Temperature and the zeroth law

Start with a probe that has not yet caught up

A small probe starts at 20 °C and is placed in a large, well-stirred cup held at 30 °C. In the assigned model below, its sensing element initially remains at 20 °C and then warms. Does that first reading mean the cup is at 20 °C? And how could the same probe establish that two different cups have the same temperature?

A contact thermometer responds to its own sensing element. To infer the sample temperature, we need a physical comparison, a relation between the sensor signal and temperature, and conditions under which the element represents the sample. Waiting can address response lag; it cannot automatically remove calibration error or steady heating of the probe.

First reading: follow the thermal-comparison example, the two temperature scales, the calibration table and the probe-response example. The published lessons “SI units and modern metrology” and “Calibration, standards, and traceability” are useful preparation for units and measurement claims. No unpublished thermodynamics lesson is required. The final return sections explain the mathematical distinctions behind the comparisons; their entropy argument is optional on a first reading.

Specify what the thermal comparison allows

Thermal contact is an arrangement that permits energy transfer associated with a temperature difference. In the ideal comparison used here, a conducting wall allows that transfer while preventing matter exchange and macroscopic boundary motion. An insulating, or adiabatic, barrier suppresses thermal transfer instead. Finding no change across an insulating barrier does not test equality of temperature.

Two initially settled bodies are in mutual thermal equilibrium when this passive contact produces no net thermal energy transfer and their macroscopic thermal states need not change. The test assumes adequate contact and time to relax, no sustained driving that maintains competing heat flows, and unchanged mechanical and material restrictions. The condition concerns the thermal degree of freedom; equal temperature does not require equal density, pressure, volume or composition.

A thermometer is the comparison body. Ideally it is small enough that bringing it into contact does not appreciably alter the sample temperature. A finite probe can perturb a small sample, and an imperfectly immersed probe can exchange heat with its surroundings as well as with the sample. A steady display is evidence to interpret with these conditions, not a universal proof of thermal equilibrium.

Use the same third-body state in both tests

The zeroth law says that if body A is in thermal equilibrium with body C, and body B is in thermal equilibrium with that same state of C, then A and B are in thermal equilibrium with each other. It makes a transferable thermal comparison possible: C can serve as a thermometer without A and B touching each other during every measurement.

In a separate assigned comparison, A, B and C are all at 293.15 K. A–C and B–C tests therefore select the same temperature class, and the A–B conclusion is consistent. A different body D at 313.15 K is not in that class. These values describe equilibrium states, not the earlier probe’s warming trajectory or a prediction of a final mixed-cup temperature.

The phrase “same state of C” matters. If C equilibrates with A at 293.15 K, is then deliberately changed to 313.15 K and equilibrates with B there, the two successful tests do not imply A and B have equal temperature. A drifting reference, or a sensor whose conditions changed between measurements, can invalidate the comparison even though each display eventually looks stable.

Give a temperature class a reproducible label

An empirical temperature is a numerical label assigned to a class of mutually thermally equilibrated states. The zeroth law supplies the consistency of the comparison. A thermometer supplies a property that changes reproducibly between classes. Its raw signal might be a resistance, a liquid-column length or a gas pressure under stated conditions; these different physical quantities need not have the same numerical value at the same temperature.

For a useful thermometer over a stated range, the chosen signal must identify temperature unambiguously once other relevant variables are controlled. A property that turns around with temperature can give two possible temperatures for one signal. Hysteresis can make the signal depend on whether the sensor was heated or cooled. Specifying a range, preparation procedure and reading conditions is therefore part of specifying the thermometer.

A calibration compares indications or sensor signals with reference temperatures and establishes their relation, including uncertainty. Adjustment changes the instrument or its settings; calibration need not do so. A large number of display digits can resolve small signal changes without establishing an equally small error in sample temperature.

Match the signal to the measurement conditions

A resistance probe uses an electrical property of its sensing material. A dilute-gas thermometer uses a relation among gas state quantities, with the amount of gas and mechanical arrangement specified. Radiation thermometry uses emitted radiation together with an optical and radiation model. These are different physical routes to a temperature estimate; the zeroth law does not assert that every signal is automatically proportional to temperature.

