The Carnot cycle is a moreof a consistency argeument than a proof. It shows that neither heat nor work are state functions in a cycle process that involves an input of heat and output of work. However, the heat transferred divided by the temperature is a state function. In order to understand the consistency argument used in the Carnot cycle we need to investigate the properties of adiabatic processes. The first segment addresses that issue. It shows how T, V and P are related during an adiabatic process.
Once we understand the relationships between T, V and P for an adiabatic process we are ready to tackle the Carnot cycle. The cycle is based on an expandions that is divided into two parts, one isothermal and one adiabatic. The compression to return to the original state is likely divided into two parts. The Carnot cycle is used to show that neither heat nor work is a state functino for a cylic pcocess that passes through these four states.
Entropy is a state function. Like internal energy and enthalpy it depends only on variables of state and not on the path taken to reach a particular state. However, we can still consider the calculation of entropy along various paths. This is of interest from the point of view of practical applications as well as for seeing how entropy actually works as a state function. The brief video below summarizes the entropy calculation along constant temperatire, volume and pressure paths.
