Electrochemistry Equations

Why This Matters

Electrochemistry sits at the intersection of thermodynamics, kinetics, and transport phenomena—three pillars you'll be tested on repeatedly in Physical Chemistry II. These equations aren't just formulas to memorize; they're tools that 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.

Here's 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? Don't just memorize these equations—know which concept each one 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

• $\Delta G = -nFE$ connects thermodynamics to electrochemistry—negative $\Delta G$ means spontaneous reaction, which corresponds to positive cell potential
• Faraday's constant ($F = 96,485$ C/mol) converts between electrical and chemical energy; $n$ represents moles of electrons transferred
• Spontaneity criterion—this equation is your go-to for determining reaction feasibility and calculating maximum electrical work

Standard Reduction Potentials

• Standard conditions define the reference point—1 M concentration, 1 atm pressure, 25°C (298 K)—for all tabulated $E°$ values
• More positive $E°$ indicates stronger oxidizing agent; the species has greater electron affinity and is more easily reduced
• Cell potential calculation—combine half-reactions using $E°_{\text{cell}} = E°_{\text{cathode}} - E°_{\text{anode}}$ to predict spontaneous direction

Nernst Equation

• $E = E° - \frac{RT}{nF}\ln Q$ adjusts cell potential for non-standard concentrations—at 25°C, this simplifies to $E = E° - \frac{0.0592}{n}\log Q$
• Reaction quotient ($Q$) drives the deviation from standard potential; as $Q$ increases, $E$ decreases for a reduction reaction
• Equilibrium connection—when $E = 0$, the system is at equilibrium and $Q = K$, linking electrochemistry to equilibrium constants

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Electrode Kinetics

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

Butler-Volmer Equation

• $j = j_0\left[\exp\left(\frac{\alpha_a F\eta}{RT}\right) - \exp\left(\frac{-\alpha_c F\eta}{RT}\right)\right]$ is the master equation for electrode kinetics, relating current density to overpotential $\eta$
• Exchange current density ($j_0$) measures intrinsic reaction speed at equilibrium—higher $j_0$ means faster kinetics and smaller overpotential needed
• Transfer coefficients ($\alpha_a$, $\alpha_c$) describe symmetry of the activation barrier; typically $\alpha_a + \alpha_c \approx 1$ for single-electron transfers

Tafel Equation

• $\eta = a + b\log j$ simplifies Butler-Volmer at high overpotentials—when one exponential term dominates, the relationship becomes linear in log(current)
• Tafel slope ($b$) reveals mechanistic information; $b = \frac{2.303RT}{\alpha nF}$ connects directly to the transfer coefficient
• Activation energy estimation—plotting $\eta$ vs. $\log j$ (Tafel plot) extracts kinetic parameters 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: $J = -D\frac{dC}{dx}$—flux is proportional to concentration gradient; the negative sign indicates flow from high to low concentration
• Second law: $\frac{\partial C}{\partial t} = D\frac{\partial^2 C}{\partial x^2}$—describes how concentration profiles evolve over time due to diffusion
• Diffusion coefficient ($D$) is the key transport parameter; typical values are $10^{-5}$ to $10^{-6}$ cm²/s for small molecules in solution

Cottrell Equation

• $I(t) = \frac{nFAD^{1/2}C}{\pi^{1/2}t^{1/2}}$ predicts current decay after a potential step—current decreases as $t^{-1/2}$ as the diffusion layer grows
• Chronoamperometry analysis—plotting $I$ vs. $t^{-1/2}$ should give a straight line; deviations indicate kinetic limitations or convection
• Diffusion coefficient determination—rearranging Cottrell lets you extract $D$ from transient current measurements

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Convective-Diffusion Systems

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

Levich Equation

• $I_{\text{lim}} = 0.62nFAD^{2/3}\omega^{1/2}\nu^{-1/6}C$ gives the limiting current at a rotating disk electrode—current increases with rotation rate $\omega$
• Kinematic viscosity ($\nu$) accounts for solution properties; the $\omega^{1/2}$ dependence is diagnostic of diffusion-limited behavior
• Koutecký-Levich analysis—plotting $1/I$ vs. $1/\omega^{1/2}$ separates kinetic and mass-transport contributions

Randles-Sevcik Equation

• $I_p = (2.69 \times 10^5)n^{3/2}AD^{1/2}C\nu^{1/2}$ relates peak current in cyclic voltammetry to scan rate $\nu$—the $\nu^{1/2}$ dependence confirms diffusion control
• Reversibility diagnostic—for a reversible system, peak current scales linearly with $\nu^{1/2}$; deviations suggest kinetic complications
• Concentration determination—once $D$ is known, this equation lets you quantify 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—how much charge produces how much product.

Faraday's Laws of Electrolysis

• First law—mass deposited ($m$) is proportional to charge passed: $m = \frac{QM}{nF}$, where $Q$ is total charge and $M$ is molar mass
• Second law—same charge deposits different masses of different substances in proportion to their equivalent weights ($M/n$)
• Coulometry applications—integrating current over time gives total charge, enabling precise determination of amount of substance transformed

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Quick Reference Table

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Self-Check Questions

1. Which two equations both contain Faraday's constant ($F$) but describe fundamentally different aspects of electrochemistry—one thermodynamic, one kinetic?

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

3. Compare and contrast the Levich and Randles-Sevcik equations: What experimental technique does each describe, and what does the characteristic exponent ($\omega^{1/2}$ vs. $\nu^{1/2}$) indicate?

4. An FRQ gives you a cell at non-standard conditions and asks whether the reaction is spontaneous. Which equation do you use first, and how does it connect to $\Delta G$?

5. Under what conditions does the Butler-Volmer equation simplify to the Tafel equation, and why is this simplification useful for analyzing experimental data?