Contact methods aim to make the relevant sensor and sample temperatures agree. The actual sensing element may lie inside a sheath, rather than at the visible tip. Immersion, contact resistance, heat conduction along leads, radiation and the surrounding flow can all affect which temperature it reaches. A reference thermometer and a test probe should sample the same sufficiently uniform region during a comparison.

Noncontact radiation measurement does not establish sensor–sample thermal equilibrium by touching the sample. It infers temperature from radiation. The emitting surface, wavelength range, emissivity or blackbody approximation, optical path and background can matter. A radiation-derived brightness temperature cannot be silently substituted for the thermodynamic temperature of an arbitrary surface.

Keep the unit size and the scale origin separate

Thermodynamic temperature is denoted T and its SI unit is the kelvin, symbol K. Celsius temperature is written t_C here to distinguish it from elapsed time. Their numerical values satisfy t_C/°C = T/K − 273.15 exactly. Thus 20 °C corresponds to 293.15 K. A change of 20 °C has the same size as a change of 20 K; temperature differences do not require adding 273.15.

A rise from 20 °C to 40 °C is a 20 K rise, but the thermodynamic-temperature ratio is 313.15/293.15 ≈ 1.068224, not 2. The Celsius numerical ratio depends on the chosen origin. Even a genuine ratio of thermodynamic temperatures does not by itself give a ratio of total internal energies: material, amount and the energy reference also matter.

Absolute zero is the T = 0 K origin, corresponding to −273.15 °C. It is included in the conversion table as a scale origin, not as a realized calibration bath. Negative Celsius readings are entirely compatible with positive kelvin temperatures. The ordinary equilibrium and thermometer cases calculated in this article all have T > 0; no special population-inverted temperature convention is being used.

Separate the SI definition from its realization

In the SI, the numerical value of the Boltzmann constant k is fixed exactly at 1.380649 × 10⁻²³ when expressed in J K⁻¹. Here J is kg·m²·s⁻², and the kilogram, metre and second are defined through h, c and the caesium frequency Δν_Cs. This fixes the unit of thermodynamic temperature without specifying one particular thermometer or calibration experiment.

The product kT is a thermal energy scale: for an assigned T = 300 K it is 4.141947 × 10⁻²¹ J. This does not say that every particle in every material has energy kT, nor that an instrument measures k directly on each reading. Connecting a signal to T requires the applicable physical model and measurements.

Realizing the kelvin means establishing a temperature value consistently with its definition and evaluating its uncertainty. Primary thermometry uses a sufficiently understood physical relation and measurements of other quantities; departures from its ideal model still require corrections and an uncertainty assessment. Absolute primary thermometry does not require a temperature fixed point; relative primary thermometry uses prior thermodynamic-temperature values at reference points to determine some parameters.

Defined practical scales provide reproducible fixed points, specified instruments and interpolation rules. ITS-90 assigns a scale temperature T₉₀ that closely approximates thermodynamic T but is a distinct quantity. Its assigned fixed-point values are exact within that scale; realizing them has uncertainty, and T − T₉₀ is not identically zero. In particular, the water triple point remains assigned 273.16 K in ITS-90, while its thermodynamic temperature is no longer the exact defining value of the SI kelvin. Ordinary ice and boiling-water preparations also depend on purity and pressure; they are not universal exact realizations of thermodynamic 0 °C and 100 °C.

Test what two calibration points leave unknown

Use an original teaching sensor with R = 100 Ω × (1 + 0.0038x − 0.0000006x²), where x = t_C/°C lies between 0 and 100. The reference bath temperatures and law are assigned exactly for this model. These coefficients are not an industrial platinum specification, measured calibration data or an ITS-90 realization.

The model gives R(0) = 100 Ω and R(100) = 137.4 Ω. A two-point straight-line indication is x_lin = 100(R − 100 Ω)/(37.4 Ω). It matches both calibration endpoints. At the assigned 50 °C check state, however, R = 118.85 Ω and the line indicates 50.401070 °C. The indication error is about +0.401070 K even though the endpoint errors are zero.

