Brief Note on the Derivation of Nernst Equation

The Nernst equation is given by a German Chemist; Walther Hermann the Nernst equation is used to compare the reduced potential of a cell in non-standard conditions to standard conditions. It is a relationship between the cell potential of the electrochemical cell and temperature. The Nernst equation can be used to calculate an electrochemical cell’s potential even when the conditions are not standard first, it is necessary to determine the standard potentials or conditions mentioned above. . Using the expression from the Nernst equation derivation, the cell potential can be calculated at any concentration of reactant, temperature and pressure for an electrochemical cell. One of the most common uses of the Nernst equation is electrochemical cell potentials. It is majorly used in batteries which are powered by electrochemical cells

What Is Nernst Equation?

The Nernst equation gives us a relation between the cell potential of an electrochemical cell, the standard cell potential, temperature, and the reaction quotient. Under even non-standard situations or timing, the cell potentials of electrochemical cells can be examined through the help of the Nernst equation. The Nernst equation defines the relationship between cell potential to standard potential and the activities of the electrically active species. The Nernst equation is used to calculate the potential of the cell at any moment during a response at conditions other than the standard state. In corrosion studies, the equation is used to examine concentration cells and in the building of Pourbaix diagrams.

Nernst Equation Derivation

The Nernst equation derivation– the metal is determined to be in contact with an aqueous solution of its salt. Now, both the metal losing an electron to become an ion and an ion taking an electron to regain its atomic state remain in an equilibrium state. This can be expressed as:

Mn++ ne–= →nM

Against Ered reduction potential, n number of moles of an electron is gained by an ion in the reduction reaction. 

1. In the movement of an electron, the work done is equal to

Wred= nFEred  (Here, F refers to Faraday= 96487 coulombs)

2. Now, this work done is combined with the equation of Gibbs free energy change. Gibbs’s free energy change indicates spontaneity. This change can be combined with  Wred as it is equal to maximum useful work. 

Wred= nFEred = -∆G 

∆G= -nFEred 

3. At standard conditions (Temperature= 298K and 1ATM pressure), the free energy change is ∆G°. Using this, the above-given equation can be written as below.

∆G°= -nFE°red  (Here, E°red is reduction potential at standard conditions)

4. The concentration continuously changes, and with the reaction rate, the potential also decreases during a reaction. But, the concentration must be maintained the same to get maximum free energy change or maximum work. To achieve this, the reaction must be carried out under reversible equilibrium. Now, van’t Hoff isotherm for reversible equilibrium says-

∆G= ∆G°+ RT ln K

Here, K= Equilibrium constant= Product/ Reactant= [M]n/ [M]n+

R= Gas Constant (8.314 J/K mole)

T= Temperature (in Kelvin)

5. Now, free energy changes are substituted in Van’t Hoff equation

-nFEred= -nFE°red + RT ln [Mn]/ [Mn+]= -nFE°red + 2.303 RT log [Mn]/ [Mn+]

6.   After dividing -nF by both sides, we get

Ered= E°red – 2.303 RT/nF1 log [Mn]/ [Mn+]

This equation can also be written as

EMn+/M= E°Mn+/M- 2.303 RT/nF1 log [Mn]/ [Mn+]

Metal’s activity is taken as unity. Now, the equation can be written as 

EMn+/M= E°Mn+/M- 2.303 RT/nF log 1/ [Mn+]

This is the Nernst equation. This equation correlates the atom/ion’s capacity to gain electrons at standard conditions to that at any conditions.

Applications Of Nernst Equation

The equation has many applications. Some of them are discussed below-

•  The pH of solutions and the solubility of sparingly soluble salts can both be determined using the Nernst equation. 
• To determine the voltage and concentration of an electrochemical cell component.
• Calculation of equilibrium constants with accuracy
• To determine if it’s possible to generate electric potential by merging single electrodes.
• Pourbaix diagram construction depicting the equilibrium potential between a metal and its numerous oxidized species as a function of pH.
• To compare a substance’s reductive and oxidative abilities.
• Calculation of a concentration cell’s potential in corrosion.
• Unknown ionic concentrations

The equation can also be used to calculate half-cell reaction potentials or cell potentials for a variety of electrochemical cells such as voltaic and galvanic cells.

Limitations Of The Nernst Equation

·In the case of dilute solutions, the potential assumption can reach infinity, which is not practical and should be ignored.

·The Nernst equation cannot be used when there is a flowing of current through the cell. This effect can cause the equation to be inaccurate or inefficient.

·The Nernst equation gets erroneous when concentrations are extremely high. Experiments are done to determine potentials when reactant or product concentrations are exceedingly high.

Conclusion

The Nernst equation defines the relationship between cell potential to standard potential and the activities of the electrically active species. The Nernst equation is used to calculate the potential of the cell at any moment during a response at conditions other than the standard state. In corrosion studies, the equation is used to examine concentration cells and in the building of Pourbaix diagrams.