Entropy from Counting: The Molecular Meaning of the Third Law
Entropy change for a reversible step at constant temperature is heat moved divided by that temperature; counting arrangements gives the molecular view through S = k ln W. The count explains why a perfect crystal at absolute zero has zero entropy.
What a learner can do afterwards
- States the Boltzmann entropy expression and applies it to a system with a known number of arrangements
- Calculates the residual entropy of a disordered solid such as carbon monoxide and compares it with the measured value
- Explains the third law in terms of a single accessible arrangement at absolute zero
- Uses S = k(ln Z + <E>/kT) to compare entropies at the same T with the same <E>, and explains why Z alone does not order entropy
1 · Read
You have met a change in entropy for a reversible step at constant temperature as heat divided by that temperature. A second view shows why mixed things tend to stay mixed. Entropy counts the arrangements your sample can adopt. Each distinct arrangement of positions and energies that fits the observed state is a microstate; W counts the equally likely ones. Boltzmann linked them with S = k ln W, where k is the Boltzmann constant.
Picture four labelled particles in left and right boxes. Counting placements gives 16 microstates. Six place two in each box, so the even split is most likely. Two place all four on one side, the least likely. N labelled particles in n boxes give n to the power N arrangements.
Cool a perfect crystal toward absolute zero to one arrangement. One arrangement means W is 1, and ln 1 is zero, so entropy is zero. That is the third law. Real carbon monoxide keeps some entropy because each frozen molecule can point either way, leaving many arrangements. Flipped nitrogen looks the same, so almost none remains.
More arrangements means larger entropy. Of the same substance, gas outranks liquid, and liquid outranks solid. S follows Z and mean energy <E> at T: S = k(ln Z + <E>/kT). An energy shift changes Z but not probabilities or S. Compare Z only at same T and <E>.
Entropy counts arrangements through S = k ln W, and a perfect crystal at absolute zero has exactly one, so its entropy is zero.
2 · Watch
Take it off screen
Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.