Entropy

Entropy $S$ is a measurement of uniformity and disorder of a system. In Clausius' words, it is the "amount of transformation of a system". In differential form it is defined as

where $Q_\text{rev}$ is the heat exchanged by a reversible thermodynamic transformation and $T$ is temperature. $S_\text{it}$ is the information theory entropy, which becomes thermodynamic entropy when weighed by the Boltzmann constant $k_{B}$. Notably, $dS$ is an Exact differential, unlike $\delta Q$, which usually isn't.

Entropy is defined up to an additive constant and the difference between two states $A$ and $B$ connected by a reversible transformation is

By Clausius' theorem, $\Delta S$ is independent of path, so long it is reversible.

Irreversible transformations#

To understand the effects of an irreversible transformation, let's define two paths, $P$ and $R$, between the same end states. $P$ is not reversible, $R$ is. The combined process $P-R$ forms an irreversible cycle. Invoking Clausius' theorem again, we know that $\int_{P-R}\delta Q/T\leq 0$, which in turn means

since the right-hand integral is the entropy variation defined above. In other words, for an irreversible path $P$ between $A$ and $B$, we have

If we are working in an isolated system, $\delta Q=0$ and so we get

which means that the entropy of an isolated system can only increase or, at most, remain constant if all transformations are reversible. Since the Universe is considered and isolated system, that means that the global entropy of the Universe can only increase.

From heat capacity#

Entropy can be measure starting from heat capacity $C(T)$, as

This is a convenient way to measure entropy differences experimentally, as we can see by inverting the formula:

With an experimental fit of $C(T)$, we can perform consistency tests on the theory. Also, if we take the second derivative of $S$ in terms of $E$ we get

The second derivative of entropy is negative if heat capacity is positive, which it almost always is¹. If $C>0$, the system is said to be thermodynamically stable. The reason is that the second law of thermodynamics requires us to maximize entropy during heat exchange. For that to be true, the post-exchange entropy needs to be at a stationary point $\frac{ \partial S }{ \partial E }=0$² and the second derivative needs to be negative, $\frac{ \partial ^{2}S }{ \partial E^{2} }<0$, which it is if the heat capacity of both participating systems is positive.

Etymology#

The word "entropy" was coined by Clausius starting from the Greek work $\tau \rho o\pi \eta$ (tropi) which means "transformation". The "en" comes from "energy". It was added as a prefix to make the word as similar as possible to energy due to them being intrinsically linked.