Thermal Equilibrium and the Zeroth Law

Bodies in contact exchange heat until they share one temperature. The zeroth law turns that into a rule: two bodies in equilibrium with a third are in equilibrium with each other. That is what makes temperature, and every thermometer, meaningful.

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Thermal equilibrium

Heat flows until T is equal

A hot body A in contact with a cold body B gives heat to B. A cools, B warms, and the flow stops when both reach one common temperature. That state, with no net heat flow, is thermal equilibrium.

Two blocks in contact settling to one temperature. — Equal masses: both settle at 50 °C.

Blocks A and B separated by an insulating wall, both resting on block C. — An insulating wall keeps A and B apart; both are in equilibrium with C.

Zeroth law

A ~ C and B ~ C ⇒ A ~ B

If A is in thermal equilibrium with C, and B is in thermal equilibrium with C, then A and B are in thermal equilibrium with each other, even if they never touch. The property they share is temperature.

Why it matters

A thermometer is body C

A thermometer that reads the same in A and in B tells us A and B are at the same temperature. The zeroth law is what lets us define temperature before the first and second laws use it. Equal temperature does not mean equal energy: a tub and a cup at 40 °C hold very different amounts.

A large tub and a small cup at 40 °C, each with a thermometer. — Same reading, same temperature, very different heat content.

Worked examples

One for every idea

Thermal Equilibrium and Zeroth Law of Thermodynamics

1. Body A is in thermal equilibrium with body C, and body B is also in thermal equilibrium with C (A and B are not in contact). Are A and B necessarily in thermal equilibrium with each other? Which law guarantees this?

Equilibrium with C means equal temperature: $T_{A} = T_{C}$ and $T_{B} = T_{C}$ .
So $T_{A} = T_{B}$ : if A and B are brought together, no heat flows.

Yes. This is the zeroth law of thermodynamics.

JEE-style question

Your turn

Bodies A and B are each in thermal equilibrium with a third body C. Which quantity must A and B have in common?

(a)Heat content

(b)Temperature

(c)Internal energy

(d)Mass

Show the answer and the traps

Temperature, option b. That is all thermal equilibrium needs.

Not heat content or internal energy: a tub and a cup at the same temperature hold very different amounts.

And not mass: equilibrium says nothing about how big the bodies are.

Watch out

Common mistakes

Thinking bodies in equilibrium have the same heat contentOnly the temperature is equal; heat content depends on mass and material.

Thinking A and B must touch to be in equilibriumThe zeroth law needs only that each is in equilibrium with a common third body.

Practice

Try these

1. A thermometer reads 36 °C in body A and 36 °C in body B. A and B are then put in contact. Which way does heat flow?

Neither way: they are at the same temperature, so they are already in thermal equilibrium (zeroth law).

2. A and B are in thermal equilibrium, but B and C are not. Can A and C be in thermal equilibrium?

No. If A ~ C as well, then $T_{B} = T_{A} = T_{C}$ , so B ~ C, which contradicts the data.