For galvanic electrochemical cells, the reduction potential of the cell or EMF can be calculated by a simple

Standard EMF = Standard Reduction Potential (SRP) of Cathode – SRP of Anode.

It is obvious from this formula that it applies where two metals are there in a cell as cathode and anode. But, there are those cells also which have only one metal but the different electrolyte concentrations in their anode and cathode chambers. Nernst equation becomes very essential there.

**Nernst Equation And Its Derivation**

The previous method to evaluate the electrode potential has some limitations especially when the temperature is not standard or the metal of the electrode is the same. To solve this problem, here comes the Nernst Equation. Walter Nernst, a German scientist proposed an equation as the relationship between the reaction coefficient of cell reaction and cell potential or more precisely, reduction cell potential.

Nernst equation is-

E_{cell} =E^{°}_{cell} – (0.0591/n) log_{10}Q ….(1)

Where E_{cell }is the reduction potential of the cell at the given temperature whereas E^{°}_{cell }is the reduction potential at standard conditions.

__Derivation Of Nernst Equation__

__Derivation Of Nernst Equation__

**As, ∆G = ∆G° + RT ln Q ….a)**

Where ∆G is the change in Gibb’s free energy and ∆G° is the change in standard Gibb’s free energy.

But, we know that the

**work done = charge × potential ….b)**

And thermodynamics states that the decrease in Gibb’s free energy of a system is equal to the reversible work or simply the maximum amount of work gained by the system if there is no significant work done in the expansion of the system’s volume.

From b),

Work done = q (charge) × E (potential)

Charge q = nF ( in terms of the charge of n mole electrons and 1 F or 1 Faraday is the charge of 1 mole of electrons)

Hence, Work done = nF × E

But, according to the previous explanation of reversible work,

Work done = ∆G

So, ∆G = -nFE ( negative sign for the decrease in Gibb’s energy)

At, standard conditions, ∆G° = -nFE°

After substituting these values in equation a),

-nFE = -nFE° + RT ln Q

E = E° – (RT/nF) ln Q ….c)

This is the Nernst equation for any temperature

** **Now, on putting values of

ideal gas constant R= 8.314 J/(mol-K)

Faraday’s Constant F= 96485 Columb

Number of electrons transferred in the redox cell reaction = n

Converting natural log to the log on base 10

And Temperature T = 25℃ or 298 K

(standard temperature),

We get equation (1).

For the temperature other than the standard one, we can use all the above-mentioned values as it is but the value of T will change.

Thus, for a redox cell reaction

pP + qQ —> rR +sS,

EMF of the cell can be calculated like this-

**E _{cell} = E°_{cell} – log_{10} [(C_{R})^{r} (C_{S})^{s}/ (C_{P})^{p} (C_{Q})^{q}]**

**Various Applications Of Nernst Equation**

Besides determining the EMF of a galvanic electrochemical cell, Nernst Equation has a wide range of applications in the chemical industry. Some of them are as follows-

**In Evaluating The Heat Of Reaction Inside A Cell**

When the charge equivalent to n Faraday flows through a cell of EMF ‘E’, then decrease in Gibb’s free energy

∆G = -nFE

Also, ∆G = ∆H – T∆S …d)

(where H is the enthalpy or heat of reaction and S is the entropy of the reaction. ∆ stands for the change.)

But, dG = VdP – SdT

From here, -dS =[∂G/ ∂T]_{P} …e)

It means Entropy change is equal to the negative partial derivative of G with respect to T at constant pressure.

On putting this value of -dS in equation d), we get

∆G = ∆H + T[∂G/ ∂T]_{P} …f)

Also, -nFE = ∆H + T[∂G/ ∂T]_{P}

-nFE = ∆H – nFT[∂E/ ∂T]_{P}

**Hence, heat of reaction inside the cell**

** ∆H = -nFE + nFT[∂E/ ∂T] _{P}**

**To Determine The Equilibrium Constant Of A Cell Reaction**

Equilibrium point, at which the rate of the forward reaction is equal to the backward reaction. More precisely, the equilibrium point here stands for equilibrium constant. Nernst equation also helps in determining the equilibrium constant of a cell reaction.

From Nernst equation,

E = E° – (0.0591/n) log_{10} Q

But, at the stage of equilibrium, the EMF of the cell is zero since no resultant transfer of electrons is happening.

So, we have to put E=0 in the above Nernst equation.

0 = E° – (0.0591/n) log_{10} K_{eqlbm}

Or, we can write as

**K _{eqlbm} = antilog_{10} [nE°/ 0.0591]**

Where K_{eqlbm} is the equilibrium constant of the cell reaction.

**In Determining The pH Of An Electrolytic Solution**

Nernst equation proves itself a troubleshooter when it comes to evaluating the pH in real conditions since the activity of hydrogen cations aC_{H+} replaces the hydrogen ion concentration term C_{H+} apart from some ideal cases.

In real cases, pH = – log_{10} aC_{H+}

Where a is the activity coefficient.

So, we’ll write a cell equation like this and calculate y.

H2 (gas, 1 atm) | Pt | H+ ( y M) || Electrodereference

And, using the required calculation in the Nernst equation, we’ll easily find y.