The maximum work $W_{max}$ that can be gained in a thermodynamic cycle from a thermal energy source that is at temperature $T_2$ by extracting in each cycle an amount of entropy $S$ and delivering it to a thermal energy sink at temperature $T_1$ is $W_{max} = S(T_2-T_1)$. This is Carnot's theorem although he used the term calorique and not Clausius who did not like that word and instead introduced the term entropy. If you wish, this is the operational meaning of entropy. The thermal energy extracted from the the reservoir is $ST_2$ while the thermal energy delivered to the sink is at least $ST_1$. It is exactly $ST_1$ if the proecss is reversible.

By cycle is meant a process in which the engine that converts thermal energy is operating in a periodic fashion meaning that all its thermodynamic parameters are periodic functions with the same period.

Almost anything that intuitively you would associate with the naïve concept of heat is what could be called "calorique", ie., entropy, except for being conserved for that would be energy. It is conserved only in a reversible cycle because the thermal energy dumped in the sink is $ST_1$, in other words the same amount of entropy $S$ is extracted at $T_2$ as it is dumped at $T_1$.

Nothing here said has anything to do with statistical mechanics although using other results following, such as Gibbs and Helmholtz potentials, etc., one can associate terms and behaviors in a statistical manner.

The maximum work $W_{max}$ that can be gained in a thermodynamic cycle from a thermal energy source that is at temperature $T_2$ by extracting in each cycle an amount of entropy $S$ and delivering it to a thermal energy sink at temperature $T_1$ is $W_{max} = S(T_2-T_1)$. This is Carnot's theorem although he used the term calorique and not Clausius who did not like that word and instead introduced the term entropy. If you wish, this is the operational meaning of entropy. The thermal energy extracted from the the reservoir is $ST_2$ while the thermal energy delivered to the sink is at least $ST_1$. It is exactly $ST_1$ if the process is reversible.

By cycle is meant a process in which the engine that converts thermal energy is operating in a periodic fashion meaning that all its thermodynamic parameters are periodic functions with the same period.

Almost anything that intuitively you would associate with the naïve concept of heat is what could be called "calorique", ie., entropy, except for being conserved for that would be energy. It is conserved only in a reversible cycle because the thermal energy dumped in the sink is $ST_1$, in other words the same amount of entropy $S$ is extracted at $T_2$ as it is dumped at $T_1$.

Nothing here said has anything to do with statistical mechanics although using other results following, such as Gibbs and Helmholtz potentials, etc., one can associate terms and behaviors in a statistical manner.