Quantum Number Electronic Configuration and Shape of Orbitals
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Quantum Number, Electronic Configuration and Shape of Orbitals
Quantum numbers and Shapes of orbitals.
Quantum numbers
(1) Each orbital in an atom is specified by a set of three quantum numbers (n, l, m) and each electron is designated by a set of four quantum numbers (n, l, m and s).
(2) Principle quantum number (n)
(i) It was proposed by Bohr’s and denoted by ‘n’.
(ii) It determines the average distance between electron and nucleus, means it is denoted the size of atom.
\[r=\frac{{{n}^{2}}}{Z}\times 0.529{\AA}\]
(iii) It determine the energy of the electron in an orbit where electron is present.
\[E=-\frac{{{Z}^{2}}}{{{n}^{2}}}\times 313.3\,Kcal\,per\,mole\]
(iv) The maximum number of an electron in an orbit represented by this quantum number as \[2{{n}^{2}}.\] No energy shell in atoms of known elements possess more than 32 electrons.
(v) It gives the information of orbit K, L, M, N------------.
(vi) The value of energy increases with the increasing value of n.
(vii) It represents the major energy shell or orbit to which the electron belongs.
(viii) Angular momentum can also be calculated using principle quantum number
\[mvr=\frac{nh}{2\pi }\]
(3) Azimuthal quantum number (l)
(i) Azimuthal quantum number is also known as angular quantum number. Proposed by Sommerfield and denoted by ‘l’.
(ii) It determines the number of sub shells or sublevels to which the electron belongs.
(iii) It tells about the shape of subshells.
(iv) It also expresses the energies of subshells \[s<p<d<f\] (increasing energy).
(v) The value of \[l=(n-1)\] always where ‘n’ is the number of principle shell.
|
(vi) Value of l |
= |
0 |
1 |
2 |
3???..(n-1) |
|
Name of subshell |
= |
s |
p |
d |
f |
|
Shape of subshell |
= |
Spherical |
Dumbbell |
Double dumbbell |
Complex |
(vii) It represent the orbital angular momentum. Which is equal to \[\frac{h}{2\pi }\sqrt{l(l+1)}\]
(viii) The maximum number of electrons in subshell \[=2(2l+1)\]
\[s-\text{subshell}\to 2\,\text{electrons}\] \[d-\text{subshell}\to 10\,\text{electrons}\]
\[p-\text{subshell}\to \text{6}\,\text{electrons}\] \[f-\text{subshell}\to 14\,\text{electrons}\text{.}\]
(ix) For a given value of ‘n’ the total value of ‘l’ is always equal to the value of ‘n’.
(x) The energy of any electron is depend on the value of n & l because total energy = (n + l). The electron enters in that sub orbit whose (n + l) value or the value of energy is less.
(4) Magnetic quantum number (m)
(i) It was proposed by Zeeman and denoted by ‘m’.
(ii) It gives the number of permitted orientation of subshells.
(iii) The value of m varies from –l to +l through zero.
(iv) It tells about the splitting of spectral lines in the magnetic field i.e. this quantum number proved the Zeeman effect.
(v) For a given value of ‘n’ the total value of ’m’ is equal to \[{{n}^{2}}.\]
(vi) For a given value of ‘l’ the total value of ‘m’ is equal to\[(2l+1).\]
(vii) Degenerate orbitals : Orbitals having the same energy are known as degenerate orbitals. e.g. for p subshell \[{{p}_{x}}\,\,{{p}_{y}}\,\,{{p}_{z}}\]
(viii) The number of degenerate orbitals of s subshell =0.
(5) Spin quantum numbers (s)
(i) It was proposed by Goldshmidt & Ulen Back and denoted by the symbol of ‘s’.
(ii) The value of \['s'\ \,\text{is }+\text{1/2}\,\ \text{and-1/2,}\]which is signifies the spin or rotation or direction of electron on it’s axis during movement.
(iii) The spin may be clockwise or anticlockwise.
(iv) It represents the value of spin angular momentum is equal to \[\frac{h}{2\pi }\sqrt{s(s+1)}.\]
(v) Maximum spin of an atom \[=1/2\times \]number of unpaired electron.
(vi) This quantum number is not the result of solution of schrodinger equation as solved for H-atom.
