The Anson equation is a fundamental equation in electrochemistry that is used to describe the relationship between charge and time at a constant potential and in depletion conditions. This equation has applications in a wide variety of fields, including electrodeposition, energy storage and sensors. In this blog post, we will explore what the Anson equation is, where it comes from, who gave its name to it and its applications in detail.

## What is the Anson Equation?

The Anson Equation is a mathematical equation used to describe the charge evolution over time in diffusion controlled coulometric measurements. It represents the integration of the Cottrell equation and is expressed as follows:

where:

Q is the charge in Coulombs,

n is the number of electrons involved in the oxidation or reduction reaction,

F is Faraday’s Constant, 96485 C/mol,

A is the area of the planar electrode in cm^{2} ,

C is the initial concentration of the analyte being reduced or oxidized, in mol/cm^{3},

D is the diffusion coefficient for the analyte in cm^{2}/s,

and t is the time in seconds.

## What is the Anson plot?

Often the Anson equation is simplified, for practical reasons, to the following expression:

where:

When using the simplified version on the Anson equation, it is very helpful to represent the collected coulometric data with an Anson plot.

The Anson plot consists in a graph that displays the charge vs sthe square root of time. As a result, the obtained plots are linearized and facilitate the analysis of the results.

## Who developed the Anson Equation?

The Anson Equation was developed by Caltech’s Emeritus Professor Fred C. Anson. His pioneering works in chronoamperometry and chronocoulometry led to the naming of Q vs t^{1/2} plots after him.

## Applications of the Anson Equation

### Biosensors

Since the equation analyzes the relationship between charge and time, it has applications in the development and analysis of enzymatic electrochemical biosensors like the glucose biosensor.

An interesting advantage of using charge units in the characterisation of enzymatic biosensors instead of the current is that the signal increases over time and the signal-to-noise ratio is diminished thanks to the integration of the current. Therefore, it is an essential tool for the development of coulometric biosensors.

### Electrodeposition

The Anson equation can be used to study electrodeposition processes. Thanks to the analysis of electrodepostion charge vs time transients, it is possible to determine how much material has been deposited, as well as determine the rate and how different factors may be affecting the process, thus enabling its optimization

### Electrochemical Energy Storage

Batteries, fuel cells and supercapacitors are all transformative devices that are and will continue to be essential in our lives as we transition away from fossil fuels. All these devices can be analyzed and improved by analyzing their characterisation data with the Anson equation, since the most important factors for their success as commercial devices is the total amount of charge they can store/deliver and at what speed that can be charged/discharged.

In conclusion, the Anson equation is a fundamental equation in electrochemistry that provides valuable insights in the charge transients in diffusion controlled redox coulometric experiments. While being simple and, essentially, an integration of the Cottrell equation, it has a wide range of applications including, but not limited to, biosensors, electrodeposition and electrochemical energy storage. Therefore, it is a must for all electrochemists to be familiar with it.