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Rutherford's planet - like model of the atom is contested by Bohr in 1913 on two grounds,  According to classical mechanics, whenever a charged particle is subjected to acceleration it emits radiation and loses energy. an electron revolving round the nucleus would therefore be continually accelerated towards the centre of the orbit and consequently emitting radiation. The result of this would be that the radius of curvature of its path would go on decreasing and due to spiral motion, the electrons will finally fall on the nucleus when all its rotational energy spent on the electromagnetic radiation and the atom would collapse. 
If the electrons lose energy continuously, the observed atomic spectra should be continuous, consisting of broad bands merging one into the other. The observed atomic, however, consists of well defined lines of definite frequencies. To resolve the anomalous position Bohr proposed several novel postulates,

Postulates of Bohr's Theory:

  1. An atom possesses several stable circular orbits in which an electron can stay. So long as an electron can stays in a particular orbit there is no emission or absorption of energy.
    These orbits are called Energy Levels or Main Energy Shells.
    These shells are numbered as 1, 2, 3, ........ starting from the nucleus and are designated as capital letters, K, L, M, ........ respectively.
    The energy associated with a certain energy level increases with the increase of its distance from the nucleus. Thus, if E1, E2, E3 ...... denotes the energy levels numbered as 1 (K - Shell), 2 (L - Shell), 3 (M - Shell) ......., these are in the order,
    E1E2E3ㄑ......
  2. An electron can jump from one orbit to another higher energy on absorption of energy and one orbit to another lower energy orbit with the emission of energy.
    Bohr's Model for Hydrogen Atom
    Emission and Absorption of Energy of an Electron.
    The amount of energy (ΔE) emitted or absorption in this type of jump of the electron is given by Plank's Equation.
    ΔE = hν
    Where ν is the frequency of the energy (radiation) emitted or absorbed and h is the Plank Constant. 
  3. The angular momentum of an electron moving in an orbit is an integral multiple of h/2π
    This is known as principle of quantisation of angular momentum according to which,
    mvr = n × (h/2π)
    Where m = mass of the electron, v = tangential velocity of electron in its orbit, r = distance between the electron and nucleus and n = a whole number which has been called principle quantum number by Bohr.

Radii of Bohr Orbit:

Bohr's Model for Hydrogen Atom
Bohr's Model
The nucleus has a mass m' and the electron has mass m. The radius of the circular orbit is r and the linear velocity of the electron is v.
Evidently on the revolving electron two types of forces are acting,
  1. Centrifugal force which is due to the motion of the electron and tends to take the electron away from the orbit. 
    Centrifugal force = (mv2/r)
    Acts outwards from the nucleus.
  2. The attractive force between the nucleus and the electron. Two attractive forces are in the operation, one being the electric force of attraction between nucleus (Proton) and the electron, the other being the Gravitational force is comparatively weak and can be neglected. It is given by Coulomb's inverse squire low and is therefore equal to,
    e × (e/r2) =  e2/r2 
    It acts towards the nucleus.
In order that the electron may keep on revolving in its orbit, these two forces, which acts in opposite direction must balanced each other.
That is, mv2/r = e2/r2
mv2 = e2/r
Now Bohr made a remarkable suggestion that the angular momentum of the system, equal to mvr, can assume certain definite values or quanta. Thus all possible r values only certain definite r values are permitted. Thus only certain, definite orbits are available to the revolving electron.
According to Bohr's theory the quantum unit of angular momentum is h/ (h being Plank's Constant).
Thus, mvr = nh/
(where n have values 1,2,3, ......∞)
or, v = n × (h/2π) × (1/mr)
Then, e2/r = mv2
= m × n2 × (h/2π)2 × (1/mr)2
= n2h2/2mr2
∴ r = n2h2/2me2 = n2 × a0 
where a0 = h2/4𝜋2me2
We thus have a solution for the radius of the permitted electron orbits in terms of quantum number n. Taking n = 1, the radius of the first orbit is r1.
r1 = 1 × h2/2me2
= {1×(6.627×10-27)2}/{4×(3.1416)2×(9.108×10-28×(4.8 × 10-10)2}
= 0.529 × 10-8 cm
= 0.529 Å = a0
∴ r = n × a0(n being 1,2,3, ......)
Thus the radius of first orbit r1 = a0,
second orbit r2 = 4 a0 and
third orbit r3 = 9 a0 and so on.
Velocity of the Electron in Bohr Orbits:
We have the following relations,
mvr = nh/2π and r = n2h2/4π2me2
Then, v = (nh/2πm) × (1/r)
= (nh/2πm) × (4π2me2/n2h2)
v = 2𝜋e2/nh
Putting the values of n (1, 2, 3, .....) we see the velocity in the second orbit will be one half of the first orbit and that in the third orbit will be one third of that in the first orbit and so on.
Calculate the velocity of the hydrogen electron in the first and third orbit. Also calculate the number of rotation of an electron per second in third orbit.
Velocity of an electron in the Bohr orbit,
= 2πe2/nh
where n = 1, 2, 3, ........
Thus, the velocity of an electron in the first orbit,
v1 = 2πe2/1 × h
= 2πe2/h (when n = 1)
v1 = {2×(3.14)×(4.8×10-10)2}/(6.626×10-27)
= 2.188 × 108 cm sec-1
Again the velocity of the third orbit,
v3 = (1/3) × v1 (when n = 3)
v3 = (2.188 × 108 cm sec-1)/3
= 7.30 × 107 cm sec-1
Radius of the third orbit,
= 32 × 0.529 × 10-8 cm
Thus the circumference of the third orbit,
= 2πr = 2 × 3.14 × 0.529 × 10-8 cm
∴ Rotation of an electron per second in third orbit,
= (7.30 × 107)/(2 × 3.14 × 9 × 0.529 × 10-8 )
= 2.44 × 1014 sec-1

