Two moles of an ideal gas undergo free expansion (expansion into vacuum) from volume $V$ to volume $10V$ at constant temperature. The entropy changes are:

Entropy (S) is one of the most profound and far-reaching concepts in all of science, with implications ranging from the efficiency of heat engines to the arrow of time to the maximum information that can be stored in a region of space. Thermodynamic definition (Clausius, 1850): dS = dq_rev/T, where dq_rev is the heat absorbed in a reversible process at absolute temperature T. The entropy is defined only as a change (ΔS = S_final - S_initial), and the third law (Nernst, 1906) provides an absolute reference point: the entropy of a perfect crystalline solid at absolute zero (0 K) is zero. Statistical definition (Boltzmann, 1877): S = k_B * ln(W), where W is the number of microstates (ways of distributing the system's energy among its particles while maintaining the same total energy) and k_B is the Boltzmann constant (1.38 × 10^-23 J/K). This definition reveals the deep connection between entropy and molecular disorder: more ways of arranging the energy → higher W → higher S. The relationship S = k_B * ln(W) is engraved on Boltzmann's gravestone in Vienna. For an ideal gas expanding from V to 2V: the number of positions available to each molecule doubles, so for N molecules, W increases by 2^N. This enormous increase in W (and therefore S) is the microscopic basis for the irreversibility of gas expansion — once the gas has expanded, it is virtually impossible for all molecules to spontaneously return to the smaller volume (though not thermodynamically forbidden, the probability is overwhelmingly small: of order (1/2)^N for N ~ 10^23 molecules).

Understanding when entropy increases or decreases in physical and chemical processes is essential for predicting spontaneity. Entropy increases when: the number of moles of gas increases in a reaction (delta n_gas > 0); a substance changes from a more ordered to a less ordered phase (melting, vaporisation, dissolution); temperature increases (more microstates available at higher thermal energy); a solid or liquid dissolves in a solvent; a gas expands into a larger volume; two or more substances mix; a polymer chain unfolds. Entropy decreases when: the reverse of the above processes occurs; a gas is compressed; a precipitate forms from solution; a polymer folds into an ordered structure. Quantitative expressions: for isothermal expansion of an ideal gas: ΔS = nRln(V_f/V_i). For reversible phase transition at temperature T_trans: ΔS = ΔH_trans/T_trans (e.g., melting: ΔS_fus = ΔH_fus/T_melt; vaporisation: ΔS_vap = ΔH_vap/T_bp). For heating at constant pressure: ΔS = nCp*ln(T_f/T_i). For a chemical reaction at standard conditions: ΔS°_rxn = Σ n_i S°_products - Σ n_j S°_reactants. The standard molar entropy S° of a substance reflects its molecular complexity, molecular mass, and physical state at 298 K (gases > liquids > solids; larger, heavier molecules > smaller, lighter ones).

The second law of thermodynamics, arguably the most fundamental law in all of science, comes in several equivalent statements that each capture a different aspect of the universal directionality (irreversibility) of natural processes. Clausius statement (1850): heat cannot flow spontaneously from a colder body to a hotter body without the expenditure of work. Kelvin-Planck statement: it is impossible to construct a heat engine that operates in a cycle and produces net work while exchanging heat with only one thermal reservoir. Entropy statement: in any spontaneous irreversible process, the total entropy of the universe (system + surroundings) increases: ΔS_universe = ΔS_sys + ΔS_surr > 0. For a reversible process: ΔS_universe = 0 (the process is at the thermodynamic limit of efficiency). For an impossible process: ΔS_universe < 0 (violates second law). The profound implication of the entropy statement of the second law is that it defines the arrow of time: processes occur in the direction of increasing total entropy. The past can be distinguished from the future thermodynamically — the state with lower entropy is in the past (e.g., the intact egg) and the state with higher entropy is in the future (scrambled egg). This asymmetry, despite the time-symmetry of the underlying microscopic laws (Newton's equations, Schrodinger equation), is explained by the overwhelming statistical improbability of low-entropy configurations: the universe started in an extremely low-entropy state (the Big Bang), and entropy has been increasing ever since, driving all irreversible processes that give time its direction.

