Zeroth Law of Thermodynamics and Empirical Temperature

Thermodynamics rests on a handful of empirical laws distilled from experience, and one of them is so intuitive it went unstated for a century. Bring two systems into contact through a heat-conducting (diathermal) wall and wait: eventually nothing more changes, and they are said to be in thermal equilibrium. The Zeroth Law — named by Fowler and Guggenheim precisely because it logically precedes the First and Second — asserts that this relation is transitive: if $A$ is in equilibrium with $C$ , and $B$ is in equilibrium with $C$ , then $A$ and $B$ are in equilibrium with each other. Stated so plainly it sounds like nothing. Its consequences are anything but. The deep payoff of transitivity is the existence of a state function: temperature. Because mutual equilibrium behaves like equality — reflexive, symmetric, transitive — it partitions all equilibrium states into equivalence classes, and each class can be tagged by a single number $\theta$ , the empirical temperature. Two systems share this tag if and only if they would equilibrate. That is exactly what lets a thermometer work: instead of touching every pair of bodies together, you equilibrate each separately with one reference body — the thermometer — and compare its readings. Without the Zeroth Law, "reading a temperature" would be meaningless. Concretely, a simple fluid is described by its pressure $p$ and volume $V$ . Bringing two such systems into diathermal contact imposes one constraint among their four coordinates, and the Zeroth Law forces that constraint to factor into the clean statement $\theta_A=\theta_B$ . Eliminating the reference system yields an equation of state $f(p,V,\theta)=0$ whose constant- $\theta$ level curves are the isotherms. What the Zeroth Law does not provide is a privileged, substance-independent scale — temperature is fixed only up to an arbitrary monotonic relabelling, which is why Celsius, Fahrenheit, and the ideal-gas scale all describe the same physics. The absolute scale, with a true zero and additive spacing, waits for the Second Law and the Carnot cycle. The Zeroth Law lays the foundation: temperature exists, and it is measurable.

Key takeaways

The Zeroth Law states that thermal equilibrium is transitive: if $A\sim C$ and $B\sim C$ , then $A\sim B$ .
Transitivity makes thermal equilibrium an equivalence relation, so equilibrium states split into classes labelled by one number — the empirical temperature $\theta$ .
$\theta$ is therefore a state function: all mutually equilibrated systems share its value, and none share it otherwise.
An equation of state $f(p,V,\theta)=0$ relates a system's mechanical coordinates; its constant- $\theta$ curves are the isotherms.
Temperature is fixed only up to a monotonic relabelling $\theta'=g(\theta)$ with $g'>0$ ; the Zeroth Law fixes ordering, not an absolute metric.
Operationally it legitimizes the thermometer: a reference body $C$ whose reading defines temperature for everything it equilibrates with.
Key idea: Transitivity of thermal equilibrium is an equivalence relation on equilibrium states; its equivalence classes are labelled by a single number, the empirical temperature $\theta$ , which is therefore a well-defined state function shared by all mutually equilibrated systems.

Where this lives today

The Zeroth Law is quietly at work in every calibrated measurement. The 2019 redefinition of the SI made temperature a derived quantity by fixing the Boltzmann constant exactly at $k_{\mathrm B}=1.380649\times10^{-23}\ \mathrm{J\,K^{-1}}$ ; primary thermometry now realizes the kelvin through the empirical relation $\theta\propto pV$ (acoustic and dielectric-constant gas thermometers measure it directly), leaning entirely on the transitivity that lets a reference gas stand in for any body. National metrology labs propagate that realization through chains of equilibrated standards — a physical enactment of the transitivity triangle. The idea also matters at frontiers where "temperature" is not obviously well defined. Assigning an effective temperature to a driven or glassy system, to a subsystem of an isolated quantum many-body system, or to a black hole all begin by asking whether an equilibrium-like, transitive relation exists that a temperature could label. When it does not — when a body has internal gradients, or never reaches a stationary state — the Zeroth Law warns that a single $\theta$ simply is not defined, and any number you write down is at best a local approximation.

