Electrochemistry Equations

Why This Matters

Electrochemistry sits at the intersection of thermodynamics, kinetics, and transport phenomena. These equations let you predict whether a reaction will happen spontaneously, how fast electrons will flow, and what limits the rate of an electrochemical process. Mastering these relationships means you can tackle everything from battery design to corrosion analysis to biosensor development.

What examiners are really testing: Can you connect cell potential to spontaneity? Do you understand when kinetics (Butler-Volmer, Tafel) versus mass transport (Fick, Cottrell, Levich) controls the current? Can you move between equilibrium descriptions and dynamic behavior? Know which concept each equation illustrates and when to apply it.

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Thermodynamic Foundations

These equations establish the energetic framework for electrochemistry. They tell you whether a reaction can happen and how much work it can perform, before you worry about how fast it goes.

Gibbs Free Energy and Cell Potential Relationship

$\Delta G = -nFE$

This is the bridge between thermodynamics and electrochemistry. A negative $\Delta G$ means a spontaneous reaction, which corresponds to a positive cell potential $E$. The quantity $n$ is the number of moles of electrons transferred in the balanced redox reaction, and Faraday's constant ($F = 96{,}485$ C/mol) converts between electrical and chemical energy.

The product $nFE$ represents the maximum non-expansion work the cell can deliver. This is your go-to equation for determining reaction feasibility and calculating the theoretical energy output of any electrochemical cell.

Standard Reduction Potentials

All tabulated $E°$ values are measured under standard conditions: 1 M effective concentration (unit activity), 1 bar pressure, and 25°C (298 K), referenced to the standard hydrogen electrode (SHE), which is assigned $E° = 0$ V by convention.

A more positive $E°$ indicates a stronger oxidizing agent: that species has a greater tendency to gain electrons and be reduced. To get the overall cell potential, combine half-reactions:

$E°_{\text{cell}} = E°_{\text{cathode}} - E°_{\text{anode}}$

A positive $E°_{\text{cell}}$ predicts a spontaneous reaction under standard conditions. Remember that when you reverse a half-reaction (to write it as an oxidation), you flip the sign of $E°$ for conceptual purposes, but the formula above already accounts for that.

Nernst Equation

$E = E° - \frac{RT}{nF}\ln Q$

This adjusts the cell potential for non-standard concentrations. At 25°C, it simplifies to:

$E = E° - \frac{0.02569}{n}\ln Q = E° - \frac{0.05916}{n}\log Q$

The reaction quotient $Q$ drives the deviation from standard potential. As $Q$ increases (more products relative to reactants), $E$ decreases for the overall cell reaction.

The equilibrium connection is important: when $E = 0$, the cell can do no work, the system is at equilibrium, and $Q = K$. Setting $E = 0$ in the Nernst equation gives $\ln K = \frac{nFE°}{RT}$, which directly links standard cell potential to the equilibrium constant.

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Electrode Kinetics

Once thermodynamics says a reaction can happen, kinetics determines how fast. These equations describe current density as a function of the driving force (overpotential) at the electrode surface.

Overpotential $\eta$ is defined as the deviation of the electrode potential from its equilibrium value: $\eta = E - E_{\text{eq}}$. It represents the extra voltage you need to apply (or that you lose) to drive the reaction at a given rate.

Butler-Volmer Equation

$j = j_0\left[\exp\left(\frac{\alpha_a F\eta}{RT}\right) - \exp\left(\frac{-\alpha_c F\eta}{RT}\right)\right]$

This is the master equation for electrode kinetics, relating current density $j$ to overpotential $\eta$. The first exponential describes the anodic (oxidation) contribution, and the second describes the cathodic (reduction) contribution.

• Exchange current density ($j_0$) measures the intrinsic reaction rate at equilibrium, where the forward and reverse currents are equal and cancel. A higher $j_0$ means faster kinetics and a smaller overpotential needed to drive net current. For example, $j_0$ for hydrogen evolution on platinum is around $10^{-3}$ A/cm², but on mercury it's roughly $10^{-12}$ A/cm².
• Transfer coefficients ($\alpha_a$, $\alpha_c$) describe the symmetry of the activation energy barrier. For a single-electron transfer elementary step, $\alpha_a + \alpha_c = 1$. A symmetric barrier gives $\alpha_a = \alpha_c = 0.5$.

At small overpotentials ($|\eta| \ll RT/F$, roughly $< 10$ mV), you can linearize the exponentials to get:

$j \approx j_0 \frac{F\eta}{RT}$

This linear regime defines the charge-transfer resistance $R_{\text{ct}} = RT/(Fj_0)$, which is measurable by impedance spectroscopy.

Tafel Equation

$\eta = a + b\log|j|$

At high overpotentials (typically $|\eta| > 50{-}100$ mV), one exponential term in Butler-Volmer dominates and the other becomes negligible. The relationship between overpotential and log of current density becomes linear. This is the Tafel equation.

The Tafel slope $b$ reveals mechanistic information:

$b = \frac{2.303RT}{\alpha nF}$

where $\alpha$ is the transfer coefficient for the dominant direction and $n$ is the number of electrons in the rate-determining step. At 25°C with $\alpha = 0.5$ and $n = 1$, the Tafel slope is about 118 mV/decade. A slope of ~60 mV/decade would suggest $\alpha n \approx 1$, pointing to a different mechanism.

Plotting $\eta$ vs. $\log|j|$ (a Tafel plot) lets you extract both $j_0$ (from the intercept) and $\alpha$ (from the slope), which are essential for comparing electrocatalysts.

