Chemical Thermodynamics

Motivation

Chemical thermodynamics is the accounting layer under all of chemistry: bookkeeping rules for energy and entropy that tell you which reactions can happen and where a system will come to rest. It is unusual among scientific fields in that its "laws" are actually laws — never observed to be violated, in the sense that no experiment has ever returned a result inconsistent with them. The catch, and it is a large one, is that "never violated" is a statement about what you can measure at equilibrium, and real chemical systems are often metastable — not at equilibrium, with the path to equilibrium blocked by a kinetic barrier (see When it breaks). Thermodynamics is the only field where the laws are laws; it is also the field whose applicability to a given moment requires the most caution.

This note exists because Corrosion cites Pourbaix diagrams and the Nernst equation without a thermodynamic foundation underneath; both are re-expressions of the Gibbs free energy, and without ΔG, K, and the bridge =ΔG = -nFE= in hand, they are uninterpretable.

The laws

Four laws, numbered zero through three — the numbering itself a small lesson in how science orders itself.

Zeroth law

If two systems are each in thermal equilibrium with a third, they are in thermal equilibrium with each other — transitivity, and what licenses temperature: a scalar constant across systems in mutual equilibrium, a thermometer being just a third system small enough not to disturb the first two. Stated fourth (1930s, Fowler) but logically prior — hence "zeroth".[fn:: Fowler's naming is a rare case where the numbering reflects epistemic order; most such renumberings in science are cosmetic.]

First law

Energy is conserved: =ΔU = q + w=, where U is internal energy, q is heat added to the system, and w is work done on the system.[fn:: Sign conventions vary; the chemist's convention (work done on the system is positive) is used throughout. Physicists often use the opposite — a persistent source of silent bugs in cross-field calculations.] Forbids perpetual motion of the first kind. Says nothing about which direction a process must go — that is the second law's job.

Second law

The entropy of an isolated system never spontaneously decreases: dS ≥ δq_rev / T, with equality only for a reversible process. The inequality is the arrow of time: for any real (irreversible) process the total entropy of system plus surroundings increases. Forbids perpetual motion of the second kind. Often misunderstood — it does not say "entropy always increases" globally, only that the entropy of an isolated system increases; local decreases (a freezer, a living cell) are paid for by larger increases elsewhere.

Third law

The entropy of a perfect crystal at absolute zero is a constant, conventionally zero — an absolute scale for entropy (unlike enthalpy and free energy, defined only up to a reference state), so S° values in tables are absolute while H° values are relative to elements in their reference states. Also implies absolute zero is unreachable in a finite number of steps (Nernst heat theorem), rarely the limiting concern in practice.

State functions

U, H, S, G, A (Helmholtz) are state functions: changes depend only on endpoints, not path. Heat and work are not — they are path-dependent process variables. This is why thermodynamics is computationally powerful: you can compute ΔH by summing any path through state space, in particular from tabulated standard enthalpies of formation even when the actual reaction proceeds by a completely different mechanism.

Enthalpy H

Absorbs the pV work term at constant pressure: H : U + pV=. At constant external pressure (the usual benchtop condition), =ΔH = q_p= — enthalpy change equals heat exchanged. This is why chemists tabulate ΔH°f; most chemistry happens at ~1 atm. For condensed-phase-only reactions ΔH ≈ ΔU, but for gas-evolving reactions the pV term matters.

Entropy S

Counts the microstates consistent with a macrostate: =S = k_B ln Ω= (Boltzmann, on his tombstone). The thermodynamic (=dS = δq_rev / T=) and statistical definitions are linked but not trivially — the bridge needs the ergodic hypothesis and equal a priori probabilities, both deeper than they look. For working chemistry: S is a state function, is absolute (third law), and the universe's entropy is what the second law constrains.

Gibbs free energy G

The free energy at constant T and p — the natural ensemble for bench chemistry: G : H - T S=. For a finite isothermal change at constant pressure, =ΔG = ΔH - T ΔS=. The spontaneity criterion at constant T, p is ΔG < 0 (spontaneous), ΔG > 0 (non-spontaneous), =ΔG = 0= (equilibrium) — the single most-used inequality in chemistry, compressing the first and second laws into one number. Note it is not the most general criterion: at constant T, V the relevant potential is the Helmholtz =A = U - TS=, and at constant S, V it is U. Choosing the wrong potential for your boundary conditions is a classic error.

