Structure of Atom
Class11ScienceChemistryChapter 4 Maharashtra State Board
Notes Part2
Topics to be Learn : Part2

Quantum mechanical model of atom :
Quantum machanics, developed in 1926 by Werner Heisenberg and Erwin Schrodinger based on uncertainty principle and wave motion, respectively.
An atomic model developed on the basis of quantum mechanics is called quantum mechanical model. Quantum mechanics is a theoretical science that deals with microscopic particles that have wave like and particle like properties.
A fundamental equation of quantum mechanics called Schrodinger equation is represented as, \(\cap{H}Ψ\) = E Ψ where \(\cap{H}\) is a mathematical operator called Hamiltonian, Ψ is a wave function and E is the total energy of the system.
Schrodinger equation : When Schrodinger equation is solved for hydrogen atom, the possible values of energy (E) that the electron may have along with the corresponding wave function (ψ) are obtained.
As a natural consequence of solving this equation, a set of three quantum numbers characteristic of the quantized energy levels and the corresponding wave functions are obtained. These are :
 Principal quantum number (n),
 azimuthal quantum number (l) and
 magnetic quantum number (m_{l}).
Splitting of spectral lines in multielectron atomic emission spectra could not be explained through above model. These were explained by George Uhlenbeck and Samuel Goudsmit (1925) who proposed the presence of the fourth quantum number called electron spin quantum number, m_{s}.
Principal quantum number (n) : This describes the orbit or shell of an atom to which the electron belongs. It is represented by ‘n’ which has integral values. The energy of an electron depends on the value of n.
Principal quantum number n  Shell symbol  Allowed number of orbitals n^{2}  Size of shell 
1  K  1  
2  L  4  
3  M  9  
4  N  16 
Azimuthal quantum number (l) : This represents the subsidiary quantum number or the subshell of the orbit to which the electron belongs. It is denoted by l. It is also called as secondary, subsidiary, orbital or angular momentum quantum number. It defines the shape of the orbital and the angular momentum of the electron. The value of l depends on principal quantum number n. It has positive values between 0 to (n — 1) :
l  0  1  2  3 
subshell  s  p  d  f 
This explains the significance of azimuthal quantum number.
Mgnetic orbital quantum number (m_{l}) : This gives the information about the relative spatial orientation of the orbitals in a given subshell defined by ’l’. For any subshell total orientations possible are (2l + 1) which range through :
m_{l} = −l, − (l−1),...O .... ..(l−1),l
The total values are 2l + 1.
The sum of orbitals in a subshell gives the total number of orbitals in a shell and is given by n^{2}.
Value of l  orbital  m_{l} values 
0  s  0 
1  p  −1, 0, +1 
2  d  −2,−1, 0, +1, +2 
3  f  −3,−2,−1, 0, +1, +2, +3 
Electron spin quantum number, (m_{s}) : An electron spins around its axis clockwise or anticlockwise and imparts spin angular momentum to it. The two spin states are represented as ↑ or ↓ with m_{s} values \(+\frac{1}{2}\) or \(\frac{1}{2}\) respectively.
An orbital can accommodate maximum two electrons and they must have opposite spins.
Subshells :
There are four subshells as follows :
(1) ssubshell : For this subshell l= 0, m_{l} = 0. This implies that ssubshell has only one orbital i.e., sorbital.
(2) pSubshell : For this subshell, l = 1, m = +1, 0, −1. Hence psubshell has three orbitals represented as p_{x}, p_{y} and p_{z} and they are respectively oriented along x, y and z axes. Therefore porbitals have threefold degeneracy.
(3) dsubshell : For this subshell, l =2, m = +2, +1, 0, −1, −2. Hence, dsubshell has five orbitals with ﬁve different orientations. They are designated as d_{xy}, d_{yz} d_{xz}, which are planar orbitals and , are axial orbitals. Therefore dorbitals have fivefold degeneracy.
(4) fsubshell : For this subshell, l = 3, m = +3, + 2, + 1, 0, — 1, — 2, —3. Hence fsubshell has seven orbitals with seven orientations in space.
They are designated as Therefore forbitals have seven fold degeneracy.
Distribution of orbitals in shells and subshells :
Below Table shows orbitals in the first four shells with the three quantum numbers for each orbital.
Q. How many orbitals make the N shell? What is the subshell wise distribution of orbitals in the N shell?
Ans : For N shell principal quantum number n =4. Hence total number of orbitals in N shell = n^{2} = 4^{2} = 16. The total number of subshells in N shell = n = 4. The four subshells with their azimuthal quantum numbers and the constituent orbital number are as shown below :
Azimuthal quantum number l  Symbol of subshell  Number of orbitals 2l + 1 
l = 0  s  (2 x 0) + 1 = 1 
l =1  p  (2 x 1) + 1 = 3 
l = 2  d  (2 x 2) + 1 = 5 
l = 3  f  (2 x 3) + 1 = 7 
Shapes of atomic orbitals :
Orbital : The three dimensional region in the space around the nucleus of an atom in which the probability of finding the electron is maximum is called orbital.