Energy of an Electron in Bohr Orbits:

The energy of an electron moving in one such Bohr orbit be calculated remembering that the total energy is the sum of the kinetic energy (T) and the Potential energy (V).
Thus, T = (1/2)mv2 and 
V is the energy due to electric attraction and is given by,
V = (e2/r2)dr
= - (e2/r)
Thus the total energy,
E = (1/2)mv2 - (e2/r)
= (1/2)mv2 - mv2
(where mv2 = e2/r)
= - (1/2)mv2
= - (1/2)(e2/r)
The energy associated with the permitted orbits is given by,
E = - (1/2)(e2/r)
= - (2π² me⁴/n2h2) = En=1/n2 [where En=1 = - (2π2me4/h2)]
The energy being governed by the value of quantum number n. As n increases the energy becomes less negative and hence the system becomes less stable. Also note that with increasing n, r also increases. Thus increasing r also makes the orbit less stable.
Thus if the energies associated with 1st, 2nd, 3rd, .... , nth orbits are E1, E2, E3 ... En, these will be in the order,
E1E2E3ㄑ........ㄑEn
Energy (E1) of the moving in the 1st Bohr orbit is obtained by putting n=1 in the energy expression of E.
Thus, E1 = - {2 × (3.14)2 × (9.109 × 10-28)×(4.8 × 10-10)4}/{12 × (6.6256 × 10-27)2
= - 21.79 × 10-12 erg
= - 13.6 eV
= - 21.79 × 10-19 Joule
= - 313.6 Kcal.
Calculate the kinetic energy of the electron in the first orbit of He+2. What will be the value if the electron is in the second orbit ?
Kinetic Energy = 1/2 mv2
= 1/2 m (2πZe2/nh)2
= 2π2mZ2e4/n2h2
Where, e = 4.8 × 10-10 esu,
Plank's Constant(h) = 6.626 × 10-27 erg sec and
m = 9.1 × 10-28 g
Kinetic Energy = 871 × 10-13 erg

The orbitals of a multi-electron atom are not likely to be quite the same as the hydrogen atom orbitals. For practical purpose, however, the number of orbitals and there shapes in multi-electron cases may be taken to be the same as for the hydrogen orbitals. In multi-electron atoms experimental studies of spectra shows that orbitals with the same vale of n but different l values have different energies.
The 3S orbital is lower energy then 3P orbitals which again are of lower energy then 3d orbitals. However orbitals belonging to a particular type ( P or d or f ) will be of equal energy (degenerate) in an atom or an ion. For example the three P orbitals or the five d orbitals originating from same n will be degenerate. The separation of orbitals of a major energy level into sub-levels is primarily due to the interaction among the many electrons.
This interaction leads to the following relative order of the energies of each type of orbitals:
1S2S〈2P3S〈3P4S〈3d 〈4P5S〈4d〈5P〈6S〈4f〈5d 6P〈7S〈5f〈6d
Admittedly it is often difficult for the readers to remember the orbital energy diagram . A trivial but distinctly more convenient way is to make give as,
Electronic Configuration of first thirty six Elements with Pictorial Representation.
Orbital Occupancy Order.
The different orbitals originating from the same principle quantum number n are written in the horizontal lines. Now inclined parallel lines are drown through the orbital according to the above picture. Filling up the different orbitals by electrons will follow these lines.
Example:
A element which contain 36 electrons the electronic configuration of this element according to the above diagram is,
1S22S22P63S23P64S23d104P6

