Bohr Atom Model
Three fundamental laws were postulated by Bohr to overcome the drawbacks in the Rutherford model.
Postulate I.
Electrons can occupy states at only certain discrete energy levels. At these discrete energy levels the electrons do not emit radiation and are said to be in stationary or non-radiating state.
Postulate II.
Emission of radiation results when an electron moves from one stationary state to another with lower energy. The frequency of radiation ' f ' is given by
f = (W2 - W1) / h (1.2)
where f is the frequency of emission in hertz
h is the Planck s constant = 6.62 x 10-34 joule second.
W1 and W2 are energies in joules.
Postulate III.
Any stationary state is determined by the condition that the Angular momentum of the electron in this state is quantized and must be a multiple of h/2p.
mvr = (nh) / 2p (1.3)
where n is an integer.
The radius of the various stationary orbits is given by
rn = (eo h2n2 ) / ( h2n2 ) / (p m q2 ) (1.4)
= 0.527 X 10-10 n2 meter(1.5)
The total energy of an electron in stationary states is given by
Wn = - (m q4 ) / (8 e o2 h2 n2) joules (1.6)
= -13.6 / n2 eV (1.7)
It should be, however, noted that the energy is negative and therefore, the energy of a electron in its orbit increases as n increases. Thus, to remove an electron from the first orbit (n=1) of the hydrogen atom, i.e., to ionise the atom, the energy required is 13.6 eV. This is known as the ionisation energy or the ionisation potential of the atom.
Velocity of revolving electrons
The velocity of the revolving electron as found from the above equations is
V = (Z e 2) / (2 e 0 n h) (1.8)
Velocity is inversely proportional to n.
Orbital Frequency
The orbital rotational frequency of an electron is
f = n / 2 p r
= (m Z2 e4 ) / (4 e 02 n3 h3) (1.9)
Electron Energy
The energy of a revolving electron is the sum of its kinetic and potential energies of the atom.
Kinetic energy of an atom is given by
KE = (m Z2 e4 ) / (8 e 02 n2 h2) (1.10)
Potential energy is given by,
PE = -(m Z2 e4 ) / (4 e 02 n2 h2) (1.11)
Therefore total energy is given by,
KE + PE = - (Z e2 ) / (8 p e 0 r) (1.12)
It is seen that En is E1/n2 where n = 1,2,3 & .
Normal, Excited and Ionised atom
Considering the case of a hydrogen atom, it is said to be in its normal state when its only electron is in its innermost orbit. If a spark is introduced into hydrogen gas, it may completely remove the electron or raise to higher orbits. If the electron is completely removed it is said to be ionised. If the electron is in a higher orbit then it is said to in an excited state. It does not remain in this state for more than 10-8 seconds and emits radiations of different frequencies when it returns to its normal state or lower states.
Electron energy levels in a Hydrogen atom
The orbital energy of an electron in the nth orbit or shell is
En = - (m Z2 e4) / (8 e 02 n2 h2)
In case of the Hydrogen atom since Z = 1,
En = - (m e4) / (8 e 02 n2 h2)
= -(( m Z2 e4) / (8 e 02 h2)) x 1 / n2
= - 21.7 x 10 - 19 / n2 joules
= -13.6 / n2 eV
This expression gives the total energy of an electron when it occupies any one of the orbits.
EXAMPLE I.0
Calculate the value of the Kinetic, Potential and total energy of the electron revolving in Bohr s first orbit in a hydrogen atom.
KE = - (me4) / (8e 02n2h2)
= ( 9.1 x 10 - 31 x (1.6 x 10 - 19) ) / (8 x (8.854 x 10-12)2 x (6.625 x 10 - 34)2)
= 13.6 eV
PE = (me4) / (4e 02n2h2)
= -43.4 x 10-19 joules
= -27.2 eV
Total Energy= KE +PE= -13.6 EV
Critical Potentials
Critical potential is defined as the least energy required to excite a free neutral atom from its ground state to a higher state. It is expressed in electron volts. There are two kinds of critical potentials.
a) Excitation potential:
The energy, in electron volts, required to raise an atom from its normal state to an excited state is called excitation potential of the state. It is also known as radiation potential or resonance potential.
b) Ionisation Potential:
The energy required to remove an electron from a given orbit to an infinite distance from the nucleus is called the ionisation potential.
The number of ionisation potentials depends on the number of electrons in an atom. For the hydorgen atom, there is only one ionisation potential and several excitation potentials. The energy required for the removal of the of the outermost valence electrons is called the first ionisation potential, the energy required to remove the second electron from the influence of the nucleus is termed the second ionisation potential and so on.
PROBLEM 1.1
Calculate the radii of the first, second and third permitted electron orbit in a Bohr s hydrogen atom.
Radius of the nth orbit for hydrogen
= (e 0 n2 h2 )/ (p m e2)
Radius of the first orbit
= ((8.854 x 10-12)2 x (6.625 x 10 - 34)2 x h2) / ( p x 9.1 x 10 - 31 x (1.6 x 10 - 19))
= 5.27 x 10 - 11 h2 metre
=5.27 x 10 - 11 x 12 metre
=0.527 AU
Radius of the second orbit = 2.108 AU
Radius of the third orbit = 4.743 AU