+2 UNIT 1 PAGE- 2
QUANTUM NUMBERS
In an atom a large number of electron orbitals are permissible. These orbitals are designated by a set of numbers known as quantum numbers. In order to specify energy, size , shape and orientation of electron orbital three quantum numbers are required. These are principal quantum number, azimuthal quantum number and magnetic quantum number. In order to designate the electron, an additional quantum number called spin quantum number is needed to specify the spin of the electron. Thus, each orbital in an atom is designated by a set of three quantum numbers and the state of an electron in an atom is specified by four quantum numbers namely n , ℓ , mℓ and ms.
1. Principal quantum number (n)
This is the most important quantum number as it determines to a large extent the energy of the electron. It also determines the average distance of an electron from the nucleus . It is denoted by the letter n . It can have any whole number value such as 1,2,3,4,....The energy levels or energy shells corresponding to these numbers are designated as K, L , M, N....... As the value of n increases, the electron gets farther away from the nucleus and its energy increases. The higher the value of n , the higher is the energy of the electron. Energy of the electron in a hydrogen atom is related to principal quantum number by the following relation.
En = 2 2 m Z2e4
n2 h2
where , m = mass of electron
e = charge on the electron : h = Planck’s constant. En = Energy of electron in nth principal shell
Z = Nuclear charge.
Substituting the value of , m, e, h in the above energy expression we get :
En = 2.178 x 1018 J atom1
n2
En = 1.312 x 103 k J mol1
n2
According to the equation for En the largest negative value of energy is obtained when n has the smallest value, i.e., when n is 1.
Thus n = 1 gives the lowest energy state (E1), of the hydrogen atom. This state is called the ground state of the electron in the atom. For hydrogen atom, all other states with n 1 are called excited states. The possible energy states of the electron in the hydrogen atom are :
E1 = 1.312 x 103 k J mol1
E2 = 3.28 x 102 k J mol1
E3 = 1.46 x 102 k J mol1
Thus, it is clear that the electron in hydrogen atom can have only certain values of energy.
2. The angular momentum quantum number ( Azimutal Quantum Number ) ( ℓ )
This quantum number determines the angular momentum of the electron. This is denoted by ' l '. The value of ' ℓ ’ gives the sub-level in which the electron is located. It also determines the shape of the orbital in which the electron is located. The number of sub-shell within a principal shell is determined by the value of 'n' for that principal energy level. Thus ' ℓ ' may have all possible whole number values from 0 to ( n 1) for each principal energy level. The various sub-levels are designated as s, p, d, f -depending upon the value of ℓ as shown below:
Value of ‘ℓ’ Designation of sub-shell
0 s
1 p
2 d
3 f
Thus with in the same principal energy level, there are different possible paths known as sub-levels. The sub-level with ℓ = 0 is called s-sub-level, that with ℓ = 1 , is called p-sub-shell and that with ℓ = 2 is called d-sub-shell and with ℓ = 3 is called f-sub-shell.
The first energy level (n = 1) has s-sub-shell only and is called 1s-sub-level, second energy level (n = 2 ) has s and p sub-levels and are called 2s and 2p sub-levels, third energy level (n = 3) has s-, p- and d-sub-levels and are called 3s, 3p, and 3d-sub-levels and the fourth energy level (n = 4) has s, p, d, and f-sub-levels and are called 4s, 4p, 4d and 4 f respectively. For a particular quantum number, the energy of various sub-levels are in the order of :
s p d f
The number of sub-levels in any quantum level is equal to the principal quantum number ( = n ) of that energy level.
3. Magnetic quantum number (mℓ )
This quantum number determines the spatial orientation of the orbital. Different orbitals have different preferred regions of space around nucleus. They are given by magnetic quantum number. It is designated by ‘mℓ ’ . The allowed value of ‘mℓ ’ can have values range from ℓ through 0 to + ℓ i.e.,
mℓ = ℓ , ( ℓ+ 1 ) ,..........0,.......,( ℓ 1 ), ℓ .
