Zeroth law of thermodynamics

Every time a nurse slides a thermometer under a tongue, or a cook checks a curry with a kitchen thermometer, they are quietly trusting a law they have never heard named. The thermometer never touches the patient's blood directly and the tongue's tissue directly at the same instant in the way you might shake two hands together — it touches the tongue, waits, and reports a number. Why should that number mean anything about a comparison the thermometer never actually performed between two other objects, say between the tongue and a cup of tea measured an hour earlier? The answer is a single, almost embarrassingly simple statement called the zeroth law of thermodynamics, and it is the quiet foundation on which the entire idea of "temperature" as a measurable, comparable quantity is built. The zeroth law says that thermal equilibrium behaves like mathematical equality: if object A matches a reference C, and object B also matches that same reference C, then A and B must match each other too — without ever being brought together. That one sentence is what licenses every thermometer, every calibration scale, and every claim that "this is hotter than that" without physically colliding the two objects. This section stays inside the CBSE Class 11 boundary: the statement of the law, why it is numbered zero, how it defines temperature as a genuine state variable, and how it underwrites the working of real thermometers such as the constant-volume gas thermometer — one clean idea, examined from every angle a board or competitive exam might probe.

Key takeaways

The zeroth law states: if system A is in thermal equilibrium with system C, and system B is also in thermal equilibrium with system C, then A and B are in thermal equilibrium with each other.
Thermal equilibrium is an equivalence relation — reflexive, symmetric, and transitive — exactly like mathematical equality, which is why grouping systems by "same temperature" makes logical sense.
The law was named zeroth not because it is trivial but because it had to be recognised after the first and second laws were already numbered, even though logically it comes before them — it is what makes "temperature" a legitimate variable in the first place.
The reference system C, in practice, is a thermometer: two objects are declared to be at the same temperature purely by comparing each separately to it, never by touching them together.
A real instrument built on this idea is the constant-volume gas thermometer, calibrated against the triple point of water at exactly $273.16\ \text{K}$ , giving temperature a numerical scale.
The zeroth law is about equilibrium, not about energy transfer — it says nothing about how much heat flows or how fast; that is the job of the first law.
Key idea: thermal equilibrium is transitive — if A and B are each separately in equilibrium with a common reference C, then A and B are in equilibrium with each other, and this transitivity is exactly what makes temperature a well-defined, measurable state variable.

The core insight

If two systems are each separately in thermal equilibrium with a third system, they are in thermal equilibrium with each other, and this transitive property of equilibrium is what allows a single number called temperature to be defined and measured with a thermometer.

Where it fits in the syllabus

In the CBSE Class 11 Thermodynamics chapter, the zeroth law is the very first formal idea, introduced right after thermal equilibrium and immediately before the first law of thermodynamics ( $\Delta Q = \Delta U + \Delta W$ ). You are expected to state the law precisely in words and symbols, explain the historical reason for its "zeroth" numbering, and use it to justify why a thermometer can compare two objects that are never placed in direct contact. This section does not touch the first or second law, the equation of state $PV = nRT$ , or the gas-process laws — those belong to their own nodes in this chapter; here the focus stays tightly on equilibrium and the definition of temperature.

The physical picture

Two systems are said to be in thermal equilibrium when they are in contact through a wall that lets heat pass (a diathermic wall) and no further change occurs in any of their measurable properties — pressure, volume, or anything else that could shift with temperature. When that state is reached, both systems share the same value of a property we call temperature. The zeroth law promotes this pairwise idea into a rule about three systems at once.

A \leftrightarrow C \ \ \text{and} \ \ B \leftrightarrow C \quad \Longrightarrow \quad A \leftrightarrow B. \qquad (1)

where: $A$ , $B$ , and $C$ are any three thermodynamic systems, and the symbol $\leftrightarrow$ means "is in thermal equilibrium with." Equation (1) reads: if A matches C, and B matches C, then A automatically matches B — without A and B ever having been brought into contact with each other. The physical reading is that thermal equilibrium behaves exactly like the equals sign in arithmetic: it is reflexive (a system is trivially in equilibrium with itself), symmetric (if A matches B, B matches A), and transitive (equation 1). Any relation with these three properties is called an equivalence relation in mathematics, and it is precisely this structure that allows every object in the universe to be sorted into equivalence classes that share one common number — the temperature.

Figure

Three blocks A B and C connected to a thermometer, with A and C shown at the same reading, B and C shown at the same reading, and a dashed arrow concluding that A and B must also match without ever touching directly

The zeroth law alone does not hand you a numerical scale — it only guarantees that a consistent notion of "same temperature" exists and can be transferred through a reference object. Building an actual number on top of that guarantee needs one more ingredient: a thermometric property, something that changes smoothly and reproducibly with hotness, such as the pressure of a fixed volume of gas.

