Electrochemical Thermodynamics

date2026-07-24tags:chem:

This note supports the rest of electrochemistry. Corrosion builds Pourbaix diagrams from the Nernst equation. OCP uses mixed-potential theory. Both rely on Nernst potentials. Reference electrodes use Nernst potentials for their internal chemistry. None of those notes derives the Nernst equation. This note produces it. It exposes the assumptions the others inherit.

Standard electrode potentials

A redox half-reaction:

$$\mathrm{Ox} + n e^- \rightleftharpoons \mathrm{Red}$$

At standard-state conditions (unit activity of all species, 298.15 K, 1 bar gas pressure), the reaction has a well-defined Gibbs energy change $\Delta G^\circ$. We define the standard electrode potential $E^\circ$ by [fn::A sign convention exists. The 1953 IUPAC Stockholm agreement set reduction potentials (written as reductions). Oxidation potentials in older texts have the same number with flipped sign. This causes the most common sign errors in textbook problems.]:

$$\Delta G^\circ = -n F E^\circ$$

The variable $n$ is the electron stoichiometry. $F = e N_A \approx 96\,485\ \mathrm{C\,mol^{-1}}$ is Faraday's constant. A spontaneous reduction (positive $E^\circ$) gives negative $\Delta G^\circ$ by convention.[fn::A cell with $E^\circ_{\text{cell}} > 0$ has $\Delta G^\circ_{\text{cell}} < 0$. It is spontaneous as written. This is the only sign rule worth memorizing.]

The standard hydrogen electrode

$E^\circ$ is a difference, not an absolute value. No voltmeter measures "the electron" directly. We need a zero point. The standard hydrogen electrode (SHE) provides it:

$$2\,\mathrm{H^+}(a = 1) + 2 e^- \rightleftharpoons \mathrm{H_2}(p = 1\ \mathrm{bar}) \qquad E^\circ = 0.000\ \mathrm{V}$$

This is defined at all temperatures.[fn::Most standard potentials shift with $T$. SHE's zero is definitional, not measured. Its practical realization drifts with $T$. This drift must fold into tabulated values of other potentials.]

The SHE is a platinized platinum electrode[fn::Platinized means coated with Pt black. This maximizes surface area. It catalyzes H⁺/H₂ exchange, which is sluggish on bare Pt.] in $a_{\mathrm{H^+}} = 1$ (≈ 1.18 M HCl, not 1.0 M, because activity differs from concentration). H₂ gas at 1 bar bubbles past. Reproducibility is about ±1 mV in the best labs. It is worse elsewhere. Nobody uses SHE day-to-day. Ag/AgCl, Hg/Hg₂Cl₂, Hg/HgO are working substitutes. Each references back to SHE by Nernst calculation.

The electrochemical series

Tabulating $E^\circ$ for half-reactions produces the electrochemical series. It ranks species by $E^\circ$. Species high in the series (F₂, MnO₄⁻ in acid, Au³⁺) are strong oxidizers. Species low in the series (Li, Na, Al) are strong reducers. The series predicts reactions:

A species oxidizes anything below it. Spontaneous cell: $E^\circ_{\text{cell}} = E^\circ_{\text{cathode}} - E^\circ_{\text{anode}} > 0$.
Larger separation means larger driving force and larger $K$.
"Displacement" reactions (Zn displaces Cu²⁺ but not vice versa) follow this rule.

The series is thermodynamic. It shows nothing about rate. Al is well below Fe in the series. Yet aluminum resists corrosion in air. The Al₂O₃ passivation layer is a kinetic barrier. The Nernst equation does not see it.[fn::$E$ tells you whether; kinetics tells you how fast. Pourbaix diagrams and polarization curves live in different universes.]

The Nernst equation, derived

We now have all the ingredients. The derivation is short. Every step carries an assumption worth naming.

Step 1: electrical work is Gibbs energy

A galvanic cell operating reversibly[fn::Reversibly means infinitesimally slowly. No overpotential. No concentration gradient at the electrodes. Measurable current drops the cell potential below the reversible value (see Tafel and ohmic regimes in the OCP note). Nernst describes the rest potential, $E_{\text{eq}}$. It never describes a working cell's terminal voltage.] extracts maximum non-expansion work per mole of reaction. This work equals the Gibbs energy change:

$$\Delta G = -n F E$$

This equation links the chemist's $\Delta G$ (a thermodynamic state function, J mol⁻¹) to the electrochemist's $E$ (a voltage, V). It defines $E$ in terms of $\Delta G$. It is not an empirical law.

Step 2: $\Delta G$ depends on composition

From chemical thermodynamics, the Gibbs energy of reaction at arbitrary composition is:

$$\Delta G = \Delta G^\circ + RT \ln Q$$

$Q$ is the reaction quotient. It is the ratio of product activities to reactant activities. Each activity is raised to its stoichiometric power at the current composition. For $\mathrm{Ox} + ne^- \rightleftharpoons \mathrm{Red}$, $Q = a_{\mathrm{Red}} / a_{\mathrm{Ox}}$. Electrons do not appear in $Q$.[fn::Electron activity $a_{e^-}$ appears in some formulations (notably Pourbaix diagrams). There the electron is a pseudo-species. pe = -log a_{e^-} is the redox analogue of pH. This is useful but not measurable.]

