Zeroth Law of Thermodynamics

The Statement and Its Meaning

Consider three systems: A, B, and C. If system A is in thermal equilibrium with system C (no heat flows between them when they are placed in contact), and system B is also in thermal equilibrium with system C, then the zeroth law guarantees that A and B are also in thermal equilibrium with each other. No heat will flow between A and B if they are brought into thermal contact.

This transitivity of thermal equilibrium is what makes thermometers possible. System C acts as the thermometer. By placing the thermometer in contact with system A until equilibrium is reached, and then with system B until equilibrium is reached, and finding the same reading both times, we conclude that A and B are at the same temperature, even if A and B have never been in direct contact.

Without the zeroth law, there would be no guarantee that two systems at the same thermometer reading would be in thermal equilibrium with each other. Temperature as a universal, transitive property would not exist, and the entire conceptual framework of thermodynamics would lack its most basic foundation.

Temperature as a Physical Property

The zeroth law establishes that temperature is a state property that determines the direction of heat flow. When two systems at different temperatures are placed in thermal contact, heat flows from the higher temperature system to the lower temperature system until equilibrium is reached. This is a consequence of the zeroth law combined with the second law, which specifies the direction of spontaneous heat transfer.

Temperature is an intensive property, meaning it does not depend on the size or amount of material in the system. A cup of water at 50 degrees Celsius has the same temperature as a swimming pool at 50 degrees Celsius, even though the pool contains vastly more thermal energy. This distinction between intensive properties (temperature, pressure, density) and extensive properties (energy, volume, mass) is central to thermodynamic reasoning.

The development of temperature scales has a long history. The Fahrenheit and Celsius scales are based on convenient reference points (freezing and boiling of water), while the Kelvin scale is anchored to absolute zero and is the fundamental temperature scale in physics. The Kelvin scale assigns zero to the lowest possible temperature and uses the same degree size as Celsius, so 0 degrees C equals 273.15 K.

Thermometers and Temperature Measurement

Every thermometer exploits some physical property that changes predictably with temperature. Mercury and alcohol thermometers use thermal expansion of a liquid. Thermocouples use the voltage generated at a junction of two different metals. Resistance thermometers use the change in electrical resistance of a wire. Infrared thermometers measure the thermal radiation emitted by a surface. Each type of thermometer is essentially a physical system that reaches thermal equilibrium with the object being measured, and the zeroth law guarantees that this process yields a meaningful result.

The International Temperature Scale (ITS-90) defines standard fixed points and interpolation methods for precise temperature measurement. These fixed points include the triple point of water (273.16 K exactly), the melting point of gallium, the freezing points of tin, zinc, aluminum, silver, gold, and copper. Between fixed points, specified thermometers and interpolation equations define the temperature scale with high precision.

At extremely low temperatures near absolute zero, temperature measurement becomes challenging because thermal contact becomes weak and equilibration times grow long. At extremely high temperatures, such as those found in plasmas and stellar interiors, temperature measurement relies on spectroscopic methods and radiation analysis rather than direct thermal contact.

Historical Context

The zeroth law was first explicitly stated by Ralph Fowler in 1931, long after the first and second laws had been established in the mid-1800s. For decades, the concept of temperature and thermal equilibrium had been used implicitly without being formally stated as a law. Fowler recognized that this assumption needed to be stated explicitly as an axiom, since the entire logical structure of thermodynamics depends on it.

The name zeroth was suggested because the law is logically prior to the first and second laws. You need the concept of temperature (established by the zeroth law) before you can discuss heat transfer and energy conservation (the first law) or the direction of natural processes (the second law). Rather than renumbering the existing laws, physicists chose the whimsical numbering scheme that has persisted ever since.

Some physicists argue that the zeroth law could be derived from statistical mechanics rather than stated as an independent axiom. In statistical mechanics, temperature emerges naturally from the condition that maximizes the total number of microstates when two systems exchange energy. Two systems are in thermal equilibrium when they have the same value of this statistical temperature. However, as a practical matter, the zeroth law remains a useful independent statement within the axiomatic framework of classical thermodynamics.