In other words, there can be ( 2 ℓ + 1) values of ‘mℓ ’ for each value of ‘ ℓ ’. For example, if ℓ = 0 ( i.e., s-subshell) , mℓ can have only one value , 0. This means that s-subshell can have only one orbital , i.e., there can be only one orientation of electrons in space.
If ℓ = 1 ( i.e., p-subshell) m can have three orientations of electrons in space viz., m = 1, 0, +1 .
If ℓ = 2 ( ie., d-subshell), m can have five values viz., m = 2, 1 , 0 , +1 , +2. This means that d-subshell can have five orbitals, ie., there can be five possible orientations of electrons in space.
If ℓ = 3 ( ie f-subshell) , m can have seven values viz., mℓ = 3, 2, 1, 0, +1, +2, +3. This means that f-subshell can have seven orbitals ie., there can be seven possible orientations.
Thus there is one orbital for n = 1 (1s ), four orbitals for n = 2 ( one 2s and three 2p orbitals) nine for n = 3 ( one 3s, three 3p and five 3d orbitals). For n = 4, there are 16 orbitals ( one 4s, three 4p, five 4d and seven 4f orbitals) .
Note
For a Pz orbital the value of mℓ has been conventionally taken to be zero, ie., mℓ =0. But for a Px or Py orbital does not designate only one value of m, but a linear combination of orbitals with mℓ = + 1 and mℓ = 1. Therefore it is not correct to designate single value for mℓ for Px orbital as +1 or 1.
The relationship between the principal quantum number (n), azimuthal quantum number (ℓ) and magnetic quantum number (mℓ) is summed in the following Table.
Quantum numbers and their significance
Quantum number Symbol Restrictions Range of values Significance
Principal quantum number n Positive integers 1,2,3…. identifies shell, determines size and energy of orbital , number of orbital in the nth shell = n2.
Azimuthal quantum number
Magnetic quantum number
Magnetic spin quantum number ℓ
mℓ
ms Positive integers less than n
Integers between ℓ and + ℓ
Half integers
+ ½ or ½ 0, 1, 2, (n 1)
Total values = n
ℓ to + ℓ including 0 .
Total possible values =
( 2 ℓ + 1 )
+ ½ or ½ Identifies sub-shell ; determines shape of orbital in a multi-electron atom along with n and total angulat momentum ie., ℓ (ℓ + 1) h .
2
determines orientaions of the orbital
determines orientaion of the spin.
4. Magnetic spin quantum number (ms)
An electron , besides charge and mass , has also spin angular momentum commonly called spin. The spin angular momentum of the electron is constant and cannot be changed. The magnitude of the spin angular momentum of the electron is
quantity and can have only two orientations relative to a chosen axis (Fig 7 )
Fig 7 . An electron spin vector of length 3/2 units ( one unit = h/2) can take only two orientations with respect to a spcified axis.
They are distinguished by magnetic quantum number ms , which can take the values + ½ and ½ . These two spin states of the electron are normally represented by two arrows (spin-up) and (spin-down) . The components of the spin angular momentum vector around the chosen axis
picture the two spin states as rotation of an electron on its axis either clockwise or counter-clockwise .
ATOMIC ORBITALS AND THEIR PICTORIAL REPRESENTATIONS
An atomic orbital is a one electron wave function ( r, , ) obtained from the solution of the Schrodinger equation for the hydrogen atom. It is a mathematical function of the three coordinates of the electron ( r, , ) and can be factorised into three separate parts each of which is a funtion of only one coordinate.
