Bohr's theory and refinements

Atomic structure

Assumptions of Bohr's theory of structure of hydrogen atom

1. An atom consists of a small positively charged nucleus surrounded by electrons. The electron in an atom revolves around the nucleus only in certain selected circular path, called orbit.
2. As long as the electron remains in a particular path, it does not lose or gain energy. This means that the energy of the electron in a particular energy level remains constant.
3. When an electron moves from one energy level to another it radiates or absorbs energy. If it moves towards nucleus energy is radiated and if it moves away from nucleus energy is absorbed.
4. For an electron to remain in its orbit, the force of attraction between electron and nucleus must be equal to centrifugal force.

∴Electrostatic force of attraction = centrifugal force

\(\frac{Ζe^2}{4\pi r^2}\)= \(\frac{mv^2}{r}\)

where, m = mass of anelectron

r = radius of anorbit

e = charge on electron

z= nuclear charges

∈₀= Permittivity of vacuum

v = velocity of electron

5. The angular momentum of electron orbiting around the nucleus is a whole number multiple of h⁄ 2π

∴ mvr = \(\frac{n h}{2\pi}\)

where, n = principle quantum number (1, 2,3....)

h = plank's constant

Atomic spectrum of hydrogen based on Bohr's model

An atom can have only certain definite energy levels. The lowest possible energy level of an atom is called ground state. When an electron absorbs energy from some external source it jumps to higher energy level and is said to be in theexcited state. The excited electron can return to the ground state emitting energy in the form of quantum equal to the difference of energies between two energy levels. If E₁ and E₂ are energies of lower and higher energy levels respectively, then frequency(ν) of the radiation emitted is given by the following relation:

E₂- E₁= hν

or,ν = E₂-E₁⁄ h

where h is plank's constant.

In an atom, two different energy levels (E₂and E₁) can have certain definite values so h being constantν can have only certain fixed values. Therefore, Bohr's model explains why there are certain discrete lines in the spectrum of hydrogen.

In ahydrogen atom, there is one electron which is present in a first orbit in the ground state. When energy is supplied, such as by passing electric discharge, this electron may be excited to some higher energy level. Since in a sample of hydrogen, there are alarge number of atoms. the electrons in different atoms absorb different amounts of energies and are excited to different higher energy levels. For example, these electrons in some atoms may jump to third energy level while in other it may be raised to fourth energy level and so on. Now, from excited states, the electron may return to theground state in one ar more jumps downwards. These different downwards jumps results in theemission of radiations of thedifferent wavelength which appear as different lines in the hydrogen spectrum. Since a large number of different type of downwards transitions take place simultaneously in a sample of hydrogen, therefore, thelarge number of lines are obtained in theemission spectrum of hydrogen.

There are five different series in hydrogen spectrum which is given below:

Lyman series: Higher energy to first energy level, n = 2,3,4,5....to n = 1, spectral region = UV

Balmer series: Higher energy level to second energy level, n = 3,4,5.6..... to n = 2, spectral region = visible

Paschen series: Higher energy level to third energy level, n = 4, 5, 6, 7.....to n = 3, spectral region = IR

Brackett series : Higher energy level to fourth energy level, n = 5, 6, 7, 8.....to n = 4, spectral region = IR

Pfund series : Higher energy level to fifth energy level, n = 6,7,8,9......to n = 5, spectral region = IR

Now, for the calculation of wave number (\(\overlineν\))of any line in the visible spectrum a simple Rydberg equation given by Balmer is used which is given below.

\(\overline ν\) =\(\frac{1}{n_1^2}\) - \(\frac{1}{n_2^2}\) where R = \(\frac{Z^2e^4m}{8∈_0^2h^3c}\)

R = Rydberg constant which is equal to 1.096776× 10⁷m^-1

e = charge on electron

z = number of charges on thenucleus

∈₀= Permittivity of vacuum

m = mass of anelectron

Refinements of Boh's theory (Sommerfeld extension of Boh's theory)

Boh's theory could not explain the presence of fine structures in the in the hydrogen atom. So Sommerfeld refined the Boh's theory. According to Sommerfeld, the electrons in an atom are so influenced by the nuclear charge that they set on the elliptical orbit rather than circular as previously thought by Bohr. In elliptical orbit, there will be a major axis and a minor axis having different lengths. As the orbit broadens, the lengths of the two axes become closer and they become equal when the orbit becomes circular as shown in the figure.

The electron moving in an elliptical orbit will have its angular momentum which is quantized and can have only limited number of values given by the factor \(\frac{kh}{2\pi}\) where k is an integer known as the azimuthal quantum number. The relation between principal quantum number and azimuthal quantum number is given by

\(\frac{n}{k}\) = length of major axis⁄ length of minor axis

Thus, k can have more than one value for any given value of n (except 1). Therefore, energy depends on not only n but on k also which gives rise to fine lines in the hydrogen spectrum.

Shortcomings of Bohr's theory

1. In thepresence of magnetic field, each spectral line of the source is further divided into a number of lines called Zeeman effect. This phenomenon could not be explained by Bohr's theory.
2. Bohr's theory could not explain the spectra of higher atomic number elements with thepresence of fine structure in the line spectra of atoms.
3. On the basis of Heisenberg principle, the exact position and momentum of a small moving particle cannot be determined simultaneously. However, postulates of Bohr that electrons revolve around in well- defined orbits around the nucleus projects that the it is possible to determine the exact position and momentum of moving particle like anelectron.

References

Lee J.D., Concise Inorganic Chemistry,5^thedition, Chapman and Hall, London, 1996

Cotton F.A., Wilkinson G., Murillo C.A., Bochman M., Advanced Inorganic Chemistry, 3^rdedition, John WIley and Sons, Pvt., Ltd., 2007

www.TheBigger.com/section/chemistry/atomic structure and wave mechanics

https://en.wikipedia.org/wiki/Bohr_model

chemed.chem.purdue.edu/genchem/history/bohr.html