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Battery Technology

Field: Electrochemistry

The technology of batteries, devices that store chemical energy and convert it to electrical energy.

Sequence of Expressions

Let EE be the cell potential, E0E^0 be the standard cell potential, TT be the absolute temperature, nn be the number of electrons transferred, FF be the Faraday constant, and α\alpha be the reaction quotient (product/reactant activities). The potential is given by:\nE=E0−RTnFln⁡αE = E^{0} - \frac{RT}{nF} \ln{\alpha}
Let ii be the net current density, i0i_0 be the exchange current density, η\eta be the overpotential, nn be the number of electrons, FF be the Faraday constant, RR be the gas constant, and TT be the absolute temperature. The current density is described by:\ni=i0nFRT(exp⁡(nFηRT)−exp⁡(−nFηRT))i = i_{0} \frac{nF}{RT} \left( \exp\left(\frac{nF\eta}{RT}\right) - \exp\left(-\frac{nF\eta}{RT}\right) \right)
Define the standard reduction potentials EoE^\text{o} for two half-reactions: Ox+ne−⇌Red\text{Ox} + n e^- \rightleftharpoons \text{Red} and Ox′+n′e−⇌Red′\text{Ox}' + n' e^- \rightleftharpoons \text{Red}'. The cell potential ΔE\Delta E for the reaction Ox+Ox′⇌Red+Red′\text{Ox} + \text{Ox}' \rightleftharpoons \text{Red} + \text{Red}' is given by the difference in these potentials:\nΔE=Eo(Ox′/Red′)−Eo(Ox/Red)orΔE=Ecathodeo−Eanodeowhere Ecathodeo>Eanodeo for a spontaneous reaction. \Delta E = E^\text{o}(\text{Ox}'/\text{Red}') - E^\text{o}(\text{Ox}/\text{Red}) \quad \text{or} \quad \Delta E = E^\text{o}_{\text{cathode}} - E^\text{o}_{\text{anode}} \quad \text{where } E^\text{o}_{\text{cathode}} > E^\text{o}_{\text{anode}} \text{ for a spontaneous reaction.}
Define the characteristic distance λ\lambda over which the electrostatic potential varies significantly in an electrolyte solution. Given the permittivity of the solvent ϵ\epsilon, the relative permittivity α\alpha, and the permittivity of free space ϵF\epsilon_F, the Debye length is mathematically defined as:\nλ=ϵα2ϵF2with units of length (m). \lambda = \sqrt{\frac{\epsilon\alpha}{2\epsilon_F^2}} \quad \text{with units of length (m).}
Let J\mathbf{J} be the ionic flux vector across a semi-permeable membrane separating two electrolyte solutions with potentials ϕ1\phi_1 and ϕ2\phi_2. The potential-driven flux J\mathbf{J} is generally modeled by the Nernst-Planck equation, which accounts for concentration gradients and potential gradients:\nJi=−Di(∇Ci+ziFRTCi∇ϕF)where Di is the diffusion coefficient, Ci is the concentration, zi is the valence, and F is the Faraday constant. \mathbf{J}_i = -D_i \left( \nabla C_i + \frac{z_i F}{RT} C_i \nabla \frac{\phi}{F} \right) \quad \text{where } D_i \text{ is the diffusion coefficient, } C_i \text{ is the concentration, } z_i \text{ is the valence, and } F \text{ is the Faraday constant.}
For an electrochemical reaction involving the transfer of nn electrons, let QQ be the total charge passed (in Coulombs), mm be the moles of substance produced, and MM be the molar mass. The relationship is defined by:\nm=QnFm = \frac{Q}{nF}
Define the change in Gibbs Free Energy (ΔG\Delta G) for a reaction under non-standard conditions, relative to the standard change (ΔG0\Delta G^0), the temperature (TT), and the reaction quotient (α\alpha):\nΔG=ΔG0+RTln⁡α\Delta G = \Delta G^{0} + RT \ln{\alpha}
Define the ionic conductivity (κ\kappa) of an electrolyte solution as the reciprocal of the solution resistance (RionR_{ion}). Alternatively, κ\kappa is related to the concentration (CiC_i) and the ion mobility (μi\mu_i) of the constituent ions:\nκ=∑iziFCiμi\kappa = \sum_{i} z_i F C_i \mu_i
Let EeqE_{eq} be the equilibrium potential and EactualE_{actual} be the measured potential under non-equilibrium conditions. The overpotential η\eta is defined as the deviation from equilibrium:\nη=Eactual−Eeqor, in terms of current density j and Tafel slope b (for high overpotentials):η=±blog⁡(∣j∣j0) \eta = E_{actual} - E_{eq} \quad \text{or, in terms of current density } j \text{ and Tafel slope } b \text{ (for high overpotentials):} \quad \eta = \pm b \log \left( \frac{|j|}{j_0} \right) \nwhere j0j_0 is the exchange current density.
Consider the rate constant kk for an electrochemical reaction. The dependence of kk on temperature TT and the activation energy EaE_a is governed by the Arrhenius equation:\nk(T)=Aexp⁡(−EakBT)where A is the pre-exponential factor, and kB is the Boltzmann constant. k(T) = A \exp\left(-\frac{E_a}{k_B T}\right) \quad \text{where } A \text{ is the pre-exponential factor, and } k_B \text{ is the Boltzmann constant.} \nAlternatively, the potential dependence can be modeled using the Tafel equation, relating the overpotential η\eta to the current density jj:\nη=βlog⁡(jj0)where β is the Tafel slope. \eta = \beta \log \left( \frac{j}{j_0} \right) \quad \text{where } \beta \text{ is the Tafel slope.}