This is interpolation error from using the wrong functional shape, not a failure of the zeroth law. Extra check points can reveal it. More calibration points and an appropriate model can improve interpolation, but unsupported extrapolation outside the checked range remains a different claim. In real use the reference temperatures, signals, repeatability and interpolation all contribute uncertainty; the deterministic table isolates only this model-shape issue.

Let a response model quantify waiting

Return to the 20 °C probe in a large bath held at 30 °C. This is a separate sensor model with a nearly uniform sensing assembly of heat capacity C_s = 0.020 J/K, connected to the bath by constant thermal conductance G = 0.010 W/K. The bath is assumed unaffected, and other heat leaks are neglected. The passive case idealizes observation as adding negligible heat. Heat capacity is the energy required per small temperature rise under the stated conditions; conductance is the heat-transfer rate per temperature difference in this linear model.

Write elapsed time as z in seconds. Let T_s be the sensor temperature, T_b the bath temperature and T_i the initial sensor temperature. With no internal power input, C_s dT_s/dz = G(T_b − T_s), so T_s(z) = T_b + (T_i − T_b)e^(−z/τ), with τ = C_s/G = 2 s. The sensing element approaches the bath temperature rather than changing instantaneously. After one τ it has completed about 63.2121% of the initial temperature step, leaving about 36.7879% of the original error.

At 10 s the model still reads about 29.932621 °C, 0.067379 K below the bath. To make the magnitude of this passive lag no more than 0.05 K, solve 10e^(−z/2) ≤ 0.05, giving z ≥ 2 ln(200) ≈ 10.596635 s. The tolerance refers only to lag in this assigned constant-bath model, not total measurement uncertainty. A real probe may have several time constants and a different response in moving liquid, still air or a thermowell.

A steady powered probe need not be at bath temperature

Electrical resistance measurement supplies power. For a bounded diagnostic, give the separate thermal-response sensor a constant effective resistance R_eff = 100 Ω and current I. This R_eff is only the power-model approximation; it is not the nonlinear calibration sensor above. The power is P = I²R_eff, and the balance becomes C_s dT_s/dz = G(T_b − T_s) + P.

The steady value is T_s,∞ = T_b + P/G. At 1 mA, P = 0.100 mW and the sensor settles 0.010 K above the bath. At 2 mA, power and steady bias are four times larger: 0.400 mW and 0.040 K. Heat continues to leave the element while electrical energy enters. The element is steady but is not in passive thermal equilibrium with the bath.

Waiting longer removes the transient relative to that powered steady value; it does not remove the steady bias. Reducing measuring current, estimating self-heating under the actual conditions or extrapolating appropriate measurements toward zero power can help, while lower current also makes electrical measurement harder. The numbers here are consequences of the assigned G and R_eff, not recommended instrument currents or a performance guarantee.

Carry the conditions into the reported result

For a useful comparison, identify the target: a stirred liquid region, the actual surface contacted by a probe, or a radiating surface in a specified optical view. Report whether the result is thermodynamic T, an ITS-90 value, or an instrument indication related to a reference scale. A calibrated instrument cannot compensate for an undefined target or for sampling a different part of a temperature gradient.

Match conditions that affect the transfer from reference calibration to use: range, immersion, excitation current, medium, thermal contact, sensor assembly and readout. Check settling, spatial uniformity, drift and hysteresis as appropriate. A displayed value can be reproducible yet biased by a persistent heat leak, and an instrument can drift after its last calibration.

A traceable result needs a documented calibration chain with stated uncertainties; a label naming a national laboratory is not the chain by itself. Separate resolution, short-term scatter, calibration uncertainty and in-use effects. For these teaching models the inputs are supplied, but an experiment would need evidence for them rather than inheriting their exactness from the arithmetic.

