In this section the concept of entropy is introduced qualitatively by Buchdahl's method.
The environment of a thermodynamic system may be hot or cold. If the system is in a container with thermally insulating walls, then changes in the environment from hot to cold, or cold to hot, do not cause any change in the equilibrium state of the system. The thermally insulating walls are called adiabatic walls. Any change of state of a system separated from its environment by adiabatic walls is called an adiabatic transition.
Imagine a thermodynamic system that consists of a fluid (e.g. liquid and vapor) in a container with adiabatic walls. Experience shows that when the state of the system is changed by slowly changing the volume it is possible to reverse the process and return the system to its initial state. In other words there exists an adiabatic transition from the initial state to the final state, and an adiabatic transition from the final state to the initial state. We then say that there exists a relation of mutual adiabatic accessibility between the two states.
The relation of mutual adiabatic accessibility between states satisfies the axioms of an equivalence relation. Let (P1,V1), (P2,V2), and (P3,V3) denote states, and let (P1,V1) ~ (P2,V2) mean that (P1,V1) and (P2,V2) are mutually adiabatically accessible from each other. Then, for all (P1,V1), (P2,V2), and (P3,V3):
This equivalence relation divides the equilibrium states into equivalence classes called isentropic sets S. Two states in the same isentropic set are mutually adiabatically accessible from each other.
Now suppose that the state of the same adiabatically enclosed system can be changed in other ways, such as by passing an electric current through a resistor in the fluid, or by stirring the fluid. Experience shows that in these cases the adiabatic transitions are irreversible; there exists an adiabatic transition from the initial state to the final state, but not from the final state to the initial state. We say that a relation of one-way adiabatic accessibility exists from the initial state to the final state. If the initial state is (P1,V1) and the final state is (P2,V2), we write (P1,V1) \ (P2,V2).
Let (P1,V1) be any state in an isentropic set S1, and let (P2,V2) be any state in a different isentropic set S2. Experience shows that either (P1,V1) \ (P2,V2) or (P2,V2) \ (P1,V1). If (P1,V1) \ (P2,V2), then every state in S2 is one-way adiabatically accessible from every state in S1 via (P1,V1) and (P2,V2). In other words, the one-way adiabatic accessibility relation holds from S1 to S2, and we write S1 \ S2.
The one-way adiabatic accessibility relation between isentropic sets satisfies the axioms for a simple order relation. For all S1, S2, and S3:
Axiom (1) is true because, as explained above, it is true for the individual states in S1 and S2. To see that axiom (2) is true note that if the conclusion is false, then there exists an adiabatic transition from every state in S3 to every state in S1. This implies that if S1 \ S2 is true then S2 \ S3 is false, and if S2 \ S3 is true then S1 \ S2 is false. In other words, the hypothesis is false.
The ordered system of isentropic sets can now be labeled with numerical entropies consistent with the ordering.