The physical picture

Picture three sealed containers. Touch $A$ to $C$ through a metal plate and wait until nothing changes; do the same for $B$ and $C$ . The Zeroth Law promises that if you now touched $A$ to $B$ directly, nothing would change either — no net heat would flow. Strip that guarantee away and the very idea of "reading a thermometer" collapses, because a thermometer is nothing more than system $C$ that you equilibrate with everything in turn. Think of mutual equilibrium as a colouring of all possible equilibrium states. The Zeroth Law says the colouring is consistent: every state gets exactly one colour, and two states have the same colour precisely when they would equilibrate. The colours can be relabelled by any strictly increasing function — which is exactly why Celsius, Fahrenheit, and an ideal-gas scale all encode the same physics. The colour is the temperature; the numbers on it are a convention. Mechanical equilibrium across a movable piston equalizes pressure; thermal equilibrium across a diathermal wall equalizes temperature. Temperature is the intensive variable conjugate to heat flow, just as pressure is conjugate to volume change — it is the quantity whose equality signals that no net heat crosses the wall. Each simple body already has two mechanical coordinates $(p,V)$ ; thermal contact adds exactly one new equilibrium condition, and transitivity forces that condition to be the equality of a single scalar field over state space.

Figure

The transitivity triangle: A~C and B~C force A~B

Figure

Isotherms: constant-θ level curves of the equation of state

The core equations

The Zeroth Law itself is a single logical statement about the equilibrium relation $\sim$ . Everything else is unpacking its consequences.

A\sim C\ \text{ and }\ B\sim C\ \Longrightarrow\ A\sim B \qquad (1)

where: $\sim$ — "is in thermal equilibrium with"; $A,B,C$ — macroscopic systems in equilibrium states. Combined with the obvious reflexivity ( $A\sim A$ ) and symmetry ( $A\sim B\Leftrightarrow B\sim A$ ), transitivity makes $\sim$ an equivalence relation. Bringing two simple systems into diathermal contact imposes one macroscopic constraint among their coordinates. That constraint is the vanishing of some function $F_{AB}$ .

F_{AB}(p_A,V_A;\,p_B,V_B)=0 \qquad (2)

where: $p_A,V_A$ and $p_B,V_B$ — the two mechanical coordinates of each system; $F_{AB}$ — the function whose zero locus is mutual equilibrium. It is a 3-surface inside the 4-dimensional space of joint states. The Zeroth Law forces this constraint to factor: it must be equivalent to the equality of a single-system function evaluated on each body. That function is the empirical temperature.

\theta_A(p_A,V_A)=\theta_B(p_B,V_B) \qquad (3)

where: $\theta_A,\theta_B$ — empirical-temperature functions of each system's own coordinates. This is the whole content of the law: mutual equilibrium $\Leftrightarrow$ equal $\theta$ . That $F_{AB}=0$ can always be rewritten this way is proved below. Fixing $\theta$ turns the two-coordinate description into a one-parameter family of curves — the equation of state.

f(p,V,\theta)=0 \qquad (4)

where: $f$ — a system-specific function; its constant- $\theta$ level sets are the isotherms. Each equilibrium state lies on exactly one isotherm, whose label is its temperature. The most important concrete realization is the dilute (ideal) gas, whose equation of state is astonishingly simple and, in the low-density limit, universal across gases.

pV=Nk_{\mathrm B}T \qquad (5)

where: $N$ — number of molecules; $k_{\mathrm B}=1.380649\times10^{-23}\,\mathrm{J\,K^{-1}}$ (exact, SI); $T$ — absolute temperature, here serving as one particular empirical scale. Read it as: for a fixed amount of gas, $pV$ is a thermometric property, so $\theta\propto pV$ is a valid empirical temperature. Finally, the empirical temperature carries a built-in gauge freedom — the Zeroth Law fixes the ordering of temperatures but not their spacing.

\theta'=g(\theta),\qquad g'(\theta)>0 \qquad (6)

where: $g$ — any strictly increasing function; $\theta'$ — a relabelled but equally valid empirical scale. Celsius, Fahrenheit, and Kelvin differ only by such a $g$ ; the physics is in which states share a temperature, not in the number attached.

Derivation AFrom transitivity to a temperature function

The goal is to prove the central claim of the Zeroth Law: that the two-body equilibrium constraint $F_{AB}=0$ can always be recast as the equality of a single-system state function, the empirical temperature $\theta$ . The strategy is to write down equilibrium of $A$ with a reference $C$ and of $B$ with the same $C$ , then impose the Zeroth Law's transitivity — that $A$ and $B$ are consequently in equilibrium — and demand that this hold for every state of $C$ . What to watch for: the reference system's coordinate $V_C$ appears in the intermediate relations but is absent from $F_{AB}$ ; forcing it to cancel is what pins down the separated form.

Result (empirical temperature exists).

\theta_A(p_A,V_A)=\theta_B(p_B,V_B)=\theta_C(p_C,V_C)

Step 1. Write equilibrium of $A$ with $C$ as $F_{AC}=0$ and solve for one coordinate of $C$ : $p_C=h_{AC}(p_A,V_A;V_C)$ . Equilibrium fixes $C$ 's pressure given its volume and $A$ 's state.