Return path: equality does not fix a unique numerical scale

Write A ~ B for mutual thermal equilibrium within the family of ordinary equilibrium states being compared. The zeroth-law transitivity, together with reflexive and symmetric comparison, makes ~ an equivalence relation: it groups states into thermal classes. Assigning the same label to each member of a class expresses equality without requiring the members to have identical pressures or sensor properties.

An equivalence relation alone does not establish every extra property we want from a real-valued thermometer. Ordering the classes, choosing a continuous scalar coordinate and making a local smooth inverse rely on additional empirical regularity and the chosen state domain. We use a range where those conditions hold. Any one-to-one relabelling preserves equality, and an increasing relabelling also preserves the hot-to-cold ordering on that range.

Consequently, the zeroth law alone cannot select Kelvin ratios or decide whether an arbitrary signal is linear in thermodynamic temperature. For T > 0, replacing T/K by (T/K)² preserves equality and order but changes numerical ratios and interval sizes. Thermodynamic temperature obtains its material-independent absolute scale from further thermodynamic structure, expressible through reversible heat ratios or entropy; the SI then fixes the kelvin size through k. A detailed engine calculation belongs to the later engine/second-law subjects and is not required for the operational comparison here.

Return path: the thermodynamic quantity behind the comparison

For an appropriate differentiable equilibrium entropy S(U,V,N,…) with energy U, volume V and particle numbers N specified, the thermodynamic definition is 1/T = (∂S/∂U)_(V,N,…). Entropy is introduced here only as the equilibrium state function whose energy derivative defines this temperature; the full second-law theory comes later. The derivative holds the other independent quantities fixed. It is not a derivative along an arbitrary heating-and-expansion path.

For two weakly coupled bodies that exchange only energy, write U_B = U_total − U_A and S_total = S_A(U_A) + S_B(U_total − U_A), with the other relevant quantities fixed. A small allowed energy transfer gives dS_total/dU_A = 1/T_A − 1/T_B. An interior stationary thermal state therefore has T_A = T_B. A stable equilibrium also requires that the relevant second variation select an entropy maximum; equality of the first derivatives alone is not a stability proof.

This return argument uses thermodynamic structure beyond the zeroth-law statement. It explains why a common thermodynamic T can be the quantity matched by different physical thermometers, while each instrument still needs a signal model and uncertainty assessment. It does not assert that equal temperature settles mechanical or chemical exchange, nor does it provide a temperature for every driven microscopic degree of freedom.

Return path: check the inverse and the response equation

For the teaching calibration law, dR/dx = 100 Ω × (0.0038 − 0.0000012x). It remains positive from 0 to 100, with its smallest value 0.368 Ω per unit x at the upper endpoint, so the selected branch is invertible over the stated range. With y = R/(100 Ω) − 1, a = 0.0038 and b = −0.0000006, a cancellation-resistant inverse is x = 2y/[a + √(a² + 4by)]. The root and domain must still be checked; another algebraic quadratic branch is not a second allowed calibration temperature here.

For an independent signal R = 119 Ω, that inverse gives about 50.401095 °C, while the endpoint line gives about 50.802139 °C. This is a different check from the 50 °C forward example: the input is now the resistance. A small resistance error contributes approximately δx = δR/(dR/dx) locally, but this sensitivity alone omits uncertainties in the reference temperatures, fitted coefficients and in-use conditions.

For the constant-power response model, put T_∞ = T_b + P/G. Then C_s d(T_s − T_∞)/dz = −G(T_s − T_∞), giving T_s(z) = T_∞ + (T_i − T_∞)e^(−z/τ). Its initial value, positive decay time and steady power balance can all be checked directly. If the sensor is not nearly uniform, the bath changes, G changes or electrical resistance varies enough to change P, this particular single-exponential model needs revision. It is not a universal response law.

Recognise when the comparison needs a richer description

A body with a temperature gradient need not have one global equilibrium temperature. Local thermodynamic temperature can still be useful when a small region equilibrates internally much faster than conditions vary across it or in time. Probe size and response then set a spatial and temporal averaging problem. This article has supplied uniform equilibrium bodies or a fixed bath, so it has not solved that problem.

