Chemistry - Explanation of the line spectrum of hydrogen using Bohr's theory

Explanation of the line spectrum of hydrogen using Bohr theory : 

The line emission spectrum obtained from atomic hydrogen can be explained qnantitatively using Bohr theory. According to second postulate of Bohr theory, radiation is emitted when electron moves from an outer orbit of higher principal quantum number (ni) to an inner orbit of lower principal quantum number (nf). The energy difference (∆E) between the initial and final orbit of the electronic transition corresponds to the energy of the emitted radiation. From the third postulate of Bohr theory ∆E can be expressed as

According to the results derived from Bohr theory the energy E of an orbit is related to its principal quantum number ‘n’ by the Eq. 

Combinig these two Eq. we get:

Substituting the value of RH in joules we get

This expression can be rewritten in terms of wavenumber of the emitted radiation in the following steps.

We know: (∆E) J = (h) J s × (v) Hz

and by definition.

combining the the above equations, we get. 

This appears like the Rydberg Equation, where

nf = n1 and ni = n2.

In other words, Bohr theory successfully accounts for the empirical Rydberg equation for the line emission spectrum of hydrogen. In the Rydberg equation ‘n1’ and ‘nf’ are integers. Bohr’s theory assigns physical meaning to them as principal quantum numbers corresponding to the concentric orbits. The integers in Rydberg equation, stand for the final orbit, nf of electronic transition and n1 for the initial orbit. The emission lines comprising the five series thus, are result of electronic transitions from the excited hydrogen atoms. The Lyman series is the result of moving of electron excited to higher orbits of n2 = ni = 2, 3, 4,.....etc. to lower orbits of n1 = nf = 1; the Balmer series results from electron from n2 = ni = 3, 4,........ to the lower orbit of n1= nf = 2, so on and so forth. The electronic transitions giving rise to different emmission line series of atomic hydrogen are shown in Figure.

Electronic transition in the hydrogen spectrum

Limitations of Bohr model :

1. Bohr’s atomic model failed to account for finer details of the hydrogen atom spectrum observed in sophisticated spectroscop experiments.
2. Bohr model was unable to explain the spectrum of atoms other than hydrogen .
3. Bohr theory could not explain the splitting of spectral lines in the presence of a magnetic field (Zeeman effect) or electric field (Stark effect).
4. Bohr theory failed to explain the ability of atoms to form molecules by chemical bonds. 

It was, therefore, thought that a better theory was needed to explain salient features of atomic structure.

Reasons for failure of the Bohr model :

With the limitations of Bohr model for hydrogen atom becoming transparent, attempts were made to develop a better and general model for atom. This was possible because of two important developments took place after the Bohr model was postulated. 

These development were :

1. de Broglie’s proposal of dual behaviour of
matter.
2. Heisenberg uncertainty principle.

In Bohr model an electron is regarded as a charged particle moving in well defined circular orbits about the nucleus. In contrast to this de Broglie proposed in 1924 that matter should exhibit a dual behaviour, that is, both particle and wave like properties. This means that electron should have momentum, p, a property of particle as well as wavelength, λ, a property of wave. He gave the following relation between λ and p of a material particle.

De Broglie’s prediction was confirmed by 

diffraction experiments (a wave property).

In the year 1927 Werner Heisenberg stated the uncertainty principle : “It is impossible to determine simultaneously, the exact position and exact momentum (or velocity) of an electron. In other words the position and momentum of an electron can not be determined with the same certainty. If the certainty of determination of one property of the two is high, it means that the uncertainty of its determination is low. In that case the uncertainty of determination of the other property is very high. Mathematically Heisenberg uncertainty principle is expressed as: 

Here ∆x is the uncertainty in position and ∆px (or∆vx) is the uncertainty in momentum. A further implication of the uncertainty principle is that for an electron having certain energy one can only determine its probability at a particular point x around the nucleus. Bohr’s model describes concentric orbits as well defined paths of the electron rotating about the nucleus and calculate energy of electron occupying these orbits. Bohr model assumes that both position and momentum, of the electron in hydrogen atom are known exactly at the same time, which is ruled out by the Heiesenberg uncertainty principle. No attempt was made to extend the Bohr model to other atoms. A different approach to atomic model which could account for particle duality of matter and consistent with Heisenberg uncertainty principle was required. This became possible with the development of quantum mechanics.