Shapes of 1s, 2s orbitals :
Figure shows the boundary surface diagram of atomic orbitals 1s and 2s, which are spherical in shape. Here, a boundary surface is drawn in space for an orbital such that the value of probability density Ψ^{2} is constant and encloses a region where the probability of ﬁnding electron is typically more than 90%. Such a boundary surface diagram is a good representation of shape of an orbital.
porbital :
 This orbital has Azimuthal quantum number, l = 1 and magnetic quantum numbers m = + 1, 0, −1.
 Hence porbital has three orientations along three coordinates namely, x, y and z. Therefore, porbitals are designated as p_{x}, p_{y} and P_{z} and they determine the geometry of molecules.
 The shape of porbital resembles a dumbbell. Hence porbital has two lobes separated by a nodal plane having zero electron density.
 These three porbitals are degenerate, i.e., they are equivalent in energy in the absence of an external magnetic ﬁeld.
 The size and energy of porbitals increase with the increase in principal energy level or n, in the order of 2p < 3p < 4p.
dorbital :
 dorbital has quantum numbers, l = 2, m = + 2, + 1, 0, — 1, — 2,
 Hence dorbitals have five orientations. They are designated as d_{xy}, d_{yz}, d_{xz},
 The shapes of d_{xy}, d_{yz} and d_{zx} orbitals resembles double dumbbell in xy, yz and xz planes.
 The shape of is also of dumbbell but orbital lies along x and y axes.
 is dumbbell shaped orbital with a doughnut shaped ring of high electron density around the nucleus in xy plane.
The shapes of 3d, 4d, 5d ..... orbitals are similar but their respective size and energies are large or they are said to be more diffused.
Energies of orbitals :
The energy of an electron in the hydrogen atom or hydrogen like species is determined by the principal quantum number alone. This is because the only interaction in these species is attraction between the electron and nucleus.
Degenerate orbitals : The orbitals with the same energy and the corresponding wave functions being different are called degenerate orbitals. In hydrogen atom 2s and 2p are degenerate orbitals.
n + l rule : The energy of an electron in a multielectron atom, depends both on the principal quantum number, n, and the azimuthal quantum number, l. The lower the sum (n + 1) for an orbital, the lower is its energy. If two orbitals have the same (n +1) values then orbital with the lower value of n is of lower energy. This is called the (n +1) rule.
From the (n +1) rule the increasing order of energy of orbitals in multielectron atoms can be written as :
1s<2s<2p<3s<3p<4s<3d<4p<5s<4d<5p<6s<4f<5d<6p<7s .......
Ground state of an atom : It is a state of the lowest total electronic energy. For example, of hydrogen atom it is 1s state.
Aufbau principle : ‘
Aufbau is a German word which means building up. This principle explains the sequence of ﬁlling up of orbitals with electrons.
Aufbau principle : It states that in the ground state of an atom, the orbitals are filled with electrons in order of the increasing energies.
The orbitals are filled in order of increasing value of (n + l). For example, 4sorbital
(n + l = 4 + 0 = 4) is filled prior to 3dorbital (n + l = 3 + 2 = 5).
Among two orbitals having same (n + l) value, that orbital with lower value of n will be ﬁlled first. For example, 3dorbital (n + l = 3 + 2 = 5) is filled prior to 4porbital (n + l = 4 + 1 = 5).
The increasing order of energy of different orbitals is as follows :
1s<2s<2p<3s<3p<4sz<3d<4p<5s<4d<5p<6s<4f<5d<6p<7s<5f<6d
Pauli exclusion principle : The capacity of an orbital to accommodate electrons is decided by Pauli exclusion principle.
Statement of Pauli’s principle : No two electrons in an atom can have all the four quantum numbers, (n, l, m and s) same.
OR
Only two electrons may exist in the given orbital having three quantum numbers same but fourth quantum number being different with opposite spins.
This principle describes the capacity of a subshell or orbital to accommodate maximum number of electrons.
Consider helium atom, which has two electrons. The four quantum numbers of two electrons in He atom will be,
This principle is illustrated with helium atom He (Z = 2). Its electronic configuration is 1s^{2} as
Hund’s rule of maximum multiplicity :
Statement of Hund’s rule of maximum multiplicty : It states that pairing of electrons in the orbitals belonging to the same subshell does not occur unless each orbital belonging to that subshell has accommodated one electron each.
 Consider filling of psubshell which has three degenerated orbitals namely
After filling three electrons, one in each with same spins, the next electrons enter with pairing
It observed that half ﬁlled and completely filled set of degenerate orbitals have extra stability.
