# Bohr radius - definition and derivation

## Introduction

Bohr’s atomic model was a major breakthrough in the field of study of atomic structure. Atoms are a basic unit of all materials. According to the Bohr atomic model atoms contain a small central core called a nucleus which is full of positive charges. Electrons round about the nucleus in particular circles named orbits. Sharon Bewick, Bohr's model, CC BY-SA 3.0

The circular path is maintained by the centripetal force derived from the electrostatic force between the electrons and the nucleus.

## Atomic Models

Multiple models were developed to explain the structure of the atom. These models are proposed by John Dalton, J.J.Thomson, Ernest Rutherford, Niels Bohr, and Erwin Schrodinger.

### Dalton Atom Model

According to Dalton, all matter consists of small particles called atoms. All atoms in the element look the same. Atoms of the different elements have different properties. But he failed to explain the structure of atoms.

### Thomson’s Atom Model

J.J.Thomson was the first person who explained the fine structure of atoms. He proposed that atoms are positively charged particles spread like a cloud. According to this, the negatively charged electrons scattered in a positively charged cloud look like watermelon or plum in a pudding. That is why it is called the plum pudding model. This model also failed due to the inability to give experimental support.

### Rutherford Atom Model

Ernest Rutherford did an α - scattering to propose his atom model. He noticed that most α particles passed through the gold foil. Few get mild deflection. Very few get deflected back in the same path. So experimentally he proposed that there is a nucleus at the center which is positively charged. Its mass is very small than the atom. Electrons revolve around the nucleus in a circular path. But it fails to explain the stability of an atom.

## Bohr’s Atom Model

Drawbacks of Rutherford’s atom model were overcome by Neil Bohr. According to his experiment result, he concluded that the atom contains a nucleus of positively charged particles. They are revolved by electrons in a circular orbit due to the force of attraction between electrons and protons. It revolves in a certain orbit with a certain energy. It does not lose or gain energy while rotating. But it emits energy when the electrons jump from one orbit to another. He makes use of the concept of quantization. To know well about the Bohr radius, it is necessary to know about the Bohr atom model and Plank’s quantum theory.

### Plank’s Quantum Theory

This theory explains the emission and absorption of radiation and the wave nature of electrons. According to this, when electrons jump between the energy levels it emits energy in the form of quanta of energy called photons. This energy is directly proportional to the frequency of radiation.

$$\mathrm{E\:\propto\:v}$$

$$\mathrm{E = hv}$$

h=planck's constant $\mathrm{=6.626×10^{−24}\:J. sec}$

## Postulates of Bohr Atom Model

Bohr’s postulates are based on Plank’s quantum theory. According to this, electrons revolve around the nucleus in a definite orbit having definite energy and it has a definite quantum number. They emit quanta of energy when electrons jump between the orbits and it is given by Plank’s equation. As the electrons move in orbit having fixed angular momentum, the angular momentum is quantized.

$$\mathrm{mvr =\frac{nh}{2\pi}\:\:\:\:\:n = 1, 2, 3, ……}$$

According to Bohr, the radius of the atom is the distance between the nucleus and the outermost shell of the atom. Precisely it is the distance between the nucleus and electron in the ground state of the hydrogen atom. The formula for Bohr radius derived by Neil Bohr is

$$\mathrm{a_0=(\frac{\epsilon_0 h^2}{\pi me^2})n^2}$$

Let us consider the electron of mass m and charge e moving around the nucleus of effective charge $\mathrm{Ze^+}$. Assume that the electron is moving with velocity v around the nucleus in an orbit of radius $\mathrm{a_0}$. According to Coulomb's law, there is a force of attraction occurs between charged particles and it is given by

$$\mathrm{F=\frac{q_1 q_2}{4\pi\epsilon_0 a_0^2}}$$

Now applying this the electrostatic force is

$$\mathrm{F =\frac{Ze^2}{4\pi\epsilon_0 a_0^2}\:\:\:\:\:………. (1)}$$

The electrons move in a circular orbit due to the centripetal force $\mathrm{(F_c)}$.