Distribution of electrons among the quantum levels
|
n |
l |
m |
s |
Designation of orbitals |
Electrons present |
Total no. of electrons |
|
1 (K shell) |
0 |
0 |
+1/2, –1/2 |
1s |
2 |
2 |
|
2 (L shell) |
0
1
|
0 +1 0 –1 |
\[\begin{align} & +1/2,\,-1/2 \\ & +1/2,\,-1/2 \\ & +1/2,\,-1/2 \\ & +1/2,\,-1/2 \\ \end{align}\] |
2s
2p
|
\[\left. \begin{align} & 2 \\ & \\ & 6 \\ \end{align} \right]\] |
8
|
|
3 (M shell)
4(N shell)
|
0
1
2
0
1
2
3
|
0 +1 0 –1
+2 +1 0 –1 –2 0 +1 0 –1
+2 +1 0 –1 –2
+3 +2 +1 +0 –1 –2 –3 |
\[\begin{align} & +1/2,-1/2 \\ & +1/2,-1/2 \\ & +1/2,-1/2 \\ \end{align}\] \[+1/2,-1/2\]
\[\left. \begin{align} & +1/2,-1/2 \\ & +1/2,-1/2 \\ & +1/2,-1/2 \\ & +1/2,-1/2 \\ & +1/2,-1/2 \\ \end{align} \right]\]
\[\begin{align} & +1/2,-1/2 \\ & \left. \begin{align} & +1/2,-1/2 \\ & +1/2,-1/2 \\ & +1/2,-1/2 \\ \end{align} \right] \\ \end{align}\]
\[\left. \begin{align} & +1/2,-1/2 \\ & +1/2,-1/2 \\ & +1/2,-1/2 \\ & +1/2,-1/2 \\ & +1/2,-1/2 \\ \end{align} \right]\]
\[\left. \begin{align} & +1/2,-1/2 \\ & +1/2,-1/2 \\ & +1/2,-1/2 \\ & +1/2,-1/2 \\ & +1/2,-1/2 \\ & +1/2,-1/2 \\ & +1/2,-1/2 \\ \end{align} \right]\] |
3s
3p
3d
4s
4p
4d
4f
|
\[\left. \begin{align} & 2 \\ & \\ & 6 \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & 10 \\ \end{align} \right]\]
\[\left. \begin{align} & 2 \\ & \\ & 6 \\ & \\ & \\ & \\ & \\ & \\ & \\ & 10 \\ & \\ & \\ & \\ & \\ & \\ & \\ & 14 \\ \end{align} \right]\] |
18
32
|
Shape of orbitals
(1) Shape of ‘s’ orbital
(i) For ‘s’ orbital l=0 & m=0 so ‘s’ orbital have only one unidirectional orientation i.e. the probability of finding the electrons is same in all directions.
(ii) The size and energy of ‘s’ orbital with increasing ‘n’ will be \[1s<2s<3s<4s.\]
(iii) It does not possess any directional property. s orbital has spherical shape.
(2) Shape of ‘p’ orbitals
(i) For ‘p’ orbital l=1, & m=+1,0,–1 means there are three ‘p’ orbitals, which is symbolised as \[{{p}_{x}},{{p}_{y}},{{p}_{z}}.\]
(ii) Shape of ‘p’ orbital is dumb bell in which the two lobes on opposite side separated by the nodal plane.
(iii) p-orbital has directional properties.
(3) Shape of ‘d’ orbital
(i) For the ‘d’ orbital l =2 then the values of ‘m’ are –2,–1,0,+1,+2. It shows that the ‘d’ orbitals has five orbitals as \[{{d}_{xy}},{{d}_{yz}},{{d}_{zx}},\,{{d}_{{{x}^{2}}-{{y}^{2}}}},{{d}_{{{z}^{2}}.}}\]
(ii) Each ‘d’ orbital identical in shape, size and energy.
(iii) The shape of d orbital is double dumb bell.
(iv) It has directional properties.
