The Four Laws

Put a hot cup of coffee on your desk. Walk away. Come back in an hour. The coffee is cold, and you knew it would be.

Every microscopic law governing the molecules in that cup is symmetric in time. Run the equations backwards and they hold just as well, so nothing at that level distinguishes cooling from reheating. The asymmetry you observe has to come from somewhere else.

Thermodynamics was built in the 1800s to answer questions like that, by people who had no settled view on whether atoms existed. They arrived at four principles governing heat and energy, numbered zeroth through third. The zeroth was identified last and then put first, because the other three quietly assume it.

Zeroth: temperature exists

If $A$ is in thermal equilibrium with $C$, and $B$ is in equilibrium with $C$, then $A$ is in equilibrium with $B$.

This sounds too obvious to be a law. But that transitivity is precisely what licenses the idea of a temperature scale, a single number you can attach to a body and compare against any other. Without it, thermometry would be meaningless.

First: energy is conserved

$$dU=\delta Q-\delta W$$

The internal energy changes only through heat in or work out.

Second: entropy never decreases

$$\Delta S_{\text{universe}}\ge 0$$

Clausius: heat never passes spontaneously from colder to warmer. Kelvin: no process has as its sole result the conversion of heat entirely into work. The two statements are equivalent, and both say entropy increases.

This is the one that gives time a direction. Everything else in physics runs equally well backwards.

Third: absolute zero is a limit, not a destination

As $T\to 0$, the entropy of a perfect crystal tends to zero. In statistical terms $\Omega \to 1$, a unique ground state.

The consequence is operational: each step of cooling removes a fraction of the remaining energy, never a fixed amount, so reaching exactly zero requires infinitely many steps. Absolute zero is unreachable in principle rather than merely difficult.

The Carnot bound

Put the first and second together for a heat engine between reservoirs at $T_H$ and $T_C$ and you get a ceiling on efficiency that no engineering can lift:

$$\eta \le 1-\frac{T_C}{T_H}$$

Nothing about the working fluid, the mechanism, or the century in which it was built enters this. It is a statement about temperature ratios alone, which is why it was correct before anyone knew what heat was.

The cycle is worth walking round once. From $A$ to $B$ the gas expands in contact with the hot reservoir, absorbing $Q_H$ at constant $T_H$. From $B$ to $C$ it expands with no heat exchange, so it cools by doing work. $C$ to $D$ compresses it against the cold reservoir, dumping $Q_C$. $D$ to $A$ compresses adiabatically back to the start.

Because entropy is a state function it must return to its initial value after a full loop, so $Q_H/T_H=Q_C/T_C$. Substituting into $\eta =1-Q_C/Q_H$ gives the bound directly. The efficiency limit is entropy bookkeeping, nothing more.