Zeroth law and temperature

Thermal equilibrium

Thermodynamics begins with the idea of equilibrium. Two systems are in thermal equilibrium if, when placed in thermal contact, there is no net transfer of energy as heat between them. Thermal contact means energy can pass between systems because of temperature difference, even if matter does not pass.

For example, a hot metal spoon placed in a cool cup of water is not initially in thermal equilibrium with the water. Energy flows from the spoon to the water. After enough time, both reach a common temperature, and the net heat transfer stops. At that point, they are in thermal equilibrium.

The zeroth law

The zeroth law of thermodynamics states: if system A is in thermal equilibrium with system B, and system B is in thermal equilibrium with system C, then system A is in thermal equilibrium with system C.

This may sound obvious, but it is foundational. It allows temperature to be defined consistently. If a thermometer is in thermal equilibrium with a cup of water, and that same thermometer reading corresponds to another object, then those objects have the same temperature.

The zeroth law is what makes thermometers possible. The thermometer does not directly measure every molecule in a system. It comes into thermal equilibrium with the system, and its own measurable property indicates temperature.

Temperature versus heat

Temperature and heat are not the same. Temperature is a state variable: it describes the thermal condition of a system at equilibrium. Heat is energy transferred because of a temperature difference.

A large bathtub of warm water may contain more internal energy than a small cup of boiling water, even though the cup has higher temperature. Temperature is not total energy; it is related to how energy is distributed among microscopic degrees of freedom.

Heat is written often as $Q$, but it is not something contained in a system. A system has internal energy, not heat. Heat is energy in transit.

Temperature and microscopic motion

In kinetic theory, temperature is related to average microscopic kinetic energy. For an ideal monatomic gas,

angle = rac{3}{2}k_B T$$ Here $m$ is molecular mass, $langle v^2 angle$ is the average squared speed, $k_B$ is Boltzmann's constant, and $T$ is absolute temperature. This equation does not mean every molecule has the same speed. Molecules have a distribution of speeds. Temperature is related to the average energy of random motion. ## State variables Temperature is a state variable, meaning it depends only on the current equilibrium state, not on how the system got there. Other state variables include pressure, volume, and internal energy. Heat and work are not state variables. They describe energy transfer processes. The same final temperature can be reached by heating, compression, friction, or mixing. The path matters for heat and work, but not for the final temperature itself. ## Thermal equilibrium and measurement To measure temperature accurately, the thermometer and system must reach thermal equilibrium, and the thermometer should not significantly disturb the system. A large thermometer placed in a tiny sample could change the sample's temperature. Good measurement requires the measuring device to interact weakly enough while still reaching equilibrium. ## The big idea The zeroth law establishes temperature as a reliable property shared by systems in thermal equilibrium. Temperature measures thermal state, not total energy, and heat is energy transferred because of temperature difference. This distinction is essential for all later thermodynamics, from engines to entropy to statistical mechanics.