## Introduction

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 cellE

_{0}stands for cell potential under standard conditionsR 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 Cu^{2+} and Zn^{2+} 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 Fe^{2+} ion may be found through titration with a strong oxidizing agent such as the solution containing the Ce^{4+} 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 Fe^{2+}. As the titrant Ce^{4+} is added, the ferrous ion is oxidized to Fe^{3+} ions as the reaction comes to an end: Fe^{2+} Ce^{4+} ((Fe^{3+} Ce^{3+}) The cell potential is measured as the Ce^{4+} 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.

## FAQs on Nernst Equation

**1. How was gravity found out and why is it important?**

Remember the story of the apple falling from a tree straight to the ground and Issac Newton who sat below the tree got hit by it? Well, that is where all the findings of gravitational forces started. Gravity is one of the four fundamental forces that was proposed by Issac Newton but it was years later that Albert Einstein, another great scientist who gave the theory of general relativity. It is exactly this gravity that holds not only humans to the ground but also the whole of the universe together. Gravity is also an important force even though it might be the weakest among all of the other forces as it is important in various actions. For example, imagine trying to kick a ball with force but all that happens is that the ball is just flying and has no effect of the force.

**2. Which force is one of the strongest of all the fundamental forces?**

Strong nuclear force has its name as strong for a reason and well that's right it is because it is one of the strongest forces in nature. It is seen that this force is above 6 thousand trillion trillion trillion times much stronger than gravity and hence it is one of the strongest not only than the four fundamental forces but among all the forces that may exist. This strong nuclear force, even though it is the strongest, is found in the smallest of atoms and has the purpose of holding all the protons and neutrons together in the nucleus.

**3. Why is it important to learn about the fundamental forces in nature?**

Fundamental forces are the ones that govern most of the activities that go on in the world. Be it the atoms that are being held together, or staying afloat on the ground, it is all because of these four fundamental forces. These forces play an important role not only in the lives of humans but also in the overall universe that is being kept together. Vedantu provides this article and a lot of other study materials on the four fundamental forces. Students can also check out the Vedantu NCERT Solutions for Physics that have a detailed outlook on the topic.

**4. Where are the electromagnetic forces useful?**

The electromagnetic force which acts between two particles that are negatively and positively charged is also called the Lorentz Force. It is known that opposite charges attract each other while the same charges tend to repel each other. This force consists of two parts just like what is mentioned in its name, one being the electric force and the other being the magnetic force. It is the electric force that acts between two charged particles while the second force that is the magnetic force tends to set them into action. There is hence a creation of a magnetic field when the electric charge is being provided. These forces are also one of the most important forces needed in friction, elasticity, and other solids.

**5. Is there any particular force that is more important than the others or are all equally important?**

All of the four fundamental forces are considered to be equally important because even if one of the forces were not present it would have been hard to maintain the universe the way it is now. Right from the smallest atom bonding to the elephant being held on earth it would have been quite impossible for all these things to happen. However, this does not mean that just because they are equally important they have equal strength. In fact, it is the strong nuclear forces that are the most strong, and then the gravitational forces which are weakest are present in the universe.