( r, , ) = R(r) () ()
where R(r) is the radial function which gives the dependence of orbital upon distance r of the electron from the nucleus and () and () are the angular functions giving angular dependence of orbital on and respectively. The radial function depends on the quantum numbers n and
ℓ , whereas the angular part depends upon quantum numbers ℓ and m and is independent on n. The total wave function may, therefore , be more explicitly written as
( r, , ) = Rn, ℓ (r) ()ℓ,m m()
Radial part Angular part
The orbital wave function has no physical significance. It is the square of the absolute value of the orbital wave function 2 which has a physical significance – it measures the electron probability density at a point in an atom. and 2 vary as function of the three coordinates r , and for different orbitals . Such representation of variations of or 2 in space would need a four dimensional graph – three dimensions for the coordinates and the fourth for or 2 . It is not possible to show such variation in a single diagram since we can draw only two-dimensional diagrams on paper. We can get around this difficulty by drawing separate diagrams for :
(i) variation of radial function and
(ii) angular function.
These plots are discussed below.
Plots of the the Radial Wave Function R
The plots of the radial wave function R , radial probability density R2 and radial probability function 4 r2R2 for 1s ( n = 1, ℓ = 0), 2s ( n = 2, ℓ = 0) and 2p ( n = 2, ℓ = 1) atomic orbitals as a function of the distance r from the nucleus are shown in Fig 8 (a-c) . Each of these plots arediscused separately.
(a)
(b)
(c)
Fig 8 The plots of (a) the radial function R : (b) the radial pronability density R2 and (c) the radial density function 4r2R2 as a function of distance r of the electron from the nucleus for 1s , 2s and 2p orbitals.
A. Radial wave function (R) (Fig 8 a)
In all cases R approaches zero as r approaches infinity. One finds that there is a node in the 2s radial function. At the node, the value of the radial function changes from positive to negative. In general, it has been found that ns-orbitals have (n 1) nodes , n p –orbitals have (n 2) nodes etc.
The importance of these plots lies in the fact that they give information about how the radial wave function changes with distance r and about the presence of nodes where the change of sign of R occurs.
B. Radial probability density (R2) ( Fig 8 b)
The square of the radial wave function R2 for an electron orbital gives the probability density of finding the electron at a point along a particular radius line. To get such a variation, the simplest procedure is to plot R2 against r (Fig 8 b). These plots give useful information about probability density or relative electron density at a point as a function of radius. It may be noted that while for s-orbitals the maximum electron density is at the nucleus, all other orbitals have zero electron density at the nucleus.
C. Radial probability functions 4 r2 R2 (Fig 8c).
The radial density R2 for an orbital , gives the probability density of finding the electron at a point at a distance r from the nucleus. Since the atoms have spherical symmetry, it more useful to discuss the probability of finding the electron in a spherical shell between the spheres of radius ( r + dr ) and r.
Fig 9 Sphereical shell of thickness dr
This probability which is independent of direction is called radial probability and is equal to 4 r2 dr R2 .
Radial probability function ( = 4 r2 R2 ) gives the probability of finding the electron at a distance r from the nucleus regardless of direction.
The radial probability distribution curves obtained by plotting radial probabability functions versus distance r from the nucleus for 1s, 2s and 2p orbitals are shown in Fig 8 ( c)
1s orbital
The radial probability function for 1s orbital initially increases with increase in distance from the nucleus. It reaches a maximum at a distance very close to the nucleus and then decreases. The maximum in the curve corresponds to the distance at which the probability of finding the electron is maximum. This distance is called radius of maximum probabilty and for hydrogen atom it is 52.9 pm.
It may be noted that while Bohr’s model restricts the electron to be at a definite orbit at a fixed distance from the nucleus, the quantum mechanical model gives merely the maximum probability of finding the electron at 52.9 pm distance from the nucleus. In the case of hydrogen atom, for instance , according to Bohr model, the electron always stay at a distance of 52.9 pm from the nucleus. According to quantum mechanical model, the electron in the hydrogen atom can be at any distance, but the most probable distance for finding the electron is 52.9 pm.
2s and 2p orbitals
The radial probability function for 2s orbital shows two maxima , a smaller one near the nucleus and a bigger one at a larger distance. In between these two maxima it passes through a zero value indicating that there is zero probability of finding the electron at that distance. The point at which the probability of finding the electron is zero is called nodal point.