Equilibrium constants

ΔG as a spontaneity criterion is a statement about direction, not rate: a reaction with =ΔG = -100 kJ/mol= at 298 K will proceed forward if it proceeds at all, but ΔG tells you nothing about how fast (see Reaction Kinetics). At equilibrium every allowed reaction has =ΔG = 0= for the equilibrium composition — the composition that minimizes G subject to stoichiometric constraints, where the derivative with respect to extent of reaction is zero. Because =ΔG = 0= is one equation in one unknown, it pins the equilibrium composition, and the magnitude of the constant that does the pinning is what we call K.

For a general reaction aA + bB ⇌ cC + dD, the reaction quotient is =Q = (a_C^c · a_D^d) / (a_A^a · a_B^b)=, where a_i are activities (dimensionless, ≈ concentration or partial pressure for dilute/ideal systems, divided by the standard value). The fundamental relation between G and Q is =ΔG = ΔG° + R T ln Q=. At equilibrium Q → K and =ΔG = 0=, so

: ΔG° = - R T ln K

This is the bridge between the thermodynamic side (free energies, measurable calorimetrically or electrochemically) and the analytical side (equilibrium constants, measurable by concentration). ΔG° is a standard-state quantity (1 M, 1 bar, 298.15 K by convention), so K is a standard-state constant; the actual reaction quotient at any other composition is Q, and the sign of =ΔG = ΔG° + RT ln Q= tells you which way the reaction will go from that composition. A useful number: at 298 K RT ln 10 ≈ 5.71 kJ/mol, so a 10-fold change in K is ≈ 5.7 kJ/mol of ΔG°, and a 1 eV change (≈ 96.5 kJ/mol) is a factor of ≈ 10^16.9 in K — electrochemical potentials and equilibrium constants are quantitatively close-coupled.

Le Chatelier's principle

If a system at equilibrium is perturbed, the equilibrium shifts to partially oppose the perturbation. A qualitative corollary of =ΔG° = -RT ln K= and the temperature dependence of K, not an independent law. Concretely:

• Adding a product shifts toward reactants (raises Q above K, so ΔG > 0

forward and reverse is favored).

• Compressing a gas-phase equilibrium shifts toward the side with fewer moles of gas.
• Heating shifts an endothermic reaction toward products

(=d ln K / dT = ΔH° / RT²=, the van 't Hoff equation). Adding inert gas at constant volume does not shift it (partial pressures unchanged) — a common misconception; at constant pressure it does.

The principle is a mnemonic, not a derivation; when in doubt go back to =ΔG = ΔG° + RT ln Q= and compute.

The electrochemical bridge

The bridge the Nernst equation and Pourbaix diagrams rest on, worth deriving.

Electrical work and Gibbs free energy

For a process at constant T and p, the maximum non-expansion work (work other than pV against the atmosphere) is =w_non-pV,max = ΔG= — a direct consequence of =G = H - TS= and the first and second laws combined (see Atkins, or Denbigh). For an electrochemical cell, the relevant non-=pV= work is electrical work: moving charge Q through a potential difference E. The work done by the cell on the surroundings is =w_elec = - Q_cell · E= (negative: work done by the system). For a redox reaction transferring n moles of electrons, =Q_cell = n F= where =F = 96485 C/mol= is Faraday's constant. Equating the two expressions for maximum non-=pV= work:

: ΔG = w_non-pV,max = - n F E

This is the electrochemical bridge: =ΔG = - n F E=. Under standard conditions, =ΔG° = - n F E°=. Combined with =ΔG° = -RT ln K= this gives =E° = (R T / n F) ln K=, and combined with =ΔG = ΔG° + RT ln Q= gives the Nernst equation in its full thermodynamic form:

: E = E° - (R T / n F) ln Q

Everything in Corrosion — Pourbaix diagrams as E vs pH maps, the Nernst equation for half-cells, the prediction of corrosion immunity / passivity / activity regions — is a geometric restatement of this one relation. Pourbaix diagrams work because Pourbaix computed E as a function of pH from tabulated ΔG°f values via =ΔG° = -nFE°= and the water/hydroxide equilibria; the lines are contours of =ΔG = 0= between adjacent stable species.

Worked example: cell potential from ΔG°f

Consider the Daniell cell: Zn(s) + Cu²⁺(aq) → Zn²⁺(aq) + Cu(s). Tabulated standard Gibbs energies of formation (298 K, kJ/mol): =ΔG°f(Cu²⁺, aq) = +65.5=, =ΔG°f(Zn²⁺, aq) = -147.2=, and elements in their reference states are zero by convention.