The Gibbs free energy G = H - TS provides a criterion for spontaneity in terms of the system's properties alone (without needing to calculate the entropy change of the surroundings explicitly). At constant temperature and pressure: ΔG = ΔH - TΔS_sys = -T(ΔS_sys + ΔS_surr) = -TΔS_universe. Therefore: spontaneous process: ΔG < 0 (equivalent to ΔS_universe > 0). Reversible (equilibrium): ΔG = 0. Non-spontaneous: ΔG > 0. This is enormously useful in chemistry and biochemistry because most processes of interest occur at constant T and P, and we can calculate ΔG from properties of the system (ΔH and ΔS) without needing to know anything about the surroundings. The temperature dependence of ΔG: ΔG = ΔH - TΔS. Four possible sign combinations: ΔH < 0, ΔS > 0: always spontaneous (ΔG < 0 at all T). ΔH > 0, ΔS < 0: never spontaneous (ΔG > 0 at all T). ΔH < 0, ΔS < 0: spontaneous at low T (enthalpy driven, ΔG < 0 when T|ΔS| < |ΔH|). ΔH > 0, ΔS > 0: spontaneous at high T (entropy driven, like protein denaturation). The crossover temperature: T_crossover = ΔH/ΔS. Below T_crossover: ΔG = ΔH - TΔS has sign of ΔH (enthalpy controls). Above T_crossover: ΔG has sign opposite to ΔS... wait, at T > T_crossover and ΔS > 0: TΔS > ΔH, so ΔG = ΔH - TΔS < 0 (spontaneous). Standard free energy and equilibrium: ΔG° = -RTlnK, where K is the equilibrium constant. Large K (reaction highly favourable): ΔG° very negative. Small K (reaction unfavourable): ΔG° very positive.

The entropy of mixing two ideal gases (or ideal solutions) at constant temperature and pressure is always positive, reflecting the increase in disorder upon mixing. For two ideal gases: ΔS_mix = -nR(x_A*ln(x_A) + x_B*ln(x_B)), where x_A and x_B are mole fractions. Since all x_i < 1, all ln(x_i) < 0, so ΔS_mix > 0 always. Maximum mixing entropy when x_A = x_B = 0.5: ΔS_mix,max = nR*ln(2) per mole of mixture. The entropy of mixing drives dissolution of gases in liquids, mixing of liquids, and dissolution of solutes. It is the entropy contribution to the Gibbs free energy of mixing: ΔG_mix = ΔH_mix - TΔS_mix. For ideal solutions: ΔH_mix = 0, so ΔG_mix = -TΔS_mix < 0 (always spontaneous). The entropy of mixing is also the basis of: colligative properties (reduction in chemical potential of solvent upon adding solute, due to entropy of mixing); osmotic pressure (tendency to dilute concentrated solution by osmosis is entropy-driven); spontaneous mixing of fluids (ink drop dispersing in water, perfume diffusing in air). The entropy of mixing has deep connections to information theory through the Shannon entropy H = -Σ pi*log(pi), which quantifies the information content of a message or the uncertainty in a probability distribution. Boltzmann's S = k_B*ln(W) and Shannon's H = -Σ pi*log(pi) are mathematically identical in appropriate units, revealing the deep connection between thermodynamic entropy and information entropy — explored by Szilárd, von Neumann, and Shannon in the mid-20th century.

The third law of thermodynamics, formulated by Walther Nernst in 1906 (Nernst heat theorem) and later extended by Max Planck: the entropy of a perfectly ordered crystalline solid approaches zero as the absolute temperature approaches zero. S(T → 0K) = 0 for a perfect crystal. Physical basis: at absolute zero, a perfect crystal has only one possible microstate — all atoms in their lowest energy positions in the regular crystal lattice. W = 1, and S = k_B*ln(1) = 0. The third law provides an absolute reference point for entropy, allowing absolute (third-law) entropies S° to be determined for substances by integrating heat capacity data from 0 K to 298 K: S°(298 K) = ∫₀²⁹⁸ (Cp/T)dT + ΔS_transition(s) (including contributions from all phase transitions between 0 and 298 K). Standard molar entropies S°(298 K) allow calculation of ΔS°_rxn for chemical reactions. Residual entropy: some real crystals have non-zero entropy at 0 K due to positional disorder frozen in at low temperatures. Examples: CO has residual entropy of ~5.8 J/mol K because CO molecules can orient in two ways (CO or OC) in the lattice and these are energetically almost equivalent; the random arrangement is frozen in as the crystal cools. Ice also has residual entropy (Pauling ice residual entropy ~3.4 J/mol K) due to the proton disorder in hydrogen bonds. These residual entropies were predicted theoretically and confirmed experimentally, providing elegant validation of the statistical mechanics approach to entropy.