Understanding the kinetics of electrochemical reactions is essential to develop applications for the real world. Besides the Butler-Volmer equation, the Tafel equation is the most used mathematical expression to study electrochemical kinetics for a wide range of applications, from corrosion to batteries. In this post we explain in detail what is the Tafel equation, how to analyze data with it, how to represent it with the Tafel plot and its applications.

## What is the Tafel Equation?

The Tafel Equation is a mathematical expression that describes the relationship between the electrochemical reaction rate and the overpotential of an electrode.

### The Tafel Equation

The Tafel Equation is expressed as follows:

where

η is the overpotential in V,

A is the Tafel slope in V,

i is the current density in A/m^{2} ,

and i_{0} is the exchange current density in A/m^{2} .

The +/- sign depicted in the equation relates to whether the analysis refers to an anodic (+) or cathodic (-) process.

### Assumptions of the Tafel Equation

The Tafel equation is a simplified equation based on empirical data to describe the relationship between reaction rates and overpotentials at an electrode surface. As such, this equation was developed with certain assumptions listed below:

- The reaction at the electrode is a reaction consisting of a single electron transfer.
- The reaction at the electrode is irreversible, thus, the reverse reaction is either inexistent or negligible.
- The reaction is controlled only by an anodic or a cathodic process, but not both.
- The overpotential appied is small, so that the reaction rate follows a linear trend with the overpotential.
- The exchange current density is constant over the potential range of interest.

### Who is the author of the Tafel Equation?

The Tafel Equation is attributed to Swiss chemist and electrochemist Julius Tafel. He first proposed it in 1905 to quantify the relationship between the rate at which an electrochemical reaction ocurred and the overpotential experienced by an electrode.

## The relationship between the Tafel Equation and Butler-Volmer’s Equation

Both the Tafel Equation and the Butler-Volmer equation are used to describe the kinetics of electrochemical reaction. The main difference is that the Tafel Equation is a simplified equation applicable to situations where a small potential range is used and only one of the reactions is significant, whereas the Butler-Volmer equation is more general and applicable to a wider range of electrochemical reactions.

## How to perform a Tafel Analysis

For the study of electrochemical reaction kinetics with the Tafel Equation, it is common to use a 3 electrode cell where the working electrode is exposed to an electrolyte. For example an acidic or a saline solution.

With this setup, a potential sweep around the open circuit potential of the cell is performed. This measurement yields the polarisation curve V vs i, that describes the relationship between the current density and the overpotential of the working electrode.

Once the data has been collected a Tafel plot is generated and the resulting graph is analyzed with the Tafel Equation.

### The Tafel plot

The Tafel plot is a specific type of polarisation curve graph in which the potential is plotted against the log of the current. This type of graph is used to visually represent the Tafel equation and it most commonly seen in corrosion studies.

#### How to interpret the Tafel plot

##### The Tafel slope

The Tafel plot typically shows a linear rleationship between the logarithm of the current and the overpotential. The slope of the represented curve represents the Tafel slope, which is related to the activation energy of the reaction. Therefore, the slope indicates the reaction rate. The steeper the slope, the higher the activation energy required for the reaction to occur and slower the reaction rate is.

##### The corrosion potential

The other important parameter to interpret a Tafel plot is the corrosion potential. This parameter defines the potential where the oxidation and reduction reactions are in balance, that is, where there is no net oxidation or reduction.

Above the corrosion potential the electrode material will start to oxidize/corrode. On the other hand, if the potential applied to the electrode is below the corrosion potential, the material will be reduced and thus protected (also known as cathodic protection). For this reason, it is a critical parameter for determining the corrosion behavior of materials.

As a rule of thumb, the higher the corrosion potential, the more resistant the material is to corrosion in the electrolyte tested. However, its absolute value is not as important when testing for corrosion protection. Instead, to ensure a good corrosion protection it is important to have a large difference between the corrosion potential and the breakdown potential (i.e. the potential at which the material starts to corrode rapidly).

## Applications of the Tafel Equation

### Corrosion studies

The Tafel equation is perhaps most known for its application in the study of corrosion. With it, it is possible to evaluate the effectiveness of corrosion inhibitors, which are significant in a wide variety of industries like consumer electronics and construction.

### Studying electrode reaction kinetics

The Tafel equation allows electrochemists to quantify the kinetics of electrochemical reactions. By looking at the slope of the Tafel equation it is possible to determine what type of reaction is ocurring:

- Activation-controlled reaction: in this case the limiting step is the transfer of electrons to the electrode surface. This type of reaction has a smalll Tafel slope between 30 and 120 mV/decade. An example of an activation controlled would be the oxidation of iron in acidic media.
- Diffusion-controlled reaction: in this case the limiting step is the transport of chemical species from or to the electrode surface. This type of reaction has large tafel slopes, typically between 120 and 240 mV/decade. An example of a diffusion controlled reaction is the reduction of oxygen to water.
- Mixed-controlled reaction: in this case we have a combination of activation and diffusion processes. This type of reaction has an intermediate Tafel slope between 60 and 120 mV/decade. An example would be the oxidation of a metal in an electrolyte with a low concentration of oxygen dissolved in it. In this case, the metal oxidation is an activation controlled reaction that is also limited by the amount of oxygen available in the electrolyte.

### Fuel cells

The Tafel equation is used in the optimization of fuel cells. During optimization studies, voltammetric data is acquired under different operation conditions, such as temperature, pressure or gas flow rates. By analyzing the resulting data with the Tafel equation it is possible to estimate if the reaction is limited by the charge transfer at the electrode or by the diffusion of reactants to the electrode surface. Thus gaining insights on the factors that affect the performance of the fuel cell.

### Batteries

In batteries, the electrochemical reactions involve the movement of charged ions between the anode and the cathode. As a result of this movement of ions in the electrolyte, electrons can flow through an external circuit. The Tafel equation can be used to study how the different materials used in the battery assembly, that is, the cathode, anode, electrolyte and separator, affect the kinetics of the battery. Thus, electrochemists can gain a deep understanding of how these components affect the overall performance of the battery and optimize its design.

We hope you enjoyed learning about the Tafel equation and how it can be used to gain a deep understanding of electrochemical systems used in a wide range of real-world applications. From protective coatings to batteries, knowing about this equation is a must for every electrochemist. If you would like to know more about electrochemistry, we publish insightful material regularly in our blog. Make sure to check it out.