Strongly driven, rapidly changing or poorly equilibrated systems may give different operational readings to instruments coupled to different physical modes. A radiation signal, a contact probe and a gas-state estimate should not be declared equivalent without checking the assumptions that connect each to the same thermodynamic quantity. A stable display, zero transfer through an insulator, or agreement at two calibration points does not supply those missing assumptions.

Use the distinctions together: the zeroth law supports a consistent comparison of thermal-equilibrium classes; a thermometric property and calibration assign an indication; thermodynamic theory fixes the absolute quantity; the SI defines its unit; and a realization and measurement procedure connect the physical sample to a value with uncertainty. Each step answers a different part of the original probe-and-cup question.

Worked examples

Try it yourself

1. A and B each pass a passive thermal-equilibrium comparison with the same unchanged C. What follows, and what does not?

   Show the answer and reasoning

   A and B are mutually in thermal equilibrium; equal pressure or composition does not follow.

   The zeroth law concerns the thermal comparison under its stated restrictions. It does not remove constraints on mechanical or material exchange.

2. Two bodies show no change when separated by an ideal insulating wall. Does that prove equal temperature?

   Show the answer and reasoning

   No.

   The wall prevents thermal transfer even when temperatures differ. A passive comparison needs a permitted thermal coupling and sufficient relaxation time.

3. Convert the change from 20 °C to 40 °C into kelvins, and find the thermodynamic-temperature ratio.

   Show the answer and reasoning

   The change is 20 K; the ratio is about 1.068224.

   Convert the endpoint values to 293.15 K and 313.15 K. Their difference is 20 K and their ratio is 313.15/293.15. The Celsius numerical ratio 2 uses an offset origin.

4. The original calibration sensor gives 119 Ω. Which temperature does its selected nonlinear inverse give, and how does the endpoint line differ?

   Show the answer and reasoning

   About 50.401095 °C from the nonlinear model, versus about 50.802139 °C from the endpoint line.

   Use the unique inverse in the checked 0–100 °C range, not the other quadratic root. The two conversions embody different interpolation models.

5. After one time constant, is the passive sensor within 1% of the original 10 K step?

   Show the answer and reasoning

   No; about 36.7879% of the original difference remains.

   The residual fraction is e⁻¹, while 1 − e⁻¹ ≈ 63.2121% is the fraction of the step completed. These complementary percentages describe different quantities.

6. In the 1 mA power diagnostic, will unlimited waiting make the sensor settle exactly at the bath temperature?

   Show the answer and reasoning

   No; its modeled steady offset is +0.010 K.

   Electrical input continuously balances heat loss at a nonzero temperature difference. Waiting removes the transient relative to that powered steady state, not the sustained bias.

7. If a fixed point is assigned an exact T₉₀ value, does that make every realization exact and force T = T₉₀?

   Show the answer and reasoning

   No to both claims.

   Exactness of the scale assignment is distinct from realization uncertainty. ITS-90 is a practical defined scale that approximates thermodynamic temperature, so their difference need not vanish.

8. On T > 0, does replacing T/K by (T/K)² preserve equilibrium equality? Does equality of the first entropy derivatives alone prove stable thermal equilibrium?

   Show the answer and reasoning

   The increasing one-to-one relabelling preserves equality and order, but the first-derivative condition alone does not prove stability.

   Numerical scale selection needs additional thermodynamic structure. In the entropy return argument, a stationary state must also satisfy the appropriate maximum/stability condition.

What to carry forward

• Thermal equilibrium is a passive comparison under specified conditions; insulation or a steady driven display is not sufficient evidence.
• The zeroth law makes unchanged-reference comparisons transitive, while raw thermometer properties and other state variables can differ.
• A kelvin interval and a Celsius interval have equal size; their numerical temperature values have different origins.
• Fixing k defines the kelvin unit; realizing a temperature and transferring a calibration still require physical models and uncertainty.
• Two calibration endpoints do not establish interior linearity, and waiting alone does not remove persistent measurement bias.
• The optional thermodynamic derivative and stability argument adds structure beyond the zeroth-law equality statement.