In this section the concept of temperature is introduced qualitatively by a new method similar to Buchdahl's method of introducing entropy.
A thermodynamic system may be in thermal contact with its environment through a thermally conducting wall. A thermally conducting wall is called a diathermal wall. A state that remains unchanged when the system is in thermal contact with its environment is said to be in thermal equilibrium with its environment. If the volume of a system in thermal contact with its environment is changed, the pressure changes spontaneously to keep the system in thermal equilibrium with the environment.
Two states in thermal equilibrium with the same environment are said to be in mutual thermal equilibrium with each other. The relation of mutual thermal equilibrium satisfies the axioms of an equivalence relation. Let (P1,V1) | (P2,V2) mean that (P1,V1) and (P2,V2) are in mutual thermal equilibrium with each other. Then, for all (P1,V1), (P2,V2), and (P3,V3):
This equivalence relation divides the set of equilibrium states into equivalence classes called isothermal sets T. Two states in the same isothermal set are in mutual thermal equilibrium with each other.
Now suppose that the volume of the system is fixed and the thermal property of the environment is changed (from hot to cold or from cold to hot). A spontaneous transition occurs from the initial state to a final state in thermal equilibrium with the second environment. When the final state is one-way adiabatically accessible from the initial state, the final state is said to be hotter than the initial state, and the second environment is hotter than the first environment; then, if the initial state is (P1,V0) and the final state is (P2,V0), we write (P1,V0) << (P2,V0).
Let (P1,V1) be any state in an isothermal set T1, and let (P2,V2) be any state in a different isothermal set T2. Let (P12,V2) be in T1 and let (P21,V1) be in T2. Experience shows that either (P1,V1) << (P21,V1) and (P12,V2) << (P2,V2), or (P21,V1) << (P1,V1) and (P2,V2) << (P12,V2). In other words, a unique hotter-than relation holds between T1 and T2 which is independent of the choice of the fixed volume.
The hotter-than relation between isothermal sets satisfies the axioms for a simple order relation. For all T1, T2, and T3:
These axioms are true for the isothermal sets because they are true for the adiabatic accessibility relation between individual states with a fixed volume.
The ordered system of isothermal sets can now be labeled with numerical temperatures consistent with the ordering.