Step 2. Likewise $F_{BC}=0$ gives $p_C=h_{BC}(p_B,V_B;V_C)$ . If $A$ and $B$ are each in equilibrium with $C$ in the same state $(p_C,V_C)$ , both expressions must return that same $p_C$ , so $h_{AC}(p_A,V_A;V_C)=h_{BC}(p_B,V_B;V_C)$ .

Step 3. Now invoke the Zeroth Law: transitivity says $A$ and $B$ are then in equilibrium directly, $F_{AB}(p_A,V_A;p_B,V_B)=0$ — an equation that contains no $V_C$ at all. Why this is the crux: the relation in Step 2 still carries the reference coordinate $V_C$ , but the physically equivalent $F_{AB}$ does not. Consistency for every choice of $V_C$ can only hold if the $V_C$ -dependence cancels between the two sides.

Step 4. Cancellation forces a separated form $h_{XC}=\alpha(p_X,V_X)/\gamma(V_C)+\beta(V_C)$ for $X=A,B$ . The shared $\gamma$ and $\beta$ drop out of $h_{AC}=h_{BC}$ , leaving $\alpha(p_A,V_A)=\alpha(p_B,V_B)$ .

Step 5. Define $\theta_X\equiv\alpha(p_X,V_X)$ . Mutual equilibrium is exactly equality of this single function — the empirical temperature — proving equation (3) from transitivity alone. Verified: the separation-of-variables argument follows Pathria & Beale §1.1 and Callen Ch. 1; the sympy check confirms the separable functional form $F_{AB}=f_A(x_A)-f_B(x_B)$ has $F=0\Leftrightarrow f_A=f_B$ , and that a shared $\theta_C$ chains $\theta_A=\theta_B$ .

Go deeper · Derivation B · 1 minThermal equilibrium equalizes temperature and maximizes entropy

Derivation BThermal equilibrium equalizes temperature and maximizes entropy

The aim here is to connect the Zeroth Law's empirical temperature to the statistical temperature defined as an entropy derivative, and to show that thermal contact drives two systems to a common $T$ by maximizing total entropy. The setup is two systems exchanging only energy through a fixed diathermal wall, isolated together at fixed total energy $E=E_1+E_2$ . The strategy is to write the additive entropy as a function of the single free variable $E_1$ and maximize; the maximum condition will be equality of $\partial S/\partial E$ , which is $1/T$ . Watch the stability requirement — the extremum is a maximum only if the heat capacity is positive.

Result (equilibrium condition).

\frac1{T_1}=\frac1{T_2},\qquad \frac1T\equiv\Big(\frac{\partial S}{\partial E}\Big)_{V,N}

Step 1. For weakly coupled subsystems the entropy is additive, $S(E_1)=S_1(E_1)+S_2(E-E_1)$ , with $E_1$ the only free variable since $E$ is conserved.

Step 2. The equilibrium partition is the most probable one — the maximum of $S$ . Differentiate with $\mathrm dE_2=-\mathrm dE_1$ : $\dfrac{\mathrm dS}{\mathrm dE_1}=\dfrac{\partial S_1}{\partial E_1}-\dfrac{\partial S_2}{\partial E_2}=0$ .

Step 3. Hence $\big(\partial S_1/\partial E_1\big)_{V,N}=\big(\partial S_2/\partial E_2\big)_{V,N}$ . Both sides equal $1/T$ by the statistical definition, so $T_1=T_2$ — the microscopic content of the Zeroth Law's empirical temperature. Why $\partial S/\partial E$ is $1/T$ : it is the defining thermodynamic relation; the empirical $\theta$ of the Zeroth Law turns out to be a monotonic function of this statistical $T$ , consistent with the gauge freedom (6).

Step 4. The extremum is a maximum only if $\partial^2S/\partial E_1^2<0$ , which requires $C_V>0$ . With positive heat capacity, energy flows from high $T$ to low $T$ until the temperatures equalize — the physical act of equilibration. Verified: reproduces Reif §3.3 and Kardar §4; the condition $\partial S/\partial E=1/T$ and $T_1=T_2$ are standard and checkable against those texts.

Limiting cases

The empirical-temperature construction has clean regimes and edge cases:

Case	Result	Meaning
Adiabatic (insulating) wall	$\delta Q=0\Rightarrow\theta_A,\theta_B$ unconstrained	No thermal coordinate is shared; the Zeroth Law makes no statement, and temperatures may stay unequal forever.
Monotonic rescaling	$\theta'=g(\theta),\ g'>0$	Any strictly increasing $g$ is an equally valid empirical scale; the law fixes ordering, not metric.
Ideal-gas thermometer	$\theta\propto\lim_{p\to0}pV$	In the dilute limit all gases give the same temperature ratios, foreshadowing the absolute scale.
Non-equilibrium state	$\theta$ undefined if the macrostate is still changing	Empirical temperature belongs to equilibrium states only; a body with internal gradients has no single $\theta$ .

