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>> Bohr Model
The main features of Rutherford's model, viz. the nucleus and the electrons orbiting it under the action of the Coulomb's law of electrostatic attraction, were retained in Bohr's theory. In addition Bohr introduced the concept of 'radiation less orbits' or 'stationary states' in which the electron revolves around the nucleus, but does not radiate contrary to the laws of electromagnetism. This was a hypothesis, but at least a working one.
Radiation occurred only when an electron made a transition from one stationary state to another. The difference between the energies of the two states was radiated as a single photon. Absorption occurred when a transition occurred from a lower stationary state to a higher stationary state.
The third principle invoked by Bohr was the correspondence principle: i.e. the spectrum is continuous and the frequency of light emitted equals the frequency of the electron. In the classical limit.
Using these general arguments and the existing body of knowledge, Bohr postulated the following:
(i) The electron in an atom has only certain definite stationary states of motion allowed to it, called as energy levels. Each energy level has a definite energy associated with it. In each of these energy levels, electrons move in circular orbit around the positive nucleus. The necessary centripetal force is provided by the electrostatic attraction of the protons in the nucleus. As one moves away from the nucleus, the energy of the states increases.
(ii) These states of allowed electronic motion are those in which the angular momentum of an electron is an integral multiple of h/2p or one can say that the angular momentum of an electron is quantized.
Angular momentum = m v r = n (h/2p)
Where m is the mass, v is the velocity, r is the radius of the orbit, h is Planck's constant and n is a positive integer.
(iii) When an atom is in one of these states, it does not radiate any energy but whenever there is a transition from one state to other, energy is emitted or absorbed depending upon the nature of transition.
(iv) When an electron jumps from higher energy state to the lower energy state, it emits radiations in form of photons or quanta. However, when an electron moves from lower energy state to a higher state, energy is absorbed, again in form of photons.
(v) The energy of a photon emitted or absorbed is given by using Planck's relation (E = hv). If E1be the energy of any lower energy state and E2 be the energy of any higher energy state then the energy of the photon (emitted or absorbed) is given as ΔE:
ΔE = E2 – E1 = h v
Where h = Planck's constant and v = frequency of radiation emitted or absorbed.
Bohr's Model is applicable only to one-electron atoms like: He+, Li++ apart from H-atom.
In the following section we will define the radius of orbits around the nucleus (Bohr's orbits), the velocity and energy of an electron in various orbits around the nucleus.
We will take
rn : radius of nth orbit
vn : velocity of electron in nth state (orbit)
En : energy of nth state
m : mass of an electron (9.1 x 10–31 Kg)
Z : atomic number (No. of Protons)
K = 1/(4pe0) = constant = 9 x 109 N m2 C–2
h : Planck's constant (6.67 x 10–34 J-s)
c : velocity of light (3 x 108 m/s)
R : Rydberg constant (1.097 x 107 m–1)
e : Charge on an electron (1.6 x 10–19 C)
v : frequency of the radiation emitted or absorbed
v : wave number of the spectral line in the atomic spectra
From first postulate
mv2 / rn = KZe2 / rn2
From second postulate
mvnrn = nh / 2π
Solving for rn and vn, we have:
radius = rn = n2h2 / 4π2Kme2Z
= 0.53 × 10–10 n2/Z m
= 0.53 n2 / Z A
velocity = vn = 2π2Kme2Z / nh
2.165 × 106 Z/n m/s
The energy of an electron in nth state En is given by:
En = KE + PE
En = 1/2 mv2 + (– KZe2 / rn) = –1/2 KZe2 / rn
=> En = –2π2K2e4Z2 / n2h2 (putting value of rn)
= –2.178 × 10–18 Z2/n2 J / atom
= –13.6 Z2/n2 eV/atom
= –1312 Z2/n2 kJ/mol
· When an electron jumps from an outer orbit (higher energy) n2 to an inner orbit (lower energy) n1, then the energy emitted in form of radiation is given by:
As we know that E = hv, c = vλ and v = 1/λ
=> v = ΔE/hc
Now this can be represented as:
This relation exactly matches with the empirical relation given by Balmer & Rydberg to account for the spectral lines in H-atom spectra. In fact the value of Rydberg constant in the empirical relation is approximately the same as calculated from the above relation (Bohr's Theory). This was the main success of Bohr's Theory i.e. to account for the experimental observations by postulating a theory.