Entropy: counting the ways

A cup of hot tea on the table only ever cools; it never draws heat from the air to get hotter — although that would conserve energy just as well. The ammonium nitrate in a cold pack absorbs heat as it dissolves, yet it dissolves by itself; open a partition and a gas rushes into a vacuum without any change in its energy. The first law only checks that energy balances; it says nothing about direction. The quantity that decides direction is entropy, and at heart it is surprisingly simple: counting.

A system can be described at two levels: a macrostate says only “this much energy in A, that much in B”; a microstate specifies how many quanta each individual oscillator holds. One macrostate usually corresponds to an astronomically large number of microstates, $W$. The basic assumption of statistical mechanics is that in an isolated system every accessible microstate is equally likely. So the macrostate with the largest $W$ is almost certain to be what we see.

Boltzmann defined entropy as $S = k\ln W$. The logarithm turns huge products into sums: for two solids together $W = W_\mathrm{A}W_\mathrm{B}$, so $S = S_\mathrm{A} + S_\mathrm{B}$. The second law of thermodynamics can then be put like this: an isolated system moves spontaneously toward macrostates with more microstates, and its total entropy never decreases. Heat does not flow from hot to cold because something pushes it; it flows because there are vastly more ways to share the energy that way.

On the bench, a system of 60 + 40 oscillators still visibly jitters; scale it up “×100” and the peak becomes tall and narrow, with q_A spending more than 99.9 % of the time within ±5 % of its most probable value (for 60 + 40 oscillators even ±20 % covers only about 88 %). A real solid has 10²³ atoms, and the chance of energy flowing back is too small to be seen even once in the age of the universe. The second law is a statistical law, but for everyday objects it is iron-clad.

Clausius gave entropy a thermodynamic definition long before its statistical meaning: for heat absorbed reversibly, $\Delta S = q_\text{rev}/T$. The same heat raises the entropy of a cold object more — a cold object holds little energy, so a little extra opens up relatively many new ways to share it.

• A phase change at its normal transition temperature is reversible, so $\Delta_\text{trs}S = \Delta_\text{trs}H/T_\text{trs}$. Many liquids have entropies of vaporization close to 85 J/(K·mol) (Trouton’s rule), because turning a liquid into gas at 1 bar multiplies the volume each molecule can roam by roughly the same factor; water, alcohols and ammonia have hydrogen-bonded, more ordered liquids and clearly larger values (109 for water).
• Isothermal expansion of an ideal gas: $\Delta S = nR\ln(V_2/V_1)$; doubling the volume adds $R\ln 2 = 5.76$ J/K per mole.
• Ideal mixing: $\Delta_\text{mix}S = -nR\sum x_i\ln x_i$; mixing equal amounts of two gases again gives 5.76 J/K per mole — the enthalpy does not change, the whole drive is entropy.

The third law of thermodynamics: a perfect crystal at 0 K has only one arrangement, $W = 1$, so $S = 0$. Heating from 0 K bit by bit and adding up $q_\text{rev}/T$ measures a substance’s absolute entropy at 298 K, the standard molar entropy $S^\circ$ (unlike enthalpy, entropy has a true zero). Some patterns:

• gases ≫ liquids > solids: $\ce{H2O(g)}$ 188.8, $\ce{H2O(l)}$ 70.0 J/(K·mol);
• heavier atoms and more complex molecules have larger S°: He 126.2, Xe 169.7; $\ce{CH4}$ 186.3, $\ce{C3H8}$ 270.3;
• hard, orderly crystals have very small S°: diamond 2.4, graphite 5.7.

A reaction’s entropy change is the difference between the S° of products and reactants. Rule of thumb: if the moles of gas increase, ΔrS° is usually positive; if they decrease, negative.