The distance of maximum probability for a 2p-electron is slightly less than that for a 2s electron. However, in contrast to 2p-curve, there is a small additional maxima in the 2s-curve. This indicates that the electron in the 2s-orbital spends some of its time near the nucleus. In other words , the 2s-electron penetrates a little closer to the nucleus and is therefore held more tightly than the 2p-electron. That is the reason why 2s-electron is more stable and has lower energy than a 2p-electron.
PLOTS OF ANGULAR WAVE FUNCTION ‘ ’
The angular wave function ‘ ’ depends only on quantum numbers ℓ and mℓ and is independent of principal quantum number n for a given type of orbital. It therefore means that all the s-orbitals will have same angular wave function. The plots of the angular wave function ‘ ’ and angular probability density 2 for s and Pz orbitals as shown in Fig 10 (a , b).
s Pz Pz
(a) (b)
Fig10 (a) Angular part of the wave function for hydrogen like s-orbitals and Pz orbital.
Fig10 (b) Angular probability function for Pz orbital. Only two dimensions of of the three dimensional function have been shown.
Discussion of these plots are given below.
A. Angular Wave Function , [Fig 10 a]
For an s-orbital , the angular part is independent of angle and is therefore , of constant value. Hence this graph is circular or , more properly in three dimensions i.e., spherical. For a Pz orbital , we get two tangent spheres as shown in Fig 10 a. Similarly , Px and Py have similar shape but are oriented in different directions along X and Y axes respectively. The angular wave function plots for d and f-orbitals are four lobed and six lobed respectively.
In the angular wave function plots , the distance from the centre is proportional to the numerical values of ‘ ’ in that direction and is not the distance from the centre of the nucleus.
B. Angular Probability density 2
[ Fig 10 b ]
The angular probability density plots can be obtained by squaring the angular function plots shown in (Fig 10 a ). On squaring , different orbitals change in different ways. For an s-orbital, the squaring causes no change in shape, since the function everywhere is same ; thus another sphere is obtained. For both p and d-orbitals , on squaring the plot tends to become more elongated as shown for Px and Py in Fig 10 (b )
Plots of Total Probability Density : Shapes of Atomic Orbitals
The problems associated with the representations of the variations of 2 in space have been circumvented by the following to approaches :
(a) Charge colud diagrams
(b) Boundary surface diagrams
A. Charge cloud diagrams
In this approach , the probability density 2 is shown as a collection of dots such that the density of dots in any region represents electron probability density in that region. Fig11 shows such plots for some orbitas. These give some idea about the shapes of the orbitals.
Fig 11 Probability density plots of some atomic orbitals. The density of the dots represents the probability of finding the electron in that region.
B. Boundary surface diagrams
In these diagrams , the shape of an orbital is defined as a surface of constant probability density that encloses some large fraction (say 90%) of the probability of finding the electron. The probability density is 2. When 2 is constant , so is . Hence is constant on the surface of orbital. Such a boundary surface for an s-orbital ( ℓ = 0) has the shape of a spherical shell on the nucleus. (Fig 12 ).
Fig 12 Boundary surface diagrams for 1s orbital
For each value of n , there is one s orbital. As n increases , there are (n 1) concentric shells like the successive layers in an onion.
Fig 13 Electron density distributions in the 1s, 2s ans 3s orbitals of an atom.
Boundary surface diagrams of the three 2p-orbitals (ℓ = 1 ) are shown Fig14.
Fig 14 Boundary suraface diagrams of the three 2p-orbitals
In these diagrams , the nucleus is at the origin. Each p-orbital consists of two sections called lobes that are on either side of the plane that passes through the nucleus. The size , shape and energy of the three orbitals are identical. They differ in the way the lobes are oriented. Since these lobes may be considered to lie along the x, y or z-axis , they are given the designations 2Px, 2Py and 2Pz. Like s- orbitals , p orbitals increase in size with increase in the principal quantum number and hence 4p > 3p > 2p.