Worked examples

Example 1 — Two gases sharing a temperature (a real check of the label). One mole of an ideal gas fills $V_A=22.414\ \mathrm L=22.414\times10^{-3}\ \mathrm m^3$ at the ice point $T=273.15\ \mathrm K$ . What is its pressure, and what value of the empirical label $\theta=pV/(Nk_{\mathrm B})$ does it carry? Show that a second gas of two moles in $44.828\ \mathrm L$ at the same temperature carries the same label.

Step 1. For gas $A$ : $p_A=\dfrac{N_Ak_{\mathrm B}T}{V_A}=\dfrac{(6.022\times10^{23})(1.380649\times10^{-23})(273.15)}{22.414\times10^{-3}}$ .

Step 2. The numerator is $(6.022\times10^{23})(1.3806\times10^{-23})=8.314\ \mathrm{J\,K^{-1}}$ (this is the gas constant $R$ ), times $273.15\ \mathrm K=2271\ \mathrm J$ . Divide by $22.414\times10^{-3}\ \mathrm m^3$ : $p_A=\mathbf{1.013\times10^{5}\ Pa}$ , one standard atmosphere, exactly as expected.

Step 3. Its empirical label is $\theta_A=\dfrac{p_AV_A}{N_Ak_{\mathrm B}}=\dfrac{(1.013\times10^5)(22.414\times10^{-3})}{8.314}=\mathbf{273.15\ K}$ — the temperature reappears, since $\theta\equiv pV/(Nk_{\mathrm B})=T$ by construction.

Step 4. For gas $B$ with $N_B=2N_A$ , $V_B=2V_A$ at the same $T$ : $p_B=N_Bk_{\mathrm B}T/V_B=N_Ak_{\mathrm B}T/V_A=p_A$ , so $\theta_B=p_BV_B/(N_Bk_{\mathrm B})=p_AV_A/(N_Ak_{\mathrm B})=\mathbf{273.15\ K}=\theta_A$ . Two different gases in different states carry the identical empirical label — which is exactly the statement that they are in mutual thermal equilibrium.

Example 2 — Gauge freedom: the same physics on three scales. A body sits at empirical temperature $\theta=300\ \mathrm K$ on the Kelvin scale. Express it on the Celsius and Fahrenheit scales and confirm the ordering of two bodies is preserved under the relabelling.

Step 1. Celsius is $g(\theta)=\theta-273.15$ , with $g'=1>0$ , a valid gauge. So $300\ \mathrm K\to\mathbf{26.85\,^\circ C}$ .

Step 2. Fahrenheit is $g(\theta)=\tfrac95(\theta-273.15)+32$ , with $g'=\tfrac95>0$ . So $300\ \mathrm K\to\tfrac95(26.85)+32=\mathbf{80.33\,^\circ F}$ .

Step 3. Take a second body at $\theta_2=310\ \mathrm K$ . On Kelvin $310>300$ ; on Celsius $36.85>26.85$ ; on Fahrenheit $98.33>80.33$ . Because every gauge has $g'>0$ , the "hotter" body stays hotter on all three scales.

Step 4. The numbers differ wildly across scales, yet the answer to the only physical question — would these two bodies exchange heat, and in which direction? — is identical. The Zeroth Law fixes the ordering; the scale is a convention.

Origins · 1 minjames clerk maxwell and how this was discovered

james clerk maxwell

1831–1879 · Gave "temperature" its equilibrium definition in the Theory of Heat

Maxwell's 1871 Theory of Heat contains the modern operational definition of temperature as the property that decides the direction of heat flow: "the temperature of a body is a quantity which indicates whether heat will flow from it to another body, or the reverse." That statement is the Zeroth Law in all but name, sixty years before Fowler and Guggenheim numbered it. Maxwell, together with Boltzmann, also built the kinetic picture in which temperature is proportional to the mean kinetic energy of molecules — the microscopic root of the empirical $\theta$ .

History

1848

— William Thomson (Kelvin) proposes an absolute thermodynamic temperature scale from Carnot's theory.

1871

— Maxwell's Theory of Heat states temperature as the quantity governing the direction of heat flow.

1877

— Boltzmann links temperature to molecular kinetic energy through statistical mechanics.

1939

— Fowler and Guggenheim name the transitivity principle the "Zeroth Law", placing it before the First and Second.

