The zeroth law came about to describe what property changes may be observed when two systems bounded by diathermic walls (e.g., metal walls) are placed in contact with each other. If no property change is observed, then the two systems are said to be in thermal equilibrium. This became the basis for what we observe as the temperature of the system measured by a thermometer. The exact language of the zeroth law as state by Atkins is:
If system A is in thermal equilibrium with system B and with system C, then systems B and C are in thermal equilibrium with each other. (Slightly paraphrased)
He then goes on to differentiate classical thermodynamics (dealing with observations of bulk properties like pressure that can be observed mechanically) from statistical thermodynamics dealing with an understanding of (mostly mathematically) bulk properties based on a statistical (average behavior) description of the behavior of the particles.
Good description of the equilibrium state of a system being that, the properties are unchanging even though atoms are still transitioning from one energy state to another. However, because there is no net change in the distribution of atoms in different energy states, the system is said to be in thermodynamic equilibrium.
In this section, he also reviews the origin of the Boltzmann constant k, as related to the beta variable where beta = 1/kT. The beta variable parametrizes the fraction of particles with a certain energy E (e^-beta*E), with respect to the measured kelvin temperature for the system. This fraction is exponential in nature and is an increasing function with respect to T and decreasing function with respect to E.
As Atkins noted early in the chapter, this law is an afterthought. Certainly, it was not something that was prominent in my mind remembering back to my early introductions to thermodynamics. Perhaps it is a way to formally define the property of a system we call temperature.
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