(4) Shape of ‘f’ orbital
(i) For the ‘f’ orbital l=3 then the values of ‘m’ are –3, –2, –1,0,+1,+2,+3. It shows that the ‘f’ orbitals have seven orientation as \[{{f}_{x({{x}^{2}}-{{y}^{2}})}},{{f}_{y({{x}^{2}}-{{y}^{2}})}},{{f}_{z({{x}^{2}}-{{y}^{2}}),}}{{f}_{xyz}},{{f}_{{{z}^{3}}}},{{f}_{y{{z}^{3}}}}\,\text{and}\,{{f}_{x{{z}^{2}}}}.\]
(ii) The ‘f’ orbital is complicated in shape.
Electronic configuration principles.
The distribution of electrons in different orbitals of atom is known as electronic configuration of the atoms.
Filling up of orbitals in the ground state of atom is governed by the following rules:
(1) Auf bau principle
(i) Auf bau is a German word, meaning ‘building up’.
(ii) According to this principle, “In the ground state, the atomic orbitals are filled in order of increasing energies i.e. in the ground state the electrons first occupy the lowest energy orbitals available”.
(iii) In fact the energy of an orbital is determined by the quantum number n and l with the help of (n+l) rule or Bohr Bury rule.
(iv) According to this rule
(a) Lower the value of n + l, lower is the energy of the orbital and such an orbital will be filled up first.
(b) When two orbitals have same value of (n+l) the orbital having lower value of “n” has lower energy and such an orbital will be filled up first.
Thus, order of filling up of orbitals is as follows:
\[1s<2s<2p<3s<3p<4s<4p<5s<4d<5p<6s<6f<5d\]
(2) Pauli’s exclusion principle
(i) According to this principle, “No two electrons in an atom can have same set of all the four quantum numbers n, l, m and s.
(ii) In an atom any two electrons may have three quantum numbers identical but fourth quantum number must be different.
(iii) Since this principle excludes certain possible combinations of quantum numbers for any two electrons in an atom, it was given the name exclusion principle. Its results are as follows:
(a) The maximum capacity of a main energy shell is equal to \[2{{n}^{2}}\]electron.
(b) The maximum capacity of a subshell is equal to 2(2l+1) electron.
(c) Number of sub-shells in a main energy shell is equal to the value of n.
(d) Number of orbitals in a main energy shell is equal to\[{{n}^{2}}.\]
(e) One orbital cannot have more than two electrons.
(iv) According to this principle an orbital can accomodate at the most two electrons with spins opposite to each other. It means that an orbital can have 0, 1, or 2 electron.
(v) If an orbital has two electrons they must be of opposite spin.
(3) Hund’s Rule of maximum multiplicity
(i) This rule provides the basis for filling up of degenerate orbitals of the same sub-shell.
(ii) According to this rule “Electron filling will not take place in orbitals of same energy until all the available orbitals of a given subshell contain one electron each with parallel spin”.
(iii) This implies that electron pairing begins with fourth, sixth and eighth electron in p, d and f orbitals of the same subshell respectively.
(iv) The reason behind this rule is related to repulsion between identical charged electron present in the same orbital.
(v) They can minimise the repulsive force between them serves by occupying different orbitals.
(vi) Moreover, according to this principle, the electron entering the different orbitals of subshell have parallel spins. This keep them farther apart and lowers the energy through electron exchange or resonance.
(vii) The term maximum multiplicity means that the total spin of unpaired \[{{e}^{-}}\]is maximum in case of correct filling of orbitals as per this rule.
Energy level diagram
The representation of relative energy levels of various atomic orbital is made in the terms of energy level diagrams.
One electron system : In this system \[1{{s}^{2}}\] level and all orbital of same principal quantum number have same energy, which is independent of (l). In this system l only determines the shape of the orbital.
Multiple electron system : The energy levels of such system not only depend upon the nuclear charge but also upon the another electron present in them.
Diagram of multi-electron atoms reveals the following points:
(i) As the distance of the shell increases from the nucleus, the energy level increases. For example energy level of 2 > 1.
(ii) The different sub shells have different energy levels which possess definite energy. For a definite shell, the subshell having higher value of l possesses higher energy level. For example in 4th shell.
Energy level order 4f > 4d > 4p > 4s
l= 3 l = 2 l = 1 l= 0
(iii) The relative energy of sub shells of different energy shell can be explained in the terms of the (n+l) rule.