Step 3: substitute and divide

Insert step 1 into step 2:

$$-n F E = -n F E^\circ + RT \ln Q$$

(Using $\Delta G^\circ = -n F E^\circ$.) Divide by $-nF$:

$$\boxed{E = E^\circ - \frac{RT}{nF} \ln Q}$$

This is the Nernst equation. At 298.15 K, with $RT/F \ln 10 = 0.05916\ \mathrm{V}$:

$$E = E^\circ - \frac{0.05916}{n} \log_{10} Q$$

This version is used in the lab.[fn::The constant 0.05916 is the most quoted and most misremembered in electrochemistry. Values like 0.0592, 0.059, 0.06 all appear. Use the full form $2.303\,RT/F$ outside 25 °C.]

The equation says: the potential drifts from its standard value by $(RT/nF)\ln Q$ as concentration drifts from unit activity. The logarithm makes reference electrodes workable. A decade change in concentration shifts $E$ by only $0.05916/n$ volts. An Ag/AgCl electrode in saturated KCl stays stable to a few mV.

Combining half-cells: the full cell

A complete galvanic cell is two half-cells with a salt bridge and external wire. Cell notation places the anode (oxidation) left, cathode (reduction) right. Phases separate with vertical bars:

$$\mathrm{Zn(s)} \mid \mathrm{Zn^{2+}}(a_1) \,\|\, \mathrm{Cu^{2+}}(a_2) \mid \mathrm{Cu(s)}$$

The double bar $\|$ is a salt bridge / liquid junction. The cell potential is:

$$E_{\text{cell}} = E_{\text{cathode}} - E_{\text{anode}}$$

Each term uses Nernst evaluation at its own composition. For the Daniell cell, with $E^\circ_{\mathrm{Cu^{2+}/Cu}} = +0.337$ V and $E^\circ_{\mathrm{Zn^{2+}/Zn}} = -0.763$ V:

$$E_{\text{cell}} = 1.100 + \frac{0.05916}{2}\log_{10}\!\left(\frac{a_{\mathrm{Cu^{2+}}}}{a_{\mathrm{Zn^{2+}}}}\right) \;\mathrm{V}$$

Both terms collapse into a cell-level Nernst equation with $Q_{\text{cell}} = a_{\mathrm{Cu^{2+}}}/a_{\mathrm{Zn^{2+}}}$. This $Q$ is the reaction quotient of the overall reaction $\mathrm{Zn + Cu^{2+} \to Zn^{2+} + Cu}$. The two-half-cell view and the overall-reaction view are the same equation at different scales.[fn::Students learn to "subtract $E^\circ$ values" before they understand they are subtracting $\Delta G^\circ/nF$. The subtraction rule works only for the same $n$ electrons. Different electron counts (e.g., MnO₄⁻/Mn²⁺ with $n=5$ and Fe²⁺/Fe with $n=1$) need a common $n$ before subtraction. Subtracting unscaled $E^\circ$ gives a wrong answer.]

Activity and the Debye-Hückel limiting law

The Nernst equation requires activities, not concentrations. For ion $i$:

$$a_i = \gamma_i \, c_i / c^\circ$$

$\gamma_i$ is the activity coefficient (dimensionless, → 1 as $c \to 0$). $c^\circ = 1\ \mathrm{M}$. In dilute solution, $\gamma_i \to 1$ and $a_i \approx c_i$. Every introductory textbook makes this. Every real solution violates it.

Where non-ideality comes from

Ions are not indifferent to each other. Each ion is surrounded by oppositely charged neighbors on average. This is the Debye-Hückel ionic atmosphere. It lowers effective free energy. Activity falls below concentration. The effect grows with ionic strength:

$$I = \tfrac{1}{2} \sum_i c_i z_i^2$$

Higher valency increases the effect (note the $z^2$).

The limiting law

Debye and Hückel (1923) showed for $I \to 0$:

$$\log_{10} \gamma_i = -A\, z_i^2 \sqrt{I}$$

$A \approx 0.509\ \mathrm{M^{-1/2}}$ for water at 25 °C. It is weakly temperature-dependent. For a mean ionic activity coefficient:

$$\log_{10} \gamma_\pm = -A\, |z_+ z_-| \sqrt{I}$$

This is the limiting law. It is exact as $I \to 0$. It worsens as $I$ grows.

When the limiting law breaks

The limiting law works up to ~$0.01\ \mathrm{M}$ for 1:1 electrolytes. It is less accurate for higher-valent electrolytes. Patches extend the range:

Extended Debye-Hückel: $\log_{10} \gamma_i = -A z_i^2 \sqrt{I} \,/\, (1 + B a_i \sqrt{I})$. $B \approx 0.328\ \mathrm{\AA^{-1}M^{-1/2}}$. $a_i$ is an ion size parameter (~3-9 Å). Accurate to ~0.1 M. Uses a fitted parameter.
Davies equation: $\log_{10} \gamma_i = -A z_i^2 \left[ \frac{\sqrt{I}}{1 + \sqrt{I}} - 0.3\, I \right]$. No fitted parameters. Good to ~0.5 M. It is the workhorse in geochemical and corrosion modeling.
Specific ion interaction theory (SIT) and Pitzer equations: extend to seawater (~0.7 M) and brines. They use fitted binary and ternary interaction parameters.

