What is Nernst Equation?
The Nernst equation provides a relation between the cell potential of an electrochemical cell, the standard cell potential, temperature, and the reaction quotient. Even under non-standard conditions, the cell potentials of electrochemical cells can be determined with the help of the Nernst equation.
The Nernst equation is often used to calculate the cell potential of an electrochemical cell at any given temperature, pressure, and reactant concentration. The equation was introduced by a German chemist named Walther Hermann Nernst.
Expression of Nernst Equation
Nernst equation is an equation relating the capacity of an atom/ion to take up one or more electrons (reduction potential) measured at any conditions to that measured at standard conditions (standard reduction potentials) of 298K and one molar or one atmospheric pressure.
Nernst Equation for Single Electrode Potential
Ecell = E0 – [RT/nF] ln Q
Where,
- Ecell = cell potential of the cell
- E0 = cell potential under standard conditions
- R = universal gas constant
- T = temperature
- n = number of electrons transferred in the redox reaction
- F = Faraday constant
- Q = reaction quotient
The calculation of single electrode reduction potential (Ered) from the standard single electrode reduction potential (E°red) for an atom/ion is given by the Nernst equation.
For a reduction reaction, Nernst equation for a single electrode reduction potential for a reduction reaction
Mn+ + ne– → nM is;
Ered = EMn+/M = EoMn+/M – [2.303RT/nF] log [1/[Mn+]]
Where,
- R is the gas constant = 8.314 J/K Mole
- T = absolute temperature,
- n = number of mole of electron involved,
- F = 96487 (≈96500) coulomb/mole = charged carried by one mole of electrons.
- [Mn+] = active mass of the ions. For simplicity, it may be taken as equal to the molar concentration of the salt.
Nernst Equation at 25oC
For measurements carried out 298K, the Nernst equation can be expressed as follows.
E = E0 – 0.0592/n log10 Q
Therefore, as per the Nernst equation, the overall potential of an electrochemical cell is dependent on the reaction quotient.
Derivation of Nernst Equation
Consider a metal in contact with its own salt aqueous solution. Reactions of metal losing an electron to become an ion and the ion gaining electron to return to the atomic state are equally feasible and are in an equilibrium state.
Mn+ + ne– → nM
In the reduction reaction, ‘n’ moles of an electron is taken up by the ion against a reduction potential of Ered.
1. The work done in the movement of electron
Wred = nFEred
Where,
- F is Faraday = 96487 coulomb = electrical charge carried by one mole of electrons
2. Change in the Gibbs free energy is an indication of the spontaneity and it is also equal to the maximum useful work (other than volume expansion) done in a process.
Combining work done and Gibbs free energy change:
Wred = nFEred = – ∆G or ∆G = – nFEred
3. Change in the free energy at standard conditions of 298K and one molar /one atmospheric pressure conditions is ∆G°. From the above relation, it can be written that
∆G° = – nFE°red
Where,
- E°red is the reduction potential measured at standard conditions.
4. During the reaction, concentration keeps changing and the potential also will decrease with the rate of reaction.
To get the maximum work or maximum free energy change, the concentrations have to be maintained the same. This is possible only by carrying out the reaction under a reversible equilibrium condition.
For a reversible equilibrium reaction, vant Hoff isotherm says:
∆G = ∆G° + RT ln K
Where,
- K is the equilibrium constant
- K = Product/Reactant = [M]n/[M]n+
- R is the Gas constant =8 .314J/K mole
- T is the temperature in Kelvin scale.
5. Substituting for free energy changes in ant Hoff equation,
Determining Equilibrium Constant with Nernst Equation
When the reactants and the products of the electrochemical cell reach equilibrium, the value of ΔG becomes 0. At this point, the reaction quotient and the equilibrium constant (Kc) are the same. Since ΔG = -nFE, the cell potential at equilibrium is also 0.
Substituting the values of Q and E into the Nernst equation, the following equation is obtained.
0 = E0cell – (RT/nF) ln Kc
The relationship between the Nernst equation, the equilibrium constant, and Gibbs energy change is illustrated below.
Nernst equation vs Equilibrium constant vs Gibbs energy change
Converting the natural logarithm into base-10 logarithm and substituting T=298K (standard temperature), the equation is transformed as follows.
E0cell = (0.0592V/n) log Kc
Rearranging this equation, the following equation can be obtained.
log Kc = (nE0cell)/0.0592V
Thus, the relationship between the standard cell potential and the equilibrium constant is obtained. When Kc is greater than 1, the value of E0cell will be greater than 0, implying that the equilibrium favours the forward reaction. Similarly, when Kc is less than 1, E0cell will hold a negative value which suggests that the reverse reaction will be favoured.
Nernst Equation Applications
The Nernst equation can be used to calculate:
- Single electrode reduction or oxidation potential at any conditions
- Standard electrode potentials
- Comparing the relative ability as a reductive or oxidative agent.
- Finding the feasibility of the combination of such single electrodes to produce electric potential.
- Emf of an electrochemical cell
- Unknown ionic concentrations
- The pH of solutions and solubility of sparingly soluble salts can be measured with the help of the Nernst equation.
Limitations of Nernst Equation
The activity of an ion in a very dilute solution is close to infinity and can, therefore, be expressed in terms of the ion concentration. However, for solutions having very high concentrations, the ion concentration is not equal to the ion activity. In order to use the Nernst equation in such cases, experimental measurements must be conducted to obtain the true activity of the ion.
Another shortcoming of this equation is that it cannot be used to measure cell potential when there is a current flowing through the electrode. This is because the flow of current affects the activity of the ions on the surface of the electrode. Also, additional factors such as resistive loss and overpotential must be considered when there is a current flowing through the electrode.