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# Nernst Equation Example for Electrochemical Systems Design

### Key Takeaways

• An important result of thermodynamics for electrochemical cells is the Nernst equation.

• The Nernst equation states how the output potential from an electrochemical cell varies with the cell temperature and reaction quotient.

• Electrochemical systems designers that need to monitor the voltage from an electrochemical cell can use the Nernst equation to relate potential, temperature, and concentration measurements. Use the Nernst equation to understand the measurements of electrochemical cells.

The Nernst equation is a fundamental chemical thermodynamics equation used in electrochemical cell design. As part of reaction monitoring, measurements in an electrochemical system can be used with the Nernst equation to determine electrochemical reaction progress in various conditions. For electrochemical systems designers, this important equation forms the basis for monitoring the electrical and environmental behavior of an electrochemical cell.

Although it has a simple form, some guides on the Nernst equation provide apparently conflicting details. Therefore, we’ve prepared a Nernst equation example to help electrochemical systems designers understand what to measure and what they need to include in their PCB layout. Here’s how the Nernst equation applies to electrochemical reaction monitoring and what designers need to consider in their designs.

## Nernst Equation Theory

The Nernst equation is a thermodynamic result for electrochemical reaction equations and is derived from the Gibbs free energy for the reaction. The Nernst equation relates the electrochemical potential of a reversible (redox) reaction, as measured between the anode and cathode, to the cell’s reaction quotient and temperature. This equation can be used in the following areas:

• Reaction equilibrium monitoring. By measuring the cell’s output potential and the temperature in the cell, one can spot when the reaction comes to equilibrium.

• Reaction environment monitoring. If the reaction temperature changes, the equilibrium balance in the cell will change. The Nernst equation states the correlation between the temperature and the equilibrium constant in the cell.

• Voltammetry. The Nernst equation forms the baseline for predicting when additional current can be coaxed from a cell with an additional potential in voltammetry measurements.

These basic monitoring tasks are vital parts of industrial automation and process control, battery design, and electrochemistry. A potentiostat is normally used to provide these measurements, but specialty potentiostat systems may integrate other important features that are needed for cell monitoring.

The Nernst equation is normally formulated for the following chemical reaction: The general reversible reaction that can be considered in the Nernst equation.

As the progress of chemical reactions depends on temperature, and the concentrations of reactants/products will change over time, the reaction can be comprehensively monitored if one has an equation describing the relationship between these quantities. This is where the Nernst equation comes into play.

### Definition

Note that the definitions shown below are defined for the above reaction equation, but they can be easily extended to reactions with more than two reactants and or products through the definition of the reaction quotient and equilibrium constant.

Starting from the definition of Gibbs free energy at standard conditions, we can determine the new Gibbs free energy at non-standard conditions using the reaction quotient. The definition and derivation for the Gibbs free energy in non-standard conditions can be found in many thermodynamics and chemistry textbooks, so it won’t be repeated here. By invoking the Faraday constant in the Gibbs free energy, which relates the electrochemical potential to the number of electrons released in a redox reaction, we arrive at the Nernst equation: Nernst equation definition.

The applicability of this equation is quite broad as long as the reaction quotient is known. For the reaction shown above, the reaction quotient is simply: Reaction quotient.

Here, by measuring the cell potential as a function of time, the reaction quotient at constant temperature can be determined as a function of time. Similarly, if some reactants or products are added to/removed from the cell, this change in the cell contents can be detected by monitoring the cell potential. Eventually, the reaction will reach its final equilibrium and the cell potential will be zero. This case also needs to be considered in the Nernst equation.

### Equilibrium

Note that for a reaction proceeding towards equilibrium, the reaction quotient will change as the reaction progresses. Eventually, once the reaction reaches equilibrium, the reaction quotient will be equal to the equilibrium constant. When this occurs, the cell potential E will be zero. Plugging this into the above equation gives a relation between the cell temperature, equilibrium constant, and standard cell potential: Cell standard potential vs. temperature and equilibrium constant.

Here, we can see how the standard cell potential is a function of temperature in two ways: by direct proportionality and due to the equilibrium constant. Note that equilibrium constants are also functions of temperature, being governed by the molecular kinetics of the reactants and products. By monitoring the cell potential at different temperatures, the equilibrium constant for the reaction as a function of temperature can be determined.

## Nernst Equation Example

As a simple example, consider a redox reaction involving charge transfer between Zn and Cu metal. Simple charge transfer reaction between Zn and Cu.

This reaction has a standard cell potential of 1.10 V. Suppose that, after 2 minutes, there is 0.1 mol of reactants and 1.8 mol of products. By using the value of the Faraday constant F, gas constant R, n = 2, and taking a temperature of 298 K, one finds that the cell potential is 1.10 V - 0.1711 V = 0.9289 V. This is the voltage one would measure between the cell cathode and anode.

Eventually, the cell will reach equilibrium and the cell voltage will saturate to a constant value. When building an electrochemical monitoring system, a simple voltage measurement could involve a Wheatstone bridge with high precision resistors. A small RTD probe or thermocouple can also be used for temperature measurement. Connecting these to an MCU with an amplifier on each sensor gives a simple electrochemical monitoring system.

If you’re designing your first electrochemical monitoring system, you can use our Nernst equation example to help you get started with your control unit and algorithm design. To get started with your schematic and PCB layout, you need the industry’s best PCB design and analysis software. Allegro PCB Editor gives you all the features you need to create advanced circuit boards for a range of applications, including electrochemical systems. You’ll also have access to a complete set of advanced design verification tools and field solver utilities to analyze the behavior of advanced electronics.