The Aufbau or Building Up Principle:

The question that arise now how many electrons can be accommodated per orbital. The answer to this follows from Pauli's Exclusion Principle. 
  • Pauli's Exclusion Principle:
No two electrons in an atom can have the same four quantum numbers.
This principle tells us that in each orbital maximum of two electrons can be allowed. The two electron have the same three quantum numbers namely the same n, same l and the same ml. Any conflict with Pauli Principle can now be avoided if one of the electrons has the spin quantum numbers is (+1/2) and (-1/2).
An alternative statement of the Pauli's Exclusion Principle is , 
No more than two electrons can be placed in one and the same orbital. 
When two electrons with opposite spins exists in an orbital, the electrons are said to be paired. These two electrons per orbital given the maximum accommodation of electron in an atom.

Capacities Of Electronic Levels:

The total number of electrons for a particular n is given by 2n².
Principle Quantum Number Azimuthal Quantum Number(l) Total No. of Electrons
n l=0 l=1 l=2 l=3 l=4 2n2
n =1 2 2
n=2 2 6 8
n=3 2 6 10 18
n=4 2 6 10 14 32
n=5 2 6 10 14 18 50
To determine the electronic configuration of elements the procedure is to feed electrons in different orbitals obeying certain rules.
The Aufbau or Building up Principle is bases on the following Rules: 
  1. Electrons are fed into orbitals in order of increasing energy (increasing n ) until all the electrons have been accommodated.
  2. Electrons will tend to maintain maximum spin. So long orbitals of similar energy are available for occupation electrons will prefer to remain unpaired.
  3. In other words electrons tend to avoid the same orbital, that is, hate to share space. This rule is Known as Hund's Rule of maximum spin Multiplicity.
  4. Spin pairing can occur only when vacant orbitals of similar energy are not available for occupation, and when the next available vacant orbital is of higher energy.
Electronic Configuration of Elements:
  • Electronic Configuration of Elements H to Ne:
Hydrogen (Atomic Number 1) has its only one electron in the 1S orbital and this electronic configuration represented below. the bar on the pictorial representation indicates the orbital and the arrow a single spinning electron in the orbital. In helium (Atomic Number 2) the second electron occupy the 1S orbital since the next 2S orbital is much higher energy.
Obeying Pauli principle the configuration 1S2 represented below.
From the above rule we can represented the electronic configuration from H to Ne:
Hydrogen(H) 1S1
1S
Helium(He) 1S2
1S
Lithium(Li) 1S22S1
1S 2S
Beryllium(Be) 1S22S2
1S 2S
Boron(B) 1S22S12P1


1S 2S 2Px 2Py 2Pz
Carbon(C) 1S22S12P2

1S 2S 2Px 2Py 2Pz
Nitrogen(N) 1S22S12P3
1S 2S 2Px 2Py 2Pz
Oxygen(O) 1S22S12P4
1S 2S 2Px 2Py 2Pz
Fluorine(F) 1S22S12P5
1S 2S 2Px 2Py 2Pz
Neon(Ne) 1S22S12P6
1S 2S 2Px 2Py 2Pz
  • Electronic Configuration of Elements Na to Ar:
After neon the next available orbital is 3S being followed by 3P. The orbitals are then progressively filled by electrons. 
Thus the electronic configuration of Na to Ar given below.
Sodium(Na) 1S22S22P63S1
1S 2S 2Px 2Py 2Pz 3S
Magnesium(Mg) 1S22S22P63S2
1S 2S 2Px 2Py 2Pz 3S
Aluminium(Al) 1S22S22P63S23P1


