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The zeroth law and temperature comparison

Key idea: If two systems are each in thermal equilibrium with a third, they are in thermal equilibrium with each other. This shared temperature makes thermometers possible.

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  • GCE A-Level H2 Physics 2027
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You can tell whether two cups of tea are at the same temperature without pouring one into the other: dip a thermometer in each and compare the readings. That everyday step relies on a law of thermodynamics so basic that it was named after the first and second laws had already been numbered: the zeroth law.

The zeroth law of thermodynamics

If two systems are both in thermal equilibrium with a third system, then they are also in thermal equilibrium with each other.

Two systems in thermal equilibrium are at the same temperature, so there is no net energy transfer by heating between them when they are in thermal contact.

This is what makes a thermometer work. The thermometer is the third system, C. Placed in system A, it comes to thermal equilibrium with A and shows A’s temperature. Placed in system B, it shows B’s temperature. If the readings are the same, A and B are both in thermal equilibrium with the thermometer at the same temperature, so by the zeroth law they are in thermal equilibrium with each other: put in contact, there would be no net transfer between them.

The law is about temperature, not internal energy. Two objects at the same temperature are in thermal equilibrium even if one is far larger and stores far more internal energy.

Exam-style question 1

The zeroth law and a thermometer

3 marks

Examination question

State the zeroth law of thermodynamics, and explain how it allows a thermometer to show whether two objects are at the same temperature. [3 marks]

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Show solution step by step

  1. The law

    1 mark

    Method

    State the law with three systems.

    Reason

    The thermometer will be the third system.

    Working

    Answer

    If two systems are each in thermal equilibrium with a third system, they are in thermal equilibrium with each other.

  2. The thermometer

    2 marks

    Working

    Answer

    The thermometer comes to thermal equilibrium with each object in turn. If it gives the same reading for both, both objects are in thermal equilibrium with it at the same temperature, so they are in thermal equilibrium with each other: there would be no net energy transfer between them.

Check your understanding 1 Equal readings, unequal samples

A thermometer reaches a stable reading of 30 °C in each of two copper blocks, one ten times the mass of the other. Decide whether the blocks are in thermal equilibrium with each other, and whether their internal energies must be equal.

Thermal equilibrium between the blocks
Must their internal energies be equal?
Show answer

They are in thermal equilibrium because each is at the thermometer’s temperature. Their internal energies need not be equal: the larger block contains more atoms. Check temperature equality separately from an energy total.

Check your understanding 2 Different readings

After settling in each system, a thermometer reads 30 °C in system A and 50 °C in system B. Are A and B in thermal equilibrium? What happens if they are put in contact?

Initial net heating when A and B touch
Show answer

No. They are at different temperatures. In contact, energy is transferred by heating from B to A until their temperatures are equal.

Check your understanding 3 An immediate reading

A cold thermometer is put in hot water. It reads 30 °C immediately and 60 °C after settling. A learner uses the first reading to claim the water is in thermal equilibrium with another sample at 30 °C. Explain the error and what reading is needed.

Show answer

The initial reading is the thermometer’s own temperature while it is still warming. The zeroth-law comparison needs the thermometer in thermal equilibrium with each sample, so use the settled 60 °C reading for the hot water. That water is not in thermal equilibrium with the 30 °C sample; in contact, net heating initially goes from water to sample. Check that the thermometer has settled before using it as the third system.

Common mistakes

  • Thinking thermal equilibrium means equal internal energies; it means equal temperatures.
  • Thinking that in thermal equilibrium the particles stop exchanging energy; the exchanges continue but balance.
  • Stating the zeroth law with only two systems.

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