Boundary surface diagrams of the d-orbitals (ℓ = 2 ) are shown in Fig 15. These have five d-orbitals which are deignated as dxy, dyz, dxz, dx2 y2 and dz2. The shape of dz2 orbital is different from that of others but all five
Fig 15 Boundary surface diagrams of five 3d-orbitals
d-orbitals are equivalent in energy. The d-orbitals for which n is greater than 3 ( 4d, 5d, …) have similar shapes.
PROBLEMS
21. What are the n, ℓ and mℓ values for 2 Px and 3 Pz electrons ?
22. What sub-shells are possible in n=4 energy level ?
23. An electron is in 3d-orbital. What possible values for the quantum numbers n, ℓ, m and s can it have ?
24. Can we have 5g sub-shell ? How many orbitals are possible for this sub-shell ?
25. If the principal quantum number is 3, what are the permitted values of the quantum numbers ℓ and mℓ ?
26. Energy of the electron in hydrogen atom is given by the expression :
En = 1.312 x 106 J mol-1
n2
(i) Calculate the amount of energy required to promote electron from first energy level to the third energy level.
(ii) What will be the ionisation energy of hydrogen atom ?
NODAL PLANE
The two lobes of a p-orbital can be separated by a plane which contains the nucleus and is perpendicular to the orbital axis.
Fig 16 A p-orbital with two lobes and a nodal plane
The electron density (i.e., probability of finding the electron) at the nucleus as well as on the plane is zero. The nucleus at which the electron density is zero is called the node and plane passing through orbital axis (at this plane as well the electron density is zero) is called nodal plane.
It can be seen that each of the three p-orbital has one such nodal plane, eg. for Px orbital YZ is the nodal plane.
Comparison of Radial Probability curves of 1s, 2s and 2p Orbitals (Fig 8 c)
The radial probability distribution curve for 2s orbital shows two maxima, a smaller one near the nucleus(40 pm) and a bigger one at a large distance (270 pm). In between these two maxima, there is a minima where there is no probability of finding the electron at that distance. The point at which the probability of finding the electron is zero is called a nodal point.
The distances of maximum probability for 2s and 2p-orbitals are approximately the same(210 pm) but are larger than the distance of maximum probability for 1s orbital. This is in keeping with the fact that 2s and 2p electrons have greater energy than that of the 1s electron.
The radius of maximum probability for a 2p electron is slightly less than that for a 2s electron. However, in contrast to 2p curve, there is a small additional maxima in the 2s-curve. This indicates that the electron in the 2s orbital spends some of its time near the nucleus. In other words, 2s electron penetrates a little closer to the nucleus, and is therefore , held more tightly than the 2p electron.
Thus, 2s electron is more stable and has lower energy than a 2p electron. It may be noted that the number of peaks in the curves for the s-orbital is equal to the ‘n’ value. i.e., 1 or 1s, 2 for 2s and 3 for 3s. In the curves for p and d orbitals , the number of peaks are (n 1) and ( n 2 ) respectively.
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PAULI’S EXCLUSION PRINCIPLE
This is an important generalisation given by Wolfgang Pauli (1925) which determines the maximum number of electrons that an energy level can accommodate. Pauli’s exclusion principle states that it is impossible for any two electrons in the same atom to have all four quantum numbers identical.
Thus in the same atom, any two electrons may have three quantum numbers identical, but not the fourth which must be different. This principle is very useful in determining the maximum number of electrons that can occur in any quantum group. For K-shell, for instance since n = 1, ‘ℓ’ can have only one value ( viz., 0 ). Hence ‘s’ can be either + ½ or ½ . Thus there are two combinations of quantum numbers, as shown below :-
n = 1 ; ℓ = 0 ; m = 0 ; s = +1/2
n = 1 ; ℓ = 0 ; m = 0 ; s = 1/2
This shows that in K-shell, there is only one sub-shell l = 0 and only two electrons of opposite spins can be accommodated.
