Boltzmann's formula

The link between the number of microstates $\Omega$ and the entropy $S$ is given by one of the most important formulas in all of physics, due to Ludwig Boltzmann.

Key formula — Boltzmann entropy

$\ev{S = k_B\,\ln\Omega}$

Here $k_B = 1{,}38\cdot 10^{-23}\;\text{J/K}$ is the Boltzmann constant and $\Omega$ is the number of microstates compatible with the macrostate. The unit of entropy is therefore joules per kelvin (J/K), the same units as $k_B$, since the logarithm is a pure number.

Why the logarithm specifically? The choice is not arbitrary. The logarithm turns a gigantic number of configurations (for a mole, $\Omega$ is of the order of $10$ raised to numbers with $23$ digits) into a manageable value. But above all it makes entropy additive: if we join two independent systems, the total number of microstates is the product of the two ($\Omega = \Omega_1 \cdot \Omega_2$, because every microstate of the first combines with every microstate of the second), and thanks to the logarithm the entropies add up:

$S = k_B\ln(\Omega_1\,\Omega_2) = k_B\ln\Omega_1 + k_B\ln\Omega_2 = S_1 + S_2$

This additivity is exactly the property we expect from an extensive quantity like energy: doubling the system doubles the entropy.

What matters is the change, not the absolute value

In practice, when solving problems we are never interested in the absolute values of $S$, only in the changes $\Delta S = S_B - S_A$ between an initial state A and a final state B. For the most common systems (ideal gas, solid bodies, thermostats) there are direct formulas that let us calculate $\Delta S$ without having to count the microstates $\Omega$ one by one.

These practical formulas are the subject of the next section: they are the real working tool for solving thermodynamics problems.

Links

Topics: Entropy and the second law Concepts: Entropy Skills: Micro-macro interpretation

Related exercises: Worked exercise — the coffee cools down · Hot shower: ΔS of the universe · Heat from hot to cold: proof