# Nernst Equation

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## Introduction: What is Nernst Equation?

For analytical chemistry as well as in important life processes such as nerve conduction and membrane potential, the Nernst equation has great utility. Electrochemical cells and hence the Nernst equation is widely used in the calculation of solution pH, solubility product, constant equilibrium and other thermodynamic properties, potentiometric titrations, and the calculation of cell membrane resting potentials.

The Nernst equation lends the relationship between the potential of the electrode and the potential of the standard electrode. It is also used to calculate free energy from the Gibbs, and to predict the spontaneity of an electrochemical reaction.

For the electrode reaction, Nernst demonstrated that:

Mn+ (aq) + ne → M(s)

The electrode potential can be represented by any concentration measured in respect of the standard hydrogen electrode:

$E_{(M^{n+} / M)}$ = $E^0_{(M^{n+} / M)} - \frac{RT}{nF}ln\frac{[M]}{[M^{n+}]}$

However, solid M concentration is taken as unity, and the above equation may be represented as:

$E_{(M^{n+} / M)}$  =  $E^0_{(M^{n+} / M)} - \frac{RT}{nF}ln\frac{[1]}{[M^{n+}]}$

In Daniel cell, the electrode potential for any given Cu2+ and Zn2+ ion concentration, the above equation can be written as:

For Cathode:

$E_ {(Cu^{2+}/Cu)}$ = $E^0_{(Cu^{2+}/Cu)} -\frac{RT}{2F}ln\frac{1}{[Cu^{2+}(aq)]}$

For Anode:

$E_{(Zn^{2+}/Zn)}$ = $E^0_{(Zn^{2+}/Zn)} -\frac{RT}{2F}ln\frac{1}{[Zn^{2+}(aq)]}$

The cell potential,

Ecell = $E_{cell}^0-ln\frac{[Zn^{2+}]}{[Cu^{2+}]}$

It is clear from the above equation that E(cell) depends on the concentration of both Cu2 + and Zn2 + ions. It increases with an increase in Cu2 + ion concentration and a decrease in the Zn2 + ion concentration. By translating the natural logarithm into the above final E(cell) equation, it reduces to base 10 and substitutes the values of R, F, and T= 298 K.

Nernst Equation Formula

Ecell = $E_{cell}^0-\frac{0.059}{2}log\frac{[Zn^{+2}]}{[Cu^{+2}]}$

The same number of electrons (n) is to be used for both the electrodes and therefore for the following cell:

Ni(s) | Ni2+ (aq) || Ag+ (aq) | Ag(s)

The equation Nernst can be described as:

Ecell = $E_{cell}^0-\frac{RT}{2F}ln\frac{[Ni^{+2}]}{[Ag^{+2}]}$

### Importance of Nernst Equation

The Nernst Equation allows for cell potential determination under non - standard conditions. It relates the measured cell potential to the quotient of the reaction and allows the exact determination of constants of equilibrium (including constants of solubility).

### Nernst Equation Applications

To Determine solubility products:

The Nernst equation can be used with the minimal error where sufficiently low concentrations of ions are in equilibrium with a sparingly soluble salt. Instead of directly measuring the concentration of the relevant ions, the more common and easier method would be to establish a cell in which one of the electrodes involves the insoluble salt that has a net cell reaction, just as the salt dissolves.

For example, we could use the silver-silver chloride electrode in the cell to calculate the Ksp for silver chloride: The question mark reflects the molarity concentration of the silver ions.

Potentiometric Titrations

In many situations, due to the presence of other ions and a lack of information on the activity coefficients of these ions, precise estimation of an ion concentration through direct measurement of the cell's potential is not feasible. In such situations, therefore, the concentration of the ions may be determined indirectly by titration with some other ion. For example, the initial concentration of an ion such as the Fe2+ ion may be found through titration with a strong oxidizing agent such as the solution containing the Ce4+ ion. The titration takes place in the left half cell that has a reference electrode in the right half cell

Pt(s) | Fe2+ , Fe3+ || Reference Electrode

The left cell originally only contains Fe2+. As the titrant Ce4+ is added, the ferrous ion is oxidized to Fe3+ ions as the reaction comes to an end: Fe2+ Ce4+ ((Fe3+ Ce3+) The cell potential is measured as the Ce4+ is added in small amounts / drops. The left half-cell potential is controlled by the Nernst equation ratio of oxidized and reduced iron ion concentrations.:

E = 0.68 - 0.059log$\frac{[Fe^{2+}]}{[Fe^{3+}]}$

Measurement of pH

Indeed, the pH of a solution is defined in terms of the activity of the hydrogen ion and not its concentration. A hydrogen electrode provides a direct indicator of hydrogen ion activity (aH+), thus pH= -log aH+. The H+ ion molarity is expressed by a question mark which is also a measure of the hydrogen ion concentration.

H2 (g, 1atm) | Pt | H+ (? M) || reference electrode