For L-shell, since n = 2 , ‘ ℓ’ can have two values ( 0 and 1 ) , m can have one value for l = 0 and three values for ℓ = 1 ( -1, 0, +1 ) as s can have two values (+ ½ and ½ ) for each value of m. These possibilities give
rise to eight combinations of four quantum numbers, keeping in view of the Exclusion Principle.
n = 2 ; ℓ = 0 ; m = 0 ; s = + ½
n = 2 ; ℓ = 0 ; m = 0 ; s = ½
n = 2 ; ℓ = 1 ; m = 1 ; s = + ½
n = 2 ; ℓ = 1; m = 1 ; s = ½
n = 2 ; ℓ = 1; m = 0 ; s = + ½
n = 2 ; ℓ = 1 ; m = 0 ; s = ½
n = 2 ; ℓ = 1 ; m = +1 ; s = + ½
n = 2 ; ℓ = 1 ; m = +1 ; s = ½
Thus L-shell can accomodate 8 electrons , 2 in the ℓ = 0 sub-shell ( s-subshell) and 6 in ℓ = 1 sub-shell ( p-subshell) . Similarly, it can be shown that the M-shell can accommodate 18 electrons ; 2 in ℓ = 0 subshell ( s-subshell) , 6 in the ℓ = 1 subshell ( p-subshell ) and 10 in ℓ =2 subshell ( d-subshell).
The N-shell can accomodate 32 electrons 2 in ℓ = 0 subshell ( s-subshell) , 6 in ℓ = 1 subshell (p-subshell) , 10 in ℓ = 2 subshell ( d-subshell) and 14 in ℓ = 3 subshell ( f-subshell).
The maximum number of electrons that can be accommodated in the first four principal energy shells as distributed in the various sub-shells
( ℓ values ) are given in the following Table
Principal energy shell ( n ) Sub-shell
l Orbitals
m Number of electrons in sub-shell Number of electrons in Principal shell
1 0
(1s sub-shell ) 0
(one orbital) 2 2
2 0
(2s-subshell)
1
(2p-subshell) 0
(one orbital)
1, 0, +1
(3 orbitals )
2
6
8
3 0
(3s-subshell)
1
(3p-subshell)
2
(3d-subshell) 0
(one orbital)
1,0,+1
(three orbitals)
21,0,+1,+2
(five orbitals)
2
6
10
18
4 0
(4s-subshell)
1
(4p-subshell)
2
(4d-subshell)
3
(4f-subshell) 0
(one orbital)
1,0,+1
(three orbitals)
2, 1, 0, +1, +2
(five orbitals)
3, 2, 1, 0, +1, +2, +3
(seven orbitals)
2
6
10
14
32
The Number of Various Atomic Orbitals
For s-shell (i.e., ℓ = 0 ) there is only one possible orientation (i.e., m has only one value). This means that there is only one s-orbital whatever may be the value of quantum number n.
For p-subshell ( i.e., when ℓ = 1) m has three values. In other words, a p-sub-shell can have three possible orientations. Thus, there can be three p-orbitals whatever be the value of principal quantum number n. The three p-orbitals are designated as Px, Py and Pz.
For d-sub-shell (i.e., when ℓ = 2), m has five values i.e., there are five possible orientations. This means that there can be five d-orbitals whatever be the value of principal quantum number n. The five d-orbitals are designated as dxy, dxz, dyz, dx2 y2 and dz2
For f-sub-shell (i.e., when ℓ = 3), m has seven possible orientations. This means that there can be seven f-orbitals whatever be the value of the principal quantum number n. The f-orbitals have complicated designations.
To sum up
There can be only one s-orbital, three p-orbital, five d-orbitals and seven f-orbitals whatever be the major energy shell.