1S 2S 2Px 2Py 2Pz 3S 3Px 3Py 3Pz
Silicon(Si) 1S22S22P63S23P2

1S 2S 2Px 2Py 2Pz 3S 3Px 3Py 3Pz
Phosphorus(P) 1S22S22P63S23P3
1S 2S 2Px 2Py 2Pz 3S 3Px 3Py 3Pz
Sulphur(S) 1S22S22P63S23P4
1S 2S 2Px 2Py 2Pz 3S 3Px 3Py 3Pz
Chlorine(Cl) 1S22S22P63S23P5
1S 2S 2Px 2Py 2Pz 3S 3Px 3Py 3Pz
Argon(Ar) 1S22S22P63S23P6
1S 2S 2Px 2Py 2Pz 3S 3Px 3Py 3Pz
  • Electronic Configuration of Elements K and Ca:
Then 4S orbital, being of lower energy then the 3d, is filled. the elements involves potassium and calcium are represented as,
Potassium(K) [Ar]184S1

[Ar]20 4S
Calcium(Ca) [Ar]184S2

[Ar]20 4S
  • Electronic Configuration of Elements Sc and Zn:
In Scandium(21) the twenty first electron goes to the 3d orbital, the next available orbital of the higher energy. There are five 3d orbital with the capacity of ten electrons. From scandium to Zinc these 3d orbitals are filled up.
Presence of partially filled d orbitals generates some some special properties of the elements. Elements with partially filled or f orbitals in the elementary state or ionic state are called transition elements.
Scandium(Sc) [Ar]184S23d1





[Ar]20 4S 3d 3d 3d 3d 3d
Titanium(Ti) [Ar]184S23d2




[Ar]20 4S 3d 3d 3d 3d 3d
Vanadium(V) [Ar]184S23d3



[Ar]20 4S 3d 3d 3d 3d 3d
Chromium(Cr) [Ar]184S23d4


[Ar]20 4S 3d 3d 3d 3d 3d
In reality experimental studies on Chromium revel their electronic configuration and better represented as:
Chromium(Cr) [Ar]184S13d5

[Ar]20 4S 3d 3d 3d 3d 3d
This reordering of electrons is due to an extra stability associated with a half filled sub-shell.
Manganese(Mn) [Ar]184S23d5

[Ar]20 4S 3d 3d 3d 3d 3d
Iron(Fe) [Ar]184S23d6

[Ar]20 4S 3d 3d 3d 3d 3d
Cobalt(Co) [Ar]184S23d7

[Ar]20 4S 3d 3d 3d 3d 3d
Nickel(Ni) [Ar]184S23d8

[Ar]20 4S 3d 3d 3d 3d 3d
Copper(Cu) [Ar]184S23d9

[Ar]20 4S 3d 3d 3d 3d 3d
Experimental Studies on Copper also reveals that their electronic configuration and are better represented as:
Copper(Cu) [Ar]184S13d10

[Ar]20 4S 3d 3d 3d 3d 3d
This reordering of electrons is due to an extra stability associated with a filled sub-shell.
Zinc(Zn) [Ar]184S23d10

[Ar]20 4S 3d 3d 3d 3d 3d
Note: These 3d orbital are represented as dxy, dxz, dyz, d, dx²-y².

  • Electronic Configuration of Elements Ga and Kr:
In Gallium(31) the thirty first electron goes to the 4P orbital, the next available orbital of the higher energy. There are three 4P orbital with the capacity of six electrons. From Gallium to Krypton these 4P orbitals are filled up.
Gallium(Ga) [Ar]184S23d104P1



[Ar]18 4S 3d 3d 3d 3d 3d 4P 4P 4P
Germanium(Ge) [Ar]184S23d104P2


[Ar]18 4S 3d 3d 3d 3d 3d 4P 4P 4P
Arsenic(As) [Ar]184S23d104P3

[Ar]18 4S 3d 3d 3d 3d 3d 4P 4P 4P
Selenium(Se) [Ar184S23d104P4

[Ar]18 4S 3d 3d 3d 3d 3d 4P 4P 4P
Bromine(Br) [Ar]184S23d104P5

[Ar]18 4S 3d 3d 3d 3d 3d 4P 4P 4P
Krypton(Kr) [Ar]184S23d104P6

[Ar]18 4S 3d 3d 3d 3d 3d 4P 4P 4P

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