Electronic configuration of atoms and its representation :
Electronic conﬁguration of an atom is the distribution of its electrons in orbitals.
The electronic configuration can be written by applying the aufbau principle.
There are two methods of representing electronic configuration:
(i) Orbital notation: ns^{a} np^{b} nd^{c}..........
For example for boron atom _{5}B 1s^{2}, 2s^{2}, 2p^{1}.
(ii) Orbital diagram :
Condensed orbital notation of electronic conﬁguration :
 The orbital notation of electronic configuration of an element with high atomic number can be condensed by dividing it into two parts.
 The electronic configuration of the preceding inert gas is a part of the electronic configuration of every element.
 In the condensed orbital notation inert gas is mentioned by writing the symbol of that inert gas in a square bracket and outside remaining configuration is written immediately after the bracket.
For example,
Consider an element _{19}K. Its electronic conﬁguration is 1s^{2} 2s^{2} 2p^{6} 3s^{2} 3p^{6} 4s^{1}
The preceding inert element Argon has electronic conﬁguration, _{18}Ar 1s^{2} 2s^{2} 2p^{6} 3s^{2} 3p^{6}
Hence the condensed orbital notation for potassium is _{19}K : [Ar] 4s^{1}
Orbital diagram method :
In the orbital diagram method each orbital in a subshell is represented by a box and the electron represented by an arrow (↑for up spin and ↓for low spin) placed in the respective boxes.
Electronic configurations of Cu and Cr :
(i) Copper (Cu) :
Copper (_{29}Cu) has electronic configuration.
_{29}Cu (Expected) : 1s^{2}, 2s^{2}, 2p^{6}, 3s^{2}, 3p^{6}, 3d^{9}, 4s^{2}
(Observed) : 1s^{2}, 2s^{2}, 2p^{6}, 3s^{2}, 3p^{6}, 3d^{10}, 4s^{1}
Explanation :
 The energy difference between the 3d and 4sorbitals is very low.
 dorbitals being degenerate, they acquire more stability when they are completely filled.
 Therefore, there arises a transfer of one electron from 4sorbital to 3dorbital in Cu giving completely filled more stable dorbital.
 Hence, the configuration of Cu is [Ar] 3d^{10} 4s^{1} and not [Ar] 3d^{9} 4s^{2}.
 This shows anomalous behaviour of copper.
(ii) Chromium (Cr) :
Chromium (_{24}Cr) has electronic conﬁguration,
_{24}Cr (Expected) : 1s^{2}, 2s^{2}, 2p^{6}, 3s^{2}, 3p^{6}, 3d^{4}, 4s^{2}
(Observed) : 1s^{2}, 2s^{2}, 2p^{6}, 3s^{2}, 3p^{6}, 3d^{5}, 4s^{1}
Explanation :
 The energy difference between 3d and 4sorbitals is very low.
 dorbitals being degenerate, they acquire more stability when they are halfﬁlled (3d^{5})
 Therefore, there arises a transfer of one electron from 4sorbital to 3dorbital in Cr giving more stable halffilled orbital. Hence, the configuration of Cr is [Ar] 3d^{5} 4s^{1} and not [Ar] 3d^{4} 4s^{2}.
 This shows anomalous behaviour of chromium.
Isoelectronic species : Atoms and ions having the same number of electrons are isoelectronic species.
Example :
Consider K^{+} formed by removal of one electron from K atom. Which has 19 electrons (Z = 19). Therefore K^{+} has 18 electrons.
Species such as Ar, Ca^{2+} containing 18 electrons are isoelectronic with K^{+} .
Electronic configuration of all these species with 18 electrons is 1s^{2}, 2s^{2}, 2p^{6}, 3s^{2}, 3p^{6}.
Species  K→ K^{+ }+ e^{−}  Ar, Ca^{2+} 
Number of electrons  19 18  18 18 
Electronic configuration of the first fifteen elements
Important formulae
Energy of an electron (E) = eV Wave number of a spectral line, \(\bar{v}=\frac{1}{λ}=R(\frac{1}{n_{1}^{2}}\frac{1}{n_{2}^{2}})\) Rydberg’s constant = R = 1.09677 x 10^{7} m^{−}^{1} Quantum numbers of an electron : n, l, m_{1} , m_{2} For photoelectric effect : hv = hv_{o} + ½ mv^{2 } De Broglie equation : λ = h/mv Heisenberg’s uncertainty principle : Δx x Δp ≥ h/4π Schrodinger’s wave equation : \(\frac{d^2ψ}{dx^2}+\frac{d^2ψ}{dy^2}+\frac{d^2ψ}{dz^2}+\frac{8π^2m}{h^2}(EV)φ=0\) 
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