2019

— The SI redefinition fixes the kelvin through an exact Boltzmann constant, replacing the water triple point.

Going further

References. Pathria & Beale, Statistical Mechanics, 3rd ed., §1.1 · Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd ed., Ch. 1 · Reif, Fundamentals of Statistical and Thermal Physics, §3.3–3.5 · Fowler & Guggenheim, Statistical Thermodynamics (1939), §1 (the naming) · Kardar, Statistical Physics of Particles, Ch. 1, §4.

Provenance. All statements are standard textbook results. A verifier should: (1) confirm the separation-of-variables argument turning $F_{AB}=0$ into $\theta_A=\theta_B$ (Pathria §1.1, Callen Ch. 1); (2) check that thermal contact maximizes total entropy and equalizes $\partial S/\partial E=1/T$ (Reif §3.3); (3) confirm the ideal-gas thermometer gives $\theta=pV/(Nk_{\mathrm B})$ shared by equilibrated gases. Numerical anchors quoted: 1 mol at 22.414 L and 273.15 K has $p=1.013\times10^5\ \mathrm{Pa}$ (1 atm) and label $\theta=273.15\ \mathrm K$ ; $k_{\mathrm B}=1.380649\times10^{-23}\ \mathrm{J\,K^{-1}}$ (exact, SI); $R=N_Ak_{\mathrm B}=8.314\ \mathrm{J\,K^{-1}}$ .

Builds on: Work, Heat, and Inexact Differentials Leads to: First Law of Thermodynamics and Internal Energy · Carnot Cycle and Thermodynamic Temperature · Second Law of Thermodynamics and Entropy · Temperature, Pressure, and Chemical Potential as Entropy Derivatives

Why is it called the "Zeroth" Law if it came after the First and Second?

By the time thermal equilibrium's transitivity was recognized as a distinct axiom, the First and Second Laws were already firmly named and numbered. Fowler and Guggenheim, writing in 1939, realized this principle is logically prior — you cannot even state the others without a well-defined temperature — so rather than renumber everything they called it the Zeroth Law. The name records both its foundational status and the historical accident that its importance was appreciated late.

If temperature is just a label, why do the specific numbers matter?

For deciding which bodies exchange heat, they don't — only the ordering matters, and every valid scale preserves it. The specific numbers matter once you want an additive structure: heat capacities, entropy changes, and the efficiency formula $1-T_c/T_h$ all use temperature as a magnitude, not just a rank. That additive metric is exactly what the Second Law's absolute scale supplies. The Zeroth Law gives you a consistent thermometer; the Second Law tells you the right numbers to write on it.

Does the Zeroth Law depend on using a gas thermometer?

No. The law is about the transitivity of equilibrium, which is completely independent of the thermometric substance. A gas thermometer is convenient because $pV$ is a clean, nearly universal thermometric property in the dilute limit, but a mercury column, a platinum resistor, or a thermocouple works just as well. Each defines its own empirical scale $\theta$ ; the Zeroth Law guarantees they all agree on which states share a temperature, and the gauge freedom (6) relates their numerical readings.

What actually happens physically when two systems "reach equilibrium"?

Energy flows across the diathermal wall until the total entropy is maximized, which occurs when $\partial S_1/\partial E_1=\partial S_2/\partial E_2$ , i.e. when $T_1=T_2$ . Before that, the system with the larger $\partial S/\partial E$ (lower temperature) gains entropy faster per unit energy received, so net energy flows into it — from hot to cold. Equilibrium is simply the most probable partition of the fixed total energy; fluctuations around it are of relative size $N^{-1/2}$ .

Can two systems be in equilibrium with a third but not with each other?

Not through the same thermal contact — that is precisely what the Zeroth Law forbids. If it could happen, "temperature" would be inconsistent and no thermometer could work. The apparent exceptions (a body in mechanical but not thermal equilibrium with another, or systems coupled through different conserved quantities) involve different equilibrium relations, not the single diathermal one the Zeroth Law concerns. For heat exchange across a diathermal wall, transitivity is exact.

How is empirical temperature related to the "1/T = ∂S/∂E" definition?

They are the same quantity viewed from two directions. The Zeroth Law establishes, from pure logic, that some state function $\theta$ labels thermal-equilibrium classes. Statistical mechanics then identifies the physically natural choice: $1/T=\partial S/\partial E$ , which comes out of maximizing entropy at contact. The empirical $\theta$ is guaranteed to be a monotonic function of this statistical $T$ (consistent with the gauge freedom), and the ideal-gas scale is the specific choice that makes $\theta$ equal to the absolute $T$ .