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Mass Transport

When electrode kinetics are fast, diffusion of reactants to the surface becomes rate-limiting. These equations describe how concentration gradients drive mass transport and limit current.

Fick's Laws of Diffusion

First law (steady-state flux):

$J = -D\frac{\partial C}{\partial x}$

Flux $J$ is proportional to the concentration gradient. The negative sign indicates flow from high to low concentration. This applies when the concentration profile isn't changing with time.

Second law (time-dependent):

$\frac{\partial C}{\partial t} = D\frac{\partial^2 C}{\partial x^2}$

This describes how concentration profiles evolve over time due to diffusion. It's the equation you solve (with appropriate boundary conditions) to derive expressions like the Cottrell equation.

The diffusion coefficient $D$ is the key transport parameter. Typical values for small molecules and ions in aqueous solution are $10^{-5}$ to $10^{-6}$ cm²/s. It depends on temperature, solvent viscosity, and the size of the diffusing species.

Cottrell Equation

$I(t) = \frac{nFAD^{1/2}C^{*}}{\pi^{1/2}t^{1/2}}$

This predicts the current response after a potential step (in chronoamperometry), where $C^{*}$ is the bulk concentration and $A$ is the electrode area. Current decays as $t^{-1/2}$ because the diffusion layer grows thicker over time, making it progressively harder for fresh reactant to reach the surface.

• Plotting $I$ vs. $t^{-1/2}$ should give a straight line through the origin. Deviations at short times suggest kinetic limitations; deviations at long times suggest convection is disrupting the diffusion layer.
• You can extract $D$ from the slope if $n$, $A$, and $C^{*}$ are known, or determine $C^{*}$ if $D$ is known.

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Convective-Diffusion Systems

When you add controlled convection (stirring, electrode rotation), mass transport becomes more predictable and you can achieve true steady-state currents. These equations describe limiting currents under well-defined hydrodynamic conditions.

Levich Equation

$I_{\text{lim}} = 0.62\, nFAD^{2/3}\omega^{1/2}\nu^{-1/6}C^{*}$

This gives the diffusion-limited current at a rotating disk electrode (RDE). Here $\omega$ is the angular rotation rate (in rad/s) and $\nu$ is the kinematic viscosity of the solution (in cm²/s).

The $\omega^{1/2}$ dependence is diagnostic: if a plot of $I_{\text{lim}}$ vs. $\omega^{1/2}$ is linear, the process is diffusion-controlled. Faster rotation thins the diffusion layer, bringing fresh reactant to the surface more quickly.

Koutecký-Levich analysis separates kinetic and mass-transport contributions by plotting $1/I$ vs. $1/\omega^{1/2}$:

$\frac{1}{I} = \frac{1}{I_{\text{kin}}} + \frac{1}{I_{\text{lev}}}$

The y-intercept gives $1/I_{\text{kin}}$ (the kinetic current, free of mass-transport limitations), and the slope contains the diffusion parameters.

Randles-Ševčík Equation

$I_p = (2.69 \times 10^5)\, n^{3/2}AD^{1/2}C^{*}\nu^{1/2}$

This relates the peak current $I_p$ in cyclic voltammetry (CV) to the scan rate $\nu$ (in V/s), for a reversible system at 25°C. The numerical prefactor applies when $I_p$ is in amperes, $A$ in cm², $D$ in cm²/s, and $C^{*}$ in mol/cm³.

• Reversibility diagnostic: for a fully reversible (Nernstian) process, $I_p$ scales linearly with $\nu^{1/2}$. If the plot of $I_p$ vs. $\nu^{1/2}$ curves or the peak-to-peak separation $\Delta E_p$ exceeds $59/n$ mV, the system has kinetic complications (quasi-reversible or irreversible behavior).
• Once $D$ is known, you can use this equation to determine unknown concentrations from CV peak heights.

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Electrolysis Quantification

Faraday's laws bridge electrical measurements to chemical amounts. They're the foundation for all quantitative electrochemistry.

Faraday's Laws of Electrolysis

First law: The mass $m$ of substance deposited or dissolved is proportional to the total charge $Q$ passed:

$m = \frac{QM}{nF}$

where $M$ is the molar mass and $n$ is the number of electrons per formula unit. Total charge is obtained by integrating current over time: $Q = \int I\, dt$, or simply $Q = It$ for constant current.

Second law: The same quantity of charge deposits different masses of different substances in proportion to their equivalent weights ($M/n$). This follows directly from the first law but emphasizes that comparing different electrolytic processes requires accounting for $n$.

In coulometry, you measure $Q$ precisely and use Faraday's law to determine the amount of analyte, making it an absolute analytical method (no calibration curve needed).

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Quick Reference Table

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Self-Check Questions

1. $\Delta G = -nFE$ and the Butler-Volmer equation both contain $F$, but one is thermodynamic and the other kinetic. Explain what each tells you and why both need Faraday's constant.

2. You observe that current decays as $t^{-1/2}$ after stepping the potential. Which equation describes this behavior, and what does the time dependence tell you about the rate-limiting process?

3. Compare the Levich and Randles-Ševčík equations: What experimental technique does each describe, and what does the characteristic square-root dependence ($\omega^{1/2}$ vs. $\nu^{1/2}$) indicate about the current-limiting process?

4. A problem gives you a cell at non-standard conditions and asks whether the reaction is spontaneous. Outline the steps: which equation do you use to find $E$, and how do you then determine $\Delta G$?

5. Under what conditions does the Butler-Volmer equation simplify to the Tafel equation? Why is this simplification useful for extracting kinetic parameters from experimental data?