The trend repeats: further from infinite dilution, more empirical the correction.

Cell potential and the equilibrium constant

At equilibrium, $\Delta G = 0$ and $E = 0$. $Q$ becomes the equilibrium constant $K$:

$$0 = E^\circ - \frac{RT}{nF} \ln K \quad\Longrightarrow\quad \boxed{E^\circ = \frac{RT}{nF} \ln K}$$

At 25 °C: $\log_{10} K = n E^\circ / 0.05916$.

This links tabulated $E^\circ$ to general-chemistry equilibrium constants. A cell with $E^\circ_{\text{cell}} = 1\ \mathrm{V}$ and $n = 2$ has $K \approx 10^{34}$. It is essentially quantitative. Even modest $E^\circ$ differences give enormous equilibrium constants. Redox reactions in the lab so often go to completion.[fn::A Daniell cell ($E^\circ \approx 1.10$ V, $n = 2$, $K \approx 10^{37}$) is effectively irreversible. The back-reaction needs absurd product:reactant ratios.]

Liquid junction potentials

At the interface between two electrolytes of different composition, a small potential develops spontaneously (typically 1-30 mV). Mechanism: ions diffuse at different rates. H⁺ is far more mobile than Cl⁻. Charge separation builds until it self-limits. This potential contaminates every measurement crossing a liquid junction. Salt bridges, frits, and reference electrode plugs are all liquid junctions.

Mitigations:

Salt bridges with equitransferent electrolytes (KCl, NH₄NO₃) where cation and anion mobilities match.
Saturated KCl in reference electrode fill. The high [K⁺] ≈ [Cl⁻] swamps the analyte contribution.
Theoretical correction via Henderson or Planck equations when accuracy demands.

Liquid junction potentials explain why "the" potential of a reference is a range, not a number.

Worked example: Zn | ZnSO₄ (0.01 M), 25 °C

Half-reaction:

$$\mathrm{Zn^{2+}} + 2 e^- \rightleftharpoons \mathrm{Zn(s)} \qquad E^\circ = -0.763\ \mathrm{V}$$

Here $n = 2$ and $Q = 1 / a_{\mathrm{Zn^{2+}}}$. Solids have unit activity. Nernst:

$$E = E^\circ + \frac{0.05916}{2} \log_{10} a_{\mathrm{Zn^{2+}}}$$

Naive: take $a = c$

$$E = -0.763 + 0.02958 \cdot \log_{10}(0.01) = -0.763 - 0.0592 = -0.822\ \mathrm{V}$$

This is the textbook answer. It is wrong by ~12 mV.

Corrected: Debye-Hückel activity

$\mathrm{ZnSO_4}$ is a 2:2 electrolyte, $c = 0.01\ \mathrm{M}$:

$$I = \tfrac{1}{2}(0.01 \cdot 4 + 0.01 \cdot 4) = 0.04\ \mathrm{M}$$

This is outside the strict limiting-law range. We use it past its warranty for 2:2 salts (a common practice for 2:2 salts where no clean alternative exists). Mean activity coefficient:

$$\log_{10} \gamma_\pm = -0.509 \cdot |2 \cdot 2| \cdot \sqrt{0.04} = -0.407 \quad\Rightarrow\quad \gamma_\pm \approx 0.392$$

So $a_{\mathrm{Zn^{2+}}} = 0.392 \cdot 0.01 = 3.92 \times 10^{-3}$:

$$E = -0.763 + 0.02958 \cdot \log_{10}(3.92\times 10^{-3}) = -0.834\ \mathrm{V}$$

Activity correction moves $E$ by 12 mV. It is comparable to the entire concentration effect from 1 M to 0.01 M. For multivalent ions at non-trivial ionic strength, the activity correction is dominant. Pourbaix diagrams with concentrations can be tens of mV off.

When it breaks

Nernst assumes equilibrium. Nernst describes $E_{\text{eq}}$, the equilibrium potential. OCP of a corroding electrode is not this. It is a mixed potential from anodic and cathodic curves. Nernet is ≈ OCP only when one half-reaction dominates.
Debye-Hückel breaks above ~0.01 M. Extended Debye-Hückel, Davies, SIT, and Pitzer are patches. None reliably handles concentrated brines or ionic liquids.
Standard potentials use activity, not concentration. Many textbooks use 1.0 M concentration. Its activity differs from 1.0. This error is systematic and common.
Side reactions and complexation shift the effective potential. Zn²⁺ complexes (Zn(NH₃)₄²⁺, ZnCl₄²⁻) lower free Zn²⁺ activity. Nernst with correct activity holds. Nernst with concentration fails silently.
Temperature enters through $RT/F$ and $E^\circ(T)$. The 0.05916 V is a 25 °C value. At 60 °C it is ~0.066 V. Field instruments hardcoding 0.0592 are measurably wrong.