(a) The sub-shell with lower values of (n + l) possess lower energy.
For 3d n = 3 l= 2 \[\therefore \] n + l = 5
For 4s n = 4 l = 0 n + l = 4
(b) If the value of (n + l) for two orbitals is same, one with lower values of ‘n’ possess lower energy level.
Extra stability of half filled and completely filled orbitals
Half-filled and completely filled sub-shell have extra stability due to the following reasons :
(i) Symmetry of orbitals
(a) It is a well kown fact that symmetry leads to stability.
(b) Thus, if the shift of an electron from one orbital to another orbital differing slightly in energy results in the symmetrical electronic configuration. It becomes more stable.
(c) For example \[{{p}^{3}},{{d}^{5}},{{f}^{7}}\]configurations are more stable than their near ones.
(ii) Exchange energy
(a) The electron in various subshells can exchange their positions, since electron in the same subshell have equal energies.
(b) The energy is released during the exchange process with in the same subshell.
(c) In case of half filled and completely filled orbitals, the exchange energy is maximum and is greater than the loss of orbital energy due to the transfer of electron from a higher to a lower sublevel e.g. from 4s to 3d orbitals in case of Cu and Cr .
(d) The greater the number of possible exchanges between the electrons of parallel spins present in the degenerate orbitals, the higher would be the amount of energy released and more will be the stability.
(e) Let us count the number of exchange that are possible in \[{{d}^{4}}\]and \[{{d}^{5}}\]configuraton among electrons with parallel spins.
To number of possible exchanges = 3 + 2 + 1 =6
To number of possible exchanges = 4 + 3 + 2 +1 = 10
Electronic configurations of Elements.
(1) On the basis of the elecronic configuration priciples the electronic configuration of various elements are given in the following table :
Electronic Configuration (E.C.) of Elements Z=1 to 36
| Element |
Atomic Number |
1s |
2s |
2p |
3s |
3p |
3d |
4s |
4p |
4d |
4f |
|
H |
1 |
1 |
|
|
|
|
|
|
|
|
|
|
He |
2 |
2 |
|
|
|
|
|
|
|
|
|
|
Li |
3 |
2 |
1 |
|
|
|
|
|
|
|
|
|
Be |
4 |
2 |
2 |
|
|
|
|
|
|
|
|
|
B |
5 |
2 |
2 |
1 |
|
|
|
|
|
|
|
|
C |
6 |
2 |
2 |
2 |
|
|
|
|
|
|
|
|
N |
7 |
2 |
2 |
3 |
|
|
|
|
|
|
|
|
O |
8 |
2 |
2 |
4 |
|
|
|
|
|
|
|
|
F |
9 |
2 |
2 |
5 |
|
|
|
|
|
|
|
|
Ne |
10 |
2 |
2 |
6 |
|
|
|
|
|
|
|
|
Na |
11 |
2 |
2 |
6 |
1 |
|
|
|
|
|
|
|
Mg |
12 |
|
|
|
2 |
|
|
|
|
|
|
|
Al |
13 |
|
|
|
2 |
1 |
|
|
|
|
|
|
Si |
14 |
10 electrons |
2 |
2 |
|
|
|
|
|
||
|
P |
15 |
2 |
3 |
|
|
|
|
|
|||
|
S |
16 |
|
|
|
2 |
4 |
|
|
|
|
|
|
Cl |
17 |
|
|
|
2 |
5 |
|
|
|
|
|
|
Ar |
18 |
2 |
2 |
6 |
2 |
6 |
|
|
|
|
|
|
K |
19 |
2 |
2 |
6 |
2 |
6 |
|
1 |
|
|
|
|
Ca |
20 |
|
|
|
|
|
|
2 |
|
|
|
|
Sc |
21 |
|
|
|
|
|
1 |
2 |
|
|
|
|
Ti |
22 |
|
|
|
|
|
2 |
2 |
|
|
|
|
V |
23 |
|
|
|
|
|
3 |
2 |
|
|
|
|
Cr |
24 |
|
|
|
|
|
5 |
1 |
|
|
|
|
Mn |
25 |
|
|
|
|
|
5 |
2 |
|
|
|
|
Fe |
26 |
|
|
|
|
|
6 |
2 |
|
|
|
|
Co |
27 |
18 electrons |
|
|
7 |
2 |
|
|
|
||
|
Ni |
28 |
|
|
8 |
2 |
|
|
|
|||
|
Cu |
29 |
|
|
|
|
|
10 |
1 |
|
|
|
|
Zn |
30 |
|
|
|
|
|
10 |
2 |
|
|
|
|
Ga |
31 |
|
|
|
|
|
10 |
2 |
1 |
|
|
|
Ge |
32 |
|
|
|
|
|
10 |
2 |
2 |
|
|
|
As |
33 |
|
|
|
|
|
10 |
2 |
3 |
|
|
|
Se |
34 |
|
|
|
|
|
10 |
2 |
4 |
|
|
|
Br |
35 |
|
|
|
|
|
10 |
2 |
5 |
|
|
|
Kr |
36 |
2 |
2 |
6 |
2 |
6 |
10 |
2 |
6 |
|
|
(2) The above method of writing the electronic configurations is quite cumbersome. Hence, usually the electronic configuration of the atom of any element is simply represented by the notation.