$$\mathrm{F_c =\frac{mv^2}{a_0}………. (2)}$$

Comparing equations (1) and (2) we get,

$$\mathrm{\frac{mv^2}{a_0}=\frac{Ze^2}{4\pi\epsilon_0 a_0^2}}$$

$$\mathrm{mv^2=\frac{Ze^2 \:a_0}{4\pi\epsilon_0 a_0^2}}$$

$$\mathrm{mv^2=\frac{Ze^2}{4\pi\epsilon_0 a_0}\:\:\:………. (3)}$$

From this we get,

$$\mathrm{a_0=\frac{Ze^2}{4\pi\epsilon_0 mv^2}}$$

This equation shows that the radius of the electron is inversely proportional to the velocity which means fast-moving electrons are in the lower orbits. According to the postulates of Bohr the angular momentum is given by,

$$\mathrm{mva_0=\frac{nh}{2\pi}}$$

$$\mathrm{v=\frac{nh}{2\pi ma_0}}$$

By squaring this equation

$$\mathrm{v^2=\frac{n^2 h^2}{4\pi^2 m^2 a_0^2}\:\:\:…………. (5)}$$

Substituting equation (5) in equation (4) we get,

$$\mathrm{a_0=\frac{Ze^2}{4\pi\epsilon_0 m(\frac{n^2 h^2}{4\pi^2 m^2 a_0^2})}}$$

$$\mathrm{a_0=\frac{Ze^24\pi^2 m^2 a_0^2}{4\pi\epsilon_0 mn^2 h^2}}$$

$$\mathrm{a_0=\frac{Ze^2 \pi ma_0^2}{\epsilon_0 n^2 h^2}}$$

$$\mathrm{a_0=\frac{\epsilon_0 n^2 h^2}{Ze^2 \pi m}}$$

For hydrogen Z=1. Thus

$$\mathrm{a_0=\frac{\epsilon_0 n^2 h^2}{e^2 \pi m}}$$

$$\mathrm{a_0=(\frac{\epsilon_0}{e^2 \pi m})n^2}$$

Here $\mathrm{\frac{\epsilon_0 h^2}{e^2 \pi m}}$ is a constant.

Thus, the Bohr radius is proportional to the square of the number of orbits. By substituting the values of constant terms

$$\mathrm{\frac{\epsilon_0 h^2}{e^2 \pi m}= 0.529\:Å}$$

Thus

$$\mathrm{a_0=0.529Å\:n^2}$$

## Application and Uses of Bohr Radius

Although it has some applications such as Bohr radius is used in the calculation of some physical quantities like atomic units and fine structure constants etc.

## Conclusion

Atoms are the basic unit of material. The structure of an atom is proposed by many scientists for many periods. Here in this article the model proposed by Neil Bohr was discussed and Bohr radius is also discussed here. The derivation for the formula of Bohr radius is done in detail.

## FAQs

Q1. What is meant by electrostatic attraction?

Ans. It is a force of attraction or repulsion that occurs between the charged particles of the atom without contact.

Q2. How do Rutherford do the scattering experiment?

Ans. Gold metal is drawn into a thin foil of a thickness of 100 nm and is bombarded by the α- particle. Scattered α- particles are collected at the circular screen.

Q3. What are the limitations of the Bohr atom model?

Ans. Bohr atom model fails to explain the Zeeman effect and stark effect and also it failed to explain Heisenberg’s uncertainty principle. These are all the limitations of the Bohr atom model.

Q4. What is Heisenberg's uncertainty principle?

Ans. Heisenberg’s Uncertainty principle describes that it is impossible to find both the position and momentum of the electron simultaneously at the same time.

Q5. What is the relation between Bohr Radius and quantum number?

Ans. Bohr radius is proportional to the quantum number. As the quantum number increases the Bohr radius is also increases.

Updated on: 18-Apr-2023

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