: ΔG°rxn = [(-147.2) + 0] - [0 + (+65.5)] = -212.7 kJ/mol

Transferring =n = 2= electrons:

: E°cell = -ΔG° / (n F) = (212.7e3) / (2 × 96485) = +1.101 V

Tabulated standard reduction potentials give =E°Cu²⁺/Cu = +0.337 V= and =E°Zn²⁺/Zn = -0.763 V=, so =E°cell = 0.337 - (-0.763) = +1.100 V=. The two routes agree to rounding — which is the point: standard potentials and standard free energies are the same information, interconverted by =ΔG° = -nFE°=. Calorimetry plus a reference electrode gives you both.

Worked example: corrosion spontaneity

For the anodic dissolution of iron in mildly acidic water, Fe(s) + 2 H⁺(aq) → Fe²⁺(aq) + H₂(g). Using ΔG°f(Fe²⁺, aq) ≈ -78.9 kJ/mol and water/proton/Hâ‚‚ standard values, ΔG°rxn ≈ -84 kJ/mol at 298 K — strongly negative, so iron thermodynamically wants to corrode to Fe²⁺ and Hâ‚‚ at =pH = 0=. The reason structural iron persists at room temperature is kinetic: the rate depends on cathodic H⁺ reduction exchange current, oxide film formation, oxygen availability, etc. — the canonical "thermodynamics says yes, kinetics says not yet" situation, and why corrosion engineering is mostly about kinetics and films, not ΔG.

When it breaks

Thermodynamics is rigorous but its applicability to a real moment is narrower than it looks. The standard caveats:

• Kinetic freezing. =ΔG = 0= at equilibrium but the system may be unable to get

there in any accessible time. Diamond at room temperature has ΔG > 0 for conversion to graphite; the reaction is not observed on human timescales because the activation barrier is enormous. Window glass is a kinetically arrested supercooled liquid nowhere near its thermodynamic ground state. Spontaneous means "in the absence of a barrier", which is almost never the actual condition.

• Activity ≠ concentration. =ΔG = ΔG° + RT ln Q= uses activities, not

concentrations. In dilute ideal solutions a_i ≈ [i]/c° and the substitution is invisible, but in real electrolytes (seawater, brines, physiological ionic strength) activity coefficients deviate substantially from 1. Debye-Hückel gives the limiting-law correction =log γ± = -A z⁺ z⁻ √I= at low ionic strength I, and its extended forms (Davies, SIT, Pitzer) cover higher I. Pourbaix diagrams computed with =a = 1= are projection diagrams; the real E / pH boundaries shift by tens of millivolts when activities are corrected.

• Metastability, not equilibrium. Real chemical systems are often

metastable: glass, passivating oxide films, many polymorphs, supercooled water. Thermodynamics tells you the endpoint of an infinitely slow, fully equilibrated process; it does not certify that your system is at that endpoint. A Pourbaix diagram says "if equilibrated, iron is immune at this E and =pH="; it does not say the iron in front of you is immune right now, because the passivating film may be defective or locally ruptured. (Related: ΔG°, E°, and K are defined against an arbitrary standard state — comparing K values across molarity vs. molality is a silent error; and ΔG°f is a bulk quantity, useless at nanoscale surfaces where reconstructions and adsorbates dominate, and where nucleation has its own ΔG* barrier.)

Meta-observation

Thermodynamics is the only field where the "laws" are actually laws — never observed to be violated. Not because they are trivially true; because they were abstracted from many measurements and have survived every test for a century and a half. But the operational content of "never violated" is exactly "consistent with everything we have measured at equilibrium", and the qualifier matters: real chemical systems are often not at equilibrium (metastable, kinetically frozen, driven, open). Thermodynamics is the ceiling of what is allowed, not a description of what is happening. The useful habit is to ask, every time, "is this system actually at the equilibrium this calculation assumes?" — usually the honest answer is "no, but close enough for the conclusion to hold", or "no, and that is the whole story" (corrosion, glass, life).

See also

• Electrochemistry — the hub; this note is its

thermodynamic prerequisite.

• Corrosion — Pourbaix diagrams are =E=-vs-=pH=

maps; this note is why they work.

• General Chemistry Foundations — bonding, redox, the

chemical primitives.

• Electrochemical Thermodynamics — Nernst from

first principles, the detailed derivation this note only sketches.

• Acid-Base Chemistry — pH, buffers,

the proton axis of Pourbaix diagrams.

• Reaction Kinetics — the rate counterpart

to thermodynamic spontaneity; without it, the "when it breaks" section has no resolution.