T = 273.16\ \text{K} \times \frac{P}{P_{tr}}. \qquad (2)

where: $T$ is the unknown temperature of a system in kelvin, $P$ is the pressure recorded by a constant-volume gas thermometer when in thermal equilibrium with that system, and $P_{tr}$ is the pressure the same thermometer reads when in thermal equilibrium with water at its triple point, defined to be exactly $273.16\ \text{K}$ . The physical reading is a straight-line proportionality: double the gas pressure at fixed volume and the kelvin temperature doubles too. Equation (2) is only trustworthy because of the zeroth law — the gas column never touches the triple-point cell and the unknown system simultaneously, yet comparing each separately to the same thermometer is enough to place both on one consistent scale.

Derivation AThe constant-volume gas thermometer scale from the zeroth law

The goal is to show, using nothing beyond the zeroth law and the idea of a linear thermometric property, how a real laboratory instrument converts a measured pressure into a temperature in kelvin, and why the zeroth law is the hidden justification that makes this conversion valid at all — even though the thermometer's gas bulb is never in contact with both the reference and the unknown system at the same time.

Step 1. Choose the thermometric property. A fixed mass of gas is sealed in a bulb of constant volume $V$ . Experiment shows that, for a dilute gas held at constant volume, pressure rises smoothly and very nearly linearly with hotness — the gas pressure $P$ can therefore serve as the thermometric property, and we assume the simplest possible relation between $P$ and temperature $T$ : a direct proportionality, $T = k P$ , for some constant $k$ that depends only on how much gas is in the bulb and its fixed volume, not on what the gas is in contact with.

Step 2. Bring the bulb into equilibrium with a single, universally reproducible reference: water at its triple point, the unique combination of pressure and temperature at which ice, liquid water, and water vapour coexist together. By international convention this state is assigned the exact value $273.16\ \text{K}$ . The bulb (system C in the language of the zeroth law) reaches thermal equilibrium with the triple-point cell (system "reference"), and the pressure gauge reads some value $P_{tr}$ . Substituting into $T = kP$ gives $273.16 = k\,P_{tr}$ , so $k = 273.16 / P_{tr}$ . This single measurement fixes the constant once and for all.

Step 3. Bring the same bulb, at the same fixed volume, into thermal equilibrium with the unknown system whose temperature is wanted. The gauge now reads a new pressure $P$ . Because the gas bulb is one and the same physical object throughout, the constant $k$ found in Step 2 still applies, so the unknown temperature is $T = kP = \dfrac{273.16}{P_{tr}}\,P$ , which rearranges to the working formula $T = 273.16\ \text{K} \times (P/P_{tr})$ — equation (2) above.

Step 4. Notice where the zeroth law actually enters. The bulb was never in contact with the triple-point cell and the unknown system at the same instant — those two equilibria happened one after another. What licenses treating both readings as points on one common scale is precisely the zeroth law: the bulb (as system C) reaches its own separate equilibrium with the triple-point reference and, later, with the unknown system, and the law guarantees that both readings genuinely measure the same underlying quantity, temperature, so that a single proportionality constant $k$ is valid for both. Without the zeroth law, there would be no logical basis for comparing the two readings at all.

Worked examples

Board level - Three blocks and one thermometer

A student has three metal blocks A, B, and C. She places a thermometer in turn on each block and finds all three read exactly $35\,^\circ\text{C}$ . She wants to know: if she places block A in direct contact with block B (with a good heat-conducting surface between them, inside an insulated enclosure), will there be any heat flow?

Step 1. Identify the three systems and the role of the thermometer. Here A and B are the two blocks whose relationship is unknown; the thermometer plays the role of the third system, call it C, in equation (1).

Step 2. Record the two known equilibria. The thermometer shows $35\,^\circ\text{C}$ when touching A, so $A \leftrightarrow C$ . The thermometer also shows $35\,^\circ\text{C}$ when touching B, so $B \leftrightarrow C$ , and this reading was taken completely separately from the first, with A nowhere near B or C.

Step 3. Apply the zeroth law. Since $A \leftrightarrow C$ and $B \leftrightarrow C$ , equation (1) gives $A \leftrightarrow B$ directly — A and B are already in thermal equilibrium, even though they have never been touched together.

Step 4. State the physical consequence. When A and B are finally brought into direct contact inside the insulated enclosure, no net heat flows between them, because they were already at the same temperature. This is the everyday, practical power of the zeroth law: a chain of comparisons through a common reference replaces the need for direct pairwise contact, and the answer — no heat flow — can be predicted with total confidence before the blocks ever touch.

Competitive level - Reading an unknown temperature on a gas thermometer

A constant-volume gas thermometer is first brought into thermal equilibrium with water at its triple point, and the gas pressure reads $P_{tr} = 6.80\ \text{kPa}$ . The same bulb, at the same fixed volume, is then brought into thermal equilibrium with a hot oil bath, and the pressure now reads $P = 9.50\ \text{kPa}$ . Find the temperature of the oil bath in kelvin and in degrees Celsius.

Step 1. Recognise which formula applies. Because the bulb never touches the triple-point cell and the oil bath at the same time, it is the zeroth law that justifies placing both readings on the same scale, using equation (2), $T = 273.16\ \text{K} \times (P/P_{tr})$ .

Step 2. Substitute the given pressures. $T = 273.16 \times \dfrac{9.50}{6.80}$ . First compute the ratio: $9.50/6.80 = 1.3971$ (to four significant figures).

