van der Waals gas.]
Extremum principles, stability and the chemical potential
[Reserved: maximum entropy and minimum energy; minimisation of the free energies at fixed intensive variables; concavity and convexity of the potentials, with positive heat capacities and compressibilities as stability conditions; the chemical potential, the Euler relation and the Gibbs–Duhem identity, all from [Gibbs:1876]; exposition following [Callen:1985].]
Phase equilibria
Coexistence and the Clausius–Clapeyron equation
[Reserved: equality of temperature, pressure and chemical potential across a phase boundary; latent heat; the Clausius–Clapeyron equation for the slope of the coexistence curve, from Clapeyron's memoir [Clapeyron:1834] as put on a first-law footing by Clausius [Clausius:1850]; applications to vapour pressure and to the melting line of ice.]
The Gibbs phase rule
[Reserved: the phase rule \(f=c-p+2\) counting the freedoms of a multicomponent, multiphase system [Gibbs:1876]; triple points, with the former role of the water triple point in the definition of the kelvin [BIPM:2019] noted; azeotropes and eutectics named as applications.]
Real gases: van der Waals and the critical point
[Reserved: Andrews's isotherms of carbon dioxide and the discovery of the critical point, with the continuity of the liquid and gaseous states [Andrews:1869]; the van der Waals equation of state from his 1873 dissertation [vanderWaals:1873]; the Maxwell equal-area construction [Maxwell:1875]; the law of corresponding states; where the mean-field picture fails near the critical point, deferred to Phase Transitions and Critical Phenomena.]
Above a definite temperature \(T_{\mathrm{c}}\), characteristic of the substance, no pressure whatever liquefies a gas. Following the coexistence curve upwards, the difference in density between the liquid and its vapour shrinks and the meniscus between them fades and disappears at a single point \((T_{\mathrm{c}},P_{\mathrm{c}}, V_{\mathrm{c}})\); beyond it the two states pass continuously into one another. Andrews established this for carbon dioxide by mapping its isotherms through the point [Andrews:1869]. Plotted in units of their own critical constants, the isotherms of chemically unrelated substances then very nearly coincide—the law of corresponding states [vanderWaals:1873].
Derivation. Van der Waals's equation of state for one mole, with \(v=V/n\),
corrects the ideal gas Equation (54.2) for the volume \(b\) the molecules exclude and for the attraction between them, which lowers the pressure by \(a/v^{2}\). At high temperature \(P\) is monotone in \(v\); below some temperature it is not, and a horizontal tie line then cuts the isotherm in three. The two behaviours are separated by the isotherm on which the stationary points merge, so the critical point is fixed by
Differentiating Equation (54.9) these read \(RT/(v-b)^{2}=2a/v^{3}\) and \(RT/(v-b)^{3}=3a/v^{4}\); dividing the second by the first gives \(1/(v-b)=3/(2v)\), hence
the last two by substituting \(v_{\mathrm{c}}\) back. A critical point therefore exists for every substance obeying Equation (54.9), which is the observed fact. Measuring \(P\), \(v\) and \(T\) in units of Equation (54.11) eliminates \(a\) and \(b\) from Equation (54.9) altogether,
so all such substances have one reduced equation of state: the law of corresponding states, together with the substance-independent prediction \(P_{\mathrm{c}}v_{\mathrm{c}}/(RT_{\mathrm{c}})=3/8\). Both are mean-field statements, and both are only approximate; how they fail in the immediate neighbourhood of the critical point is the subject of Phase Transitions and Critical Phenomena.
∎The Joule–Thomson effect
[Reserved: throttling through a porous plug at constant enthalpy; the Joule–Thomson coefficient, its inversion temperature, and its vanishing for the ideal gas, from the Joule–Thomson experiments of 1852–1862 [Joule:1852], whose apparatus and data are treated in Experiment: The Mechanical Equivalent of Heat; gas liquefaction as the application.]
The third law
[Reserved: Nernst's heat theorem, that entropy differences vanish as the temperature goes to zero [Nernst:1906]; the unattainability of absolute zero in finitely many steps; consequences—vanishing heat capacities and expansivities at low temperature—whose observed forms are quantum-statistical and are derived in Quantum Statistics.]
As the temperature is lowered towards absolute zero, the entropy change accompanying any isothermal change of the remaining variables tends to zero:
for any two values \(x_{1},x_{2}\) of those variables. Nernst extracted this from the low-temperature behaviour of the heats and free energies of chemical reactions, where the two quantities were found to converge on one another [Nernst:1906]. Two consequences are themselves observed: the heat capacities of condensed matter fall away as \(T\to0\), in flat contradiction of the classical constant of Kinetic Theory of Gases, and absolute zero is never reached, each successive stage of any cooling technique yielding a smaller temperature drop than the last.
Derivation. Both consequences follow from Equation (54.13) and the positivity of heat capacities. For the first, integrate \(\dd S=C_{x}\dd T/T\) at fixed \(x\) from zero:
The left side is finite, so the integral converges at its lower limit; a heat capacity tending to any nonzero constant would make it diverge logarithmically. Hence \(C_{x}(T)\to0\) as \(T\to0\), and the Dulong–Petit value cannot survive to low temperature.
For the second, suppose absolute zero were reached in one adiabatic step, carried out reversibly and therefore at constant entropy, from a state \((T,x_{1})\) with \(T>0\) to a state \((0,x_{2})\). That requires \(S(T,x_{1})=S(0,x_{2})\). But Equation (54.13) makes \(S(0,x_{1})=S(0,x_{2})\), while Equation (54.14) with \(C_{x}>0\) makes \(S(\cdot,x_{1})\) strictly increasing, so \(S(T,x_{1})>S(0,x_{1}) =S(0,x_{2})\) for every \(T>0\)—a contradiction. No such step exists, and since the argument applies to each step separately, no finite sequence of them reaches zero either. Only the microscopic ground of Equation (54.13) itself lies outside thermodynamics; it is derived from the non-degeneracy of the ground state in Quantum Statistics.
∎The bridge to the molecular theory
[Reserved: what the phenomenology cannot fix—the values of heat capacities, the entropy constant, the size of fluctuations—and where each is settled: the kinetic theory of Kinetic Theory of Gases, the ensembles of Statistical Mechanics, and the fluctuation phenomena made visible by Brownian motion in Experiment: Brownian Motion and Avogadro's Number; Boltzmann's statistical reading of entropy [Boltzmann:1877] anticipated here.]