HUND’S RULE OF MAXIMUM MULTIPILICITY
This rule states that the paring of electrons in the orbitals of a particular sub-shell ( p, d, f ) does not take
place until all the orbitals of the same sub-shell are singly occupied. Moreover, the singly occupied orbitals must have
the electrons with parallel spins. If we represent three orbitals as boxes and electrons by arrows ( or + ½ s and for ½ s ) then if two electrons are added to sub-shell , there exists two possibilities and II
In the first arrangement , two electrons are in the same orbital ( i.e., Px) but it is not permitted by Hund’s rule, so that arrangement (II) is right as this satisfies the rule. If we add third electron to the sub-shell, this will go to Pz (the third empty orbital) rather than to Px and Py , as in arrangement III . Pairing will be there if the fourth electron is added as shown in IV.
Hund’s rule is the guiding principle to arrive at the correct configuration of various elements. For example, consider the carbon atom, whose atomic number is 6. Two electrons each occupy in 1s and 2s-orbitals respectively. The remaining two electrons are to occupy the three orbitals 2Px , 2 Py and 2Pz. Let the fifth electron enters the 2Px-orbital, then the sixth electron can enter either 2Py or 2Pz and not 2Px orbital. Thus the correct electronic configuration of carbon is :
6C : 1s2, 2s2 , 2Px1, 2Py1, 2Pz0
and not 6C : 1s2, 2s2 , 2Px2,2Py0, 2Pz0
The electronic configuration of Nitrogen is :
7N : 1s2, 2s2 , 2Px1, 2Py1, 2Pz1
ENERGY LEVEL DIAGRAMS OF HYDROGEN ATOM
In hydrogen atom there is only one electron which is present in 1s orbital in the ground state. However, in excited state, the electron may jump to any of the orbitals belonging to higher energy levels.
Energies of the higher levels for hydrogen and hydrogen like species is given by :
En = 1.312 x 103 Z2 k J mol1
n2
For hydrogen atom nuclear charge Z = 1, whereas for hydrogen like species He+, Li2+, Be3+ , Z is equal to 2, 3 and 4 respectively. The energies of various orbitals belonging to these energy levels can also be calculated. The energies of these orbitals have been represented in Fig 17 .
Fig 17 Energy level diagram for hydrogen atom
It may be noted that all the orbitals of a particular energy level have the same energy. It is also because for an atom having a single electron, the principal quantum number, n is most important in determining the energy of the orbital. The value of ℓ (angular quantum number ) merely determines the shape of the orbital.
ENERGYLEVEL DIAGRAM FOR MULTI ELECTRON ATOM
In the multi-electron atoms the energies of various orbitals depend not only upon the nuclear charge but also upon the outer electrons present in the atom. It is impossible to calculate the exact energies of various orbitals in a multi-electron atom. However, approximate values of their energies can be obtained from the spectral data. Relative energies of various orbitals in all multielectron atoms is same and is illustrated in Fig 18.
Fig 18 Energy level diagrams for multi-electron atom
The main characteristics of this diagram are :
i) The different sub-shells of a particular energy level may have different energies. For example, energy of 2s-subshell is different from energy of 2p-sub-shell.
ii) In a particular energy level, the sub-shell having higher value of ‘ℓ ' has higher energy. For example, energy of 2p-sub-shell (ℓ= 1 ) is higher than energy of 2s-sub-shell (ℓ = 0 ) . In general, increasing order of energies of different types of sub-shell in a particular energy level is :
s p d f
iii) As the value of n increases some sub-shell of lower energy level may have energy higher than the energy of some sub-shell of higher energy level. For example, energy of 3d is higher than the energy level of 4s although the latter belongs to higher main energy level.
iv) The increasing order of energies of various sub-shells is :
1s 2s < 2p 3s 3p 4s 3d 4p 5s
4d 5p ………
In multielectron atom energy of electron is determined not only by principal quantum number (n) but also by azimuthal quantum number (ℓ). The relative order of energies of various sub-shells in a multi-electron atom can be predicted with the help of ( n + ℓ ) Rule. According to this rule :-
i) In neutral atoms a sub-shell with lower value of ( n + ℓ ) has lower energy.
ii) If two sub-shells have equal value of ( n + ℓ ) , the sub-shell with lower value of n has lower energy.
RULES FOR FILLLING OF ORBITALS IN AN ATOM
An atom in its lowest energy state is said to be in the ground state. The ground state is the most stable state of an atom. The filling of the orbitals in the ground state is in accordance with the following rules.
1. Aufbau Principle
The Aufbau principle states that in the ground state of an atom an electron enters the orbital of lower energy first and subsequent electrons are fed in the order of increasing energies. The following order is observed for filling of electrons in various orbitals.
1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s,.....
The following figure( Fig 19) shows a simple memory aid for remembering the increasing order of various orbitals.
2. Pauli’s Exclusion Principle
This is an important generalisation given by Wolfgang Pauli (1925) which determines the maximum number of electrons that an energy level can accommodate. Pauli’s exclusion principle states that it is impossible for any two electrons in the same atom to have all four quantum numbers identical. Thus in the same atom, any two electrons may have three quantum numbers identical, but not the fourth which must be different. This principle is very useful in determining the maximum number of electrons that can occur in any quantum group.
For K-shell, for instance since n = 1, ‘ℓ’ can have only one value ( viz., 0 ). Hence ‘s’ can be either + ½ or ½ . Thus there are two combinations of quantum numbers, as shown below:-
n = 1 ; ℓ = 0 ; m = 0 ; s = + ½
n = 1 ; ℓ = 0 ; m = 0 ; s = ½
This shows that in K-shell, there is only one sub-shell ℓ = 0 and only two electrons of opposite spins can be accommodated.
For L-shell, since n = 2 , ℓ can have two values ( 0 and 1 ) , m can have one value for ℓ = 0 and three values for ℓ = 1 ( 1, 0, +1 ) as s can have two values (+ ½ and ½) for each value of m. These possibilities give rise to eight combinations of four quantum numbers, keeping in view of the exclusion principle.
n = 2 ; ℓ = 0 ; m = 0 ; s = + ½
n = 2 ; ℓ = 0 ; m = 0 ; s = ½
n = 2 ; ℓ = 1 ; m = 1 ; s = + ½
n = 2 ; ℓ = 1 ; m = 1 ; s = ½
n = 2 ; ℓ = 1 ; m = 0 ; s = + ½
n = 2 ; ℓ = 1 ; m = 0 ; s = ½
n = 2 ; ℓ = 1 ; m = +1 ; s = + ½
n = 2 ; ℓ = 1 ; m = +1 ; s = ½
Thus L-shell can accommodate 8 electrons, 2 in ℓ = 0 sub-shell (s-subshell) and 6 in ℓ = 1 sub-shell (p-sub-shell) . Similarly it can be shown that the M-shell can accommodate 18 electrons ; 2 in l = 0 subshell (s-sub-shell) , 6 in ℓ =1 subshell (p-subshell) and 10 in l = 2 sub-shell (d-subshell).
The N-shell can accommodate 32 electrons 2 in ℓ = 0 subshell ( s-subshell) , 6 in ℓ = 1 subshell (p-sub-shell)10 in ℓ = 2 sub-shell and 14 in ℓ = 3 subshell (f-subshell)
The maximum number of electrons that can be accommodated in the first four principal energy shells as distributed in the various sub-shells (ℓ values ) are given in the following Table.
Distribution of Electrons in Various Energy Shells and Sub-shells.
Principal energy shell (n) Sub-shell
ℓ Orbitals
m Number of electrons in the sub-shell Number of electrons in principal shell
1 0
(1s-subshell) 0
(one orbital) 2 2
2 0
(2s-subshell) 0
(one orbital)
1, 0, +1
(three orbitals)
2
6
8
3 0
(3s-subshell)
1
(3p-subshell)
2
(3d-subshell) 0
(one orbital)
