User:Cemes4/Voltammetry

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Summary Notes from peer review to consider:


 * Add more figures/ captions to current ones
 * Check citation formatting
 * Put types of voltammetry and definitions into a table (improve clarity)
 * Add equations
 * Include more references throughout paragraphs
 * Condense some of the sentences (brogan)
 * Link words to other wiki pages

Theory [edit]
Voltammetry is the study of current as a function of applied potential. Voltammetric methods involve electrochemical cells, and investigate the reactions occurring at electrode/electrolyte interfaces. The reactivity of analytes in these half-cells is used to determine their concentration. It is considered a dynamic electrochemical method as the applied potential is varied over time and the corresponding changes in current are measured. Most experiments control the potential (volts) of an electrode in contact with the analyte while measuring the resulting current (amperes).

Electrochemical Cells
Electrochemical cells are used in voltammetric experiments to drive the redox reaction of the analyte. Like other electrochemical cells, two half-cells are required, one to facilitate reduction and the other oxidation. The cell consists of an analyte solution, an ionic electrolyte, and two or three electrodes, with oxidation and reduction reactions occurring at the electrode/electrolyte interfaces. As a species is oxidized, the electrons produced pass through an external electric circuit and generate a current, acting as an electron source for reduction. The generated currents are Faradaic currents, which follow Faraday’s law. As Faraday’s law states that the” number of moles of a substance, m, produced or consumed during an electrode process is proportional to the electric charge passed through the electron” the faradaic currents allow analyte concentrations to be determined. Whether the analyte is reduced or oxidized depends on the analyte, but its reaction always occurs at the working/indicator electrode. Therefore, the working electrode potential varies as a function of the analyte concentration. A second auxiliary electrode completes the electric circuit. A third reference electrode provides a constant, baseline potential reading for the other two electrode potentials to be compared to.

Three Electrode System
- Copy over from original wikipedia page

Voltammograms
A voltammogram is a graph that measures the current of an electrochemical cell as a function of the potential applied. This graph is used to determine the concentration and the standard potential of the analyte. To determine the concentration, values such as the limiting or peak current are read from the graph and applied to various mathematical models. After determining the concentration, the applied standard potential can be identified using the Nernst equation.

There are three main shapes for voltammograms. The first shape is dependent on the diffusion layer. If the analyte is continuously stirred, the diffusion later will be a constant width and produce a voltammogram that reaches a constant current. The graph takes this shape as the current increases from the background residual to reach the limiting current (il). If the mixture is not stirred, the width of the diffusion layer eventually increases. This can be observed by the maximum peak current (ip), and is identified by the highest point on the graph. The third common shape for a voltammogram measures the sample for change in current rather than current applied. A maximum current is still observed, but represents the maximum change in current.

Mathematical Models
To determine analyte concentrations, mathematical models are required to link the applied potential and current measured over time. The Nernst equation relates electrochemical cell potential to the concentration ratio of the reduced and oxidized species in a logarithmic relationship. The Nernst equation is as follows:

$$E= E^0 - \frac{R T}{z F} \ln Q$$

Where:


 * $$E$$ Reduction potential
 * $$E^0$$: standard potential
 * $$R $$ universal gas constant
 * $$T$$: temperature in kelvin
 * $$z$$: ion charge (moles of electrons)
 * $$F$$: Faraday constant
 * $$Q$$: Reaction quotient

This equation describes how the changes in applied potential will alter the concentration ratio. However, the Nernst equation is limited, as it is modeled without a time component and voltammetric experiments vary applied potential as a function of time. Other mathematical models, primarily the Butler-Volmer equation, the Tafel equation, and Fick’s law address the time dependence.

The Butler–Volmer equation relates concentration, potential, and current as a function of time. It describes the non-linear relationship between the electrode and electrolyte voltage difference and the electrical current. It helps make predictions about the how the forward and backward redox reactions affect potential and influence the reactivity of the cell. This function includes a rate constant which accounts for the kinetics of the reaction. A compact version of the Butler-Volmer equation is as follows:

$${\displaystyle j=j_{0}\cdot \left\{\exp \left[{\frac {\alpha _{\rm {a}}zF\eta }{RT}}\right]-\exp \left[-{\frac {\alpha _{\rm {c}}zF\eta }{RT}}\right]\right\}}$$

Where:

{\displaystyle \eta =E-E_{\rm {eq}}}$$). At high overpotentials, the Butler–Volmer equation simplifies to the Tafel equation. The Tafel equation relates the electrochemical currents to the overpotential exponentially, and is used to calculate the reaction rate. The overpotential is calculated at each electrode separately, and related to the voltammogram data to determine reaction rates. The Tafel equation for a single electrode is:
 * $$j$$: electrode current density, A/m2 (defined as j = I/S)
 * $$j_0$$: exchange current density, A/m2
 * $$E$$: electrode potential, V
 * $${\displaystyle E_{\rm {eq}}}$$: equilibrium potential, V
 * $$T$$: absolute temperature, K
 * $$z$$: number of electrons involved in the electrode reaction
 * $$F$$: Faraday constant
 * $$R$$: universal gas constant
 * $$\alpha_c$$: so-called cathodic charge transfer coefficient, dimensionless
 * $$\alpha_a$$: so-called anodic charge transfer coefficient, dimensionless
 * $$\eta$$: activation overpotential (defined as $$

$$ {\displaystyle \eta =\pm A\cdot \log _{10}\left({\frac {i}{i_{0}}}\right)}$$

Where:

As the redox species are oxidized and reduced at the electrodes, material accumulates at the electrode/electrolyte interface. Material accumulation creates a concentration gradient between the interface and the bulk solution. Fick's laws of diffusion is used to relate the diffusion of oxidized and reduced species to the faradaic current used to describe redox processes. Fick's law is most commonly written in terms of moles, and is as follows:
 * the plus sign under the exponent refers to an anodic reaction, and a minus sign to a cathodic reaction
 * $$\eta$$: overpotential, V
 * A: "Tafel slope", V
 * $$i$$: current density, A/m2
 * $$i_0$$: "exchange current density", A/m2.

$${\displaystyle J=-D{\frac {d\varphi }{dx}}}$$

Where:


 * J: diffusion flux (in amount of substance per unit area per unit time)
 * D: diffusion coefficient or diffusivity. (in area per unit time)
 * φ: concentration (in amount of substance per unit volume)
 * x: position (in length)