what is entropy
The equality Carnot left, QHTH = QCTC, shows that a hidden quantity returns to itself after one reversible loop. That quantity is entropy, written dS = dQrev/T, and for the universe as a whole it does not shrink. Time therefore runs only one way.
Put heat Q into a body at temperature T and the entropy rises by QT. The same heat produces a larger rise when the temperature is lower, because a cold quiet system is shaken far more than one that is already hot, so the low temperature buys more disorder. The weight sits at 1/T, which is why entropy and temperature do not come apart.
On one reversible loop the gas gains +QHTH and loses -QCTC, and the two cancel. Coming back to the same value means entropy is a state quantity: it hangs on the state, not on the path. Even when the real process cannot be undone, the change can be counted along a convenient reversible path, and that is the trick engineers keep using.
When heat Q flows from the hot side to the cold side, the hot side loses QTH and the cold side gains QTC, and because the cold temperature is lower the sum is positive. As the two temperatures draw together, the rise made by one parcel of heat approaches 0, so a flow across almost no gap is that much closer to reversible. An efficient exchanger keeps the two temperatures as near as it can, and however far you close the gap the rise stays above 0.
Statistically, entropy counts the arrangements that make one state. S = k ln Ω, and a gas that reaches more cells has more microstates and more entropy. Boltzmann counting locks onto the thermodynamic dQrev/T, so the two definitions become one, and a system drifts toward the arrangement with the most microstates simply because there are overwhelmingly more ways to be there.
Three things are worth keeping. Entropy is a state quantity (dS = dQrev/T), the entropy of the universe does not shrink (ΔStotal ≥ 0), and equality holds only for a change that can be undone. The following chapter puts the three to work on real cycles. Entropy made by a step that cannot be undone shows up as lost work, so cutting that production is how a machine yields more useful work.