(3) (i) Elements with atomic number 24(Cr), 42(Mo) and 74(W) have \[n{{s}^{1}}(n-1)\,{{d}^{5}}\] configuration and not \[n{{s}^{2}}(n-1)\,{{d}^{4}}\] due to extra stability of these atoms.
(ii) Elements with atomic number 29(Cu), 47(Ag) and 79(Au) have \[n{{s}^{1}}(n-1)\,{{d}^{10}}\] configuration instead of \[n{{s}^{2}}(n-1)\,{{d}^{9}}\] due to extra stability of these atoms.
(4) In the formation of ion, electrons of the outer most orbit are lost. Hence, whenever you are required to write electronic configuration of the ion, first write electronic configuration of its atom and take electron from outermost orbit. If we write electronic configuration of Fe\[^{2+}(Z=26,\,\,24\,{{e}^{-}}),\] it will not be similar to Cr (with \[24\,{{e}^{-}}\]) but quite different.
\[\left. \begin{align}
& \,\,\,\,\,Fe\left[ Ar \right]\,\,4{{s}^{2}}\,3{{d}^{6}} \\
& F{{e}^{2+}}\left[ Ar \right]\,\,4{{s}^{\circ }}\,3{{d}^{6}} \\
\end{align} \right\}\,\,\]outer most orbit is 4th shell hence, electrons from 4s have been removed to make \[F{{e}^{2+}}\].
(5) Ion/atom will be paramagnetic if there are unpaired electrons. Magnetic moment (spin only) is \[\mu =\sqrt{n(n+2)}\] BM (Bohr Magneton).\[(1BM=9.27\times {{10}^{-24}}J/T)\] where n is the number of unpaired electrons.
(6) Ion with unpaired electron in \[d\] or \[f\]orbital will be coloured. Thus, \[C{{u}^{+}}\] with electronic configuration \[\left[ Ar \right]\,3{{d}^{10}}\] is colourless and \[C{{u}^{2+}}\] with electronic configuration \[\left[ Ar \right]\,3{{d}^{9}}\](one unpaired electron in 3d) is coloured (blue).
(7) Position of the element in periodic table on the basis of electronic configuration can be determined as,
(i) If last electron enters into s-subshell, p-subshell, penultimate d-subshell and anti-penultimate f-subshell then the element belongs to s, p, d and f – block respectively.
(ii) Principle quantum number (n) of outermost shell gives the number of period of the element.
(iii) If the last shell contains 1 or 2 electrons (i.e. for s-block elements having the configuration\[n{{s}^{1-2}}\]), the group number is 1 in the first case and 2 in the second case.
(iv) If the last shell contains 3 or more than 3 electrons (i.e. for p-block elements having the configuration\[n{{s}^{2}}\,n{{p}^{1-6}}\]), the group number is the total number of electrons in the last shell plus 10.
(v) If the electrons are present in the (n –1)d orbital in addition to those in the ns orbital (i.e. for d-block elements having the configuration (n –1) \[{{d}^{1-10}}n{{s}^{1-2}}\]), the group number is equal to the total number of electrons present in the (n –1)d orbital and ns orbital.
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