In this section, we start from the experimental fact that heat engines are never 100% efficient, and also that there is a natural direction to spontaneous processes (such as the cooling of a cup of coffee), and derive the existence of a new function of state, entropy.
In classical thermodynamics, entropy ``completes the set'' of relevant thermodynamic variables and, starting with the fundamental thermodynamic relation and using Maxwell's relations, enables us to work out what is happening to systems during a variety of processes.
The underlying meaning of entropy however is only clarified when we progress to statistical physics.
(Some of the early part of this section will be revision, but the overall approach to entropy is new.)