Step 3. Multiply out. $T = 273.16 \times 1.3971 = 381.65\ \text{K}$ , so the oil bath is at $T \approx 381.7\ \text{K}$ .

Step 4. Convert to Celsius for a sense check. $T(^\circ\text{C}) = T(\text{K}) - 273.15 = 381.65 - 273.15 = 108.5\,^\circ\text{C}$ , so the oil bath is at approximately $108.5\,^\circ\text{C}$ , comfortably above the boiling point of water at standard pressure, which is physically reasonable for a heated oil bath used in a laboratory. Timing note: this is a two-line substitution once the formula is recalled — budget under a minute in a competitive exam. The only place marks are lost is sign or unit slips: always keep pressures in the same unit on both sides (kPa here, so they cancel cleanly in the ratio) and never forget that $273.16\ \text{K}$ , not $273\ \text{K}$ , is the exact triple-point value used in the defining formula.

Common traps

Thinking the zeroth law is about energy transfer. It says nothing about how much heat flows or how fast — only that if two systems are each in equilibrium with a third, they are in equilibrium with each other. It is a statement about equilibrium, not energy exchange; that bookkeeping is the job of the first law, $\Delta Q = \Delta U + \Delta W$ , covered separately.
Assuming "zeroth" means "least important" or "trivial." It is called zeroth only because of the historical order in which it was formally stated, after laws 1 and 2 already had their names — not because of low importance. Logically it is the most foundational, since it is what makes "temperature" a legitimate variable at all.
Forgetting the third system must be in equilibrium with BOTH others independently, not just observed once or compared informally.
Skipping the calibration step in gas-thermometer numericals. A very common slip is plugging pressures straight into $T = kP$ without first finding $k$ from the triple-point reading — always compute the ratio $P/P_{tr}$ and multiply by $273.16\ \text{K}$ .
Mixing up $273\ \text{K}$ and $273.16\ \text{K}$ . The ice point is close to $273.15\ \text{K}$ , but the triple point that defines the kelvin scale is exactly $273.16\ \text{K}$ — using the wrong constant in equation (2) gives a small but markable error.

You should be able to…

State the zeroth law of thermodynamics precisely, in both words and the symbolic form of equation (1).
Explain why it is called "zeroth" despite being logically prior to the first law.
Identify thermal equilibrium as reflexive, symmetric, and transitive — an equivalence relation.
Justify why thermometers work, using the zeroth law, without ever needing direct contact between the two objects being compared.
Apply the constant-volume gas thermometer formula $T = 273.16\ \text{K} \times (P/P_{tr})$ to compute an unknown temperature from a pressure reading.
Distinguish the zeroth law's role (defining equilibrium and temperature) from the first law's role (conservation of energy) in any mixed conceptual question.

Ask the tutor: Want more practice turning three-system equilibrium scenarios into quick zeroth-law conclusions, or extra gas-thermometer numericals before your test? Ask the AI tutor for a fresh set of problems.

Why do we even need a zeroth law? Isn't it just obvious that equal things are equal?

It feels obvious once stated, which is exactly why it went unnamed for so long — early physicists used it unconsciously every time they trusted a thermometer, without writing it down as an axiom. But physics needs its assumptions made explicit: nothing in the definitions of pressure, volume, or heat automatically guarantees that "thermal equilibrium" behaves like mathematical equality. It could, in principle, have been intransitive. The zeroth law is the experimental fact, elevated to a postulate, that it is not — and every thermometer in the world implicitly relies on that fact being true.

Does the zeroth law apply to things other than temperature, like pressure or speed?

The zeroth law as stated is specifically about thermal equilibrium and temperature. Mechanical equilibrium of pressures, or equality of speeds, are governed by their own physical laws (Newton's laws, fluid statics) and are not automatically transitive for the same reason — you would need to check separately whether "equal pressure with a common reference" implies "equal pressure with each other," and in many real situations with valves or barriers it does not. The elegance of the zeroth law is that it is a genuinely independent postulate, verified by experience, specifically for thermal equilibrium.

How is the zeroth law different from saying two objects are at the same temperature because a formula says so?

The zeroth law comes first, logically. Before you can even write down a temperature formula like $T = 273.16\,(P/P_{tr})$ , you need a guarantee that comparing two objects to a common reference thermometer, one after the other, tells you something valid about comparing those two objects to each other. That guarantee is the zeroth law. The formula in equation (2) is a later, practical step that turns the qualitative guarantee ("A and B share some common property") into an actual number on the kelvin scale, using a chosen thermometric property such as gas pressure.

Is the zeroth law tested with numerical problems in board exams, or only conceptual ones?

Mostly conceptual — expect to state the law precisely, explain the "zeroth" naming, and reason through scenarios like the three-block example without any calculation. Numerical questions on this exact topic are less common at board level but do appear at the competitive level (JEE/NEET-style), usually dressed up as a constant-volume gas thermometer problem like the worked example above, where you compute an unknown temperature from a pressure ratio using the $273.16\ \text{K}$ triple-point calibration. Knowing both the words and the one formula covers essentially everything asked.