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. 

Terms and What They stand for in the Nernst Equation

• E_cell stands for cell potential of the cell

• E₀  stands for cell potential under standard conditions

• R stands for the universal gas constant

• T stands for temperature

• n stands for the number of electrons transferred in the redox reaction

• F stands for the Faraday constant

• Q stands for the reaction quotient

Nernst Equation Demonstration

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:

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

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

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

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

For Cathode:

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

For Anode:

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

The cell potential,

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

It is clear from the above equation that E(cell) depends on the concentration of both Cu^{2 +} and Zn^{2 +} ions. It increases with an increase in Cu^{2 +} ion concentration and a decrease in the Zn^{2 +} 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

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

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:

\[E_{cell}=E^{0}_{cell}-\frac{RF}{2F}ln\frac{[Ni^{2+}]}{[Ag^{2+}]}\]

Determination of the Equilibrium Constant Using Nernst Equation

When the reactants and the products taken as part of the chemical reaction reach the point of equilibrium, the value of ΔG becomes 0. This means that there is no change in Gibbs free energy anymore. Consequently, the reaction quotient and the equilibrium constant (K_c) become the same. As we all know that ΔG is equal to -nFE, the cell potential at equilibrium is thus 0.

By substituting the values of Q and E into the Nernst equation, we reach the equation given below:

0 = E0cell – (RT/nF) ln Kc

After further substitution, we reach:

E0cell = (0.0592V/n) log Kc

Ultimately, the equation can be presented in this form:

log Kc = (nE0cell)/0.0592V

Via this method, the relationship between the standard cell potential and the equilibrium constant is established and demonstrated. When K_c is greater than 1, the value of E_0cell will be greater than 0. This suggests that the equilibrium will shift in the forward direction. In contrast to this, when K_c is less than 1, E_0cell will turn out to be negative. This implies that the backward reaction will be favored.

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 ([Fe2+]/[Fe3+ ])

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.

Limitations of the Nernst Equation

We are aware that the activity of an ion in a very dilute solution is nearly infinite. This activity can thus be expressed in terms of ion concentration. However, it is important to note that for solutions having very high concentrations, the ion concentration is not equal to the ion activity. To be able to use the Nernst equation in such cases, the true activity of the ion has to be determined using various experiments. 

Although the Nernst equation is quite useful, it has another limitation. The equation cannot be used to measure cell potential when there is electricity flowing between the 2 electrodes given. The flow of current affects the activity of the ions that have accumulated on top of the electrode.