Thermodynamics

Entropy (S) measures the dispersal of energy and matter in a system. Formally, it is related to the number of microstates (W) — the number of different ways particles and energy can be arranged while producing the same macroscopic state: S = k ln W, where k is Boltzmann’s constant.

Higher entropy means more microstates are accessible. Gases have much higher entropy than liquids, which in turn have higher entropy than solids, because particles in the gas phase have far more possible positions and energy distributions. Dissolving a solid in a solvent generally increases entropy because the solute particles become dispersed throughout the solution.

The third law of thermodynamics establishes an absolute reference point: the entropy of a perfect crystal at 0 K is exactly zero (W = 1). This allows chemists to tabulate standard molar entropies (S°) as absolute values, unlike enthalpy where only changes are measured.

You can predict the sign of ΔS for a physical or chemical process using straightforward rules:

• ΔS > 0 (entropy increases): phase changes from more ordered to less ordered (solid → liquid → gas); dissolving a solid or liquid; increasing temperature; reactions that produce more moles of gas than they consume.
• ΔS < 0 (entropy decreases): condensation or freezing; crystallization from solution; reactions that produce fewer moles of gas.

The number of moles of gas is the strongest predictor because gases have enormously more microstates than condensed phases. For example, the decomposition of CaCO₃(s) into CaO(s) + CO₂(g) has ΔS > 0 because one mole of gas is produced from zero moles of gas in the reactants.

Standard molar entropies generally increase with molar mass and molecular complexity, reflecting the greater number of ways energy can be distributed among more atoms and bonds.

The second law states that the total entropy of the universe always increases for any spontaneous process:

ΔS_universe = ΔS_system + ΔS_surroundings > 0

A process can decrease the entropy of the system (as in freezing water) as long as the surroundings gain even more entropy. At constant pressure and temperature, the entropy change of the surroundings is: ΔS_surroundings = −ΔH_system / T. An exothermic reaction releases heat, warming the surroundings and increasing their entropy; an endothermic reaction absorbs heat, decreasing the surroundings’ entropy.

This explains why some endothermic processes are spontaneous (the large positive ΔS_system outweighs the negative ΔS_surroundings) and why some exothermic processes are nonspontaneous (the entropy decrease of the system is too large to overcome).

The standard entropy change for a reaction is calculated from tabulated S° values using a products-minus-reactants approach:

ΔS°_rxn = ∑nS°(products) − ∑nS°(reactants)

Each S° value is multiplied by the stoichiometric coefficient (n) of that substance in the balanced equation. Unlike ΔH°_f values, standard molar entropies are never zero for elements in their standard states — every substance at temperatures above 0 K has a positive entropy.

Typical S° values at 298 K: H₂(g) = 130.7 J/(mol·K), O₂(g) = 205.2, H₂O(l) = 69.9, H₂O(g) = 188.8. Note the dramatic difference between liquid and gaseous water, reflecting the much greater disorder in the vapor phase. Units are J/(mol·K), not kJ, so be careful to convert when combining with ΔH values (which are in kJ).

Gibbs free energy (G) unifies enthalpy and entropy into a single criterion for spontaneity at constant temperature and pressure:

ΔG = ΔH − TΔS

• ΔG < 0: the process is spontaneous (thermodynamically favorable).
• ΔG > 0: the process is nonspontaneous (the reverse is spontaneous).
• ΔG = 0: the system is at equilibrium.

Four scenarios arise from the signs of ΔH and ΔS: (1) ΔH < 0 and ΔS > 0 — always spontaneous; (2) ΔH > 0 and ΔS < 0 — never spontaneous; (3) ΔH < 0 and ΔS < 0 — spontaneous at low T; (4) ΔH > 0 and ΔS > 0 — spontaneous at high T. The crossover temperature where ΔG = 0 is T = ΔH/ΔS.

ΔG°_rxn can also be computed from standard free energies of formation: ΔG° = ∑nΔG°_f(products) − ∑nΔG°_f(reactants).

The standard free energy change is directly related to the equilibrium constant:

ΔG° = −RT ln K

where R = 8.314 J/(mol·K) and T is in kelvins. This equation reveals that:

• K > 1 (ΔG° < 0): products predominate at equilibrium.
• K < 1 (ΔG° > 0): reactants predominate at equilibrium.
• K = 1 (ΔG° = 0): neither side is favored.

A large positive E°_cell in electrochemistry corresponds to a large negative ΔG° and a large K, linking all three thermodynamic indicators of reaction favorability.

Under nonstandard conditions (concentrations other than 1 M, pressures other than 1 bar), the free energy change depends on the reaction quotient Q:

ΔG = ΔG° + RT ln Q

When Q < K, ΔG is negative and the reaction proceeds forward to produce more products. When Q > K, ΔG is positive and the reaction proceeds in reverse. At equilibrium, Q = K and ΔG = 0.

This equation explains why a reaction with a positive ΔG° can still proceed forward if the concentrations are far from equilibrium (Q << K). Conversely, a reaction with negative ΔG° can be driven backward if products are already in great excess (Q >> K).

The distinction between ΔG° (fixed value for a reaction at a given temperature) and ΔG (varies with composition) is essential: ΔG° tells you which direction is favored from standard conditions, while ΔG tells you the direction from the current state of the system.

Students perform best when thermodynamics is treated as a decision chain:

1. Predict sign of ΔS qualitatively from phase and particle changes.
2. Use ΔG = ΔH − TΔS for spontaneity at the stated temperature.
3. Connect to equilibrium with ΔG° = −RT lnK.
4. For nonstandard conditions, evaluate ΔG = ΔG° + RT lnQ.

This links macroscopic direction, temperature dependence, and equilibrium composition in one model. Quick checks: if K ≫ 1 then ΔG° should be negative; if Q < K then ΔG should be negative (forward-favored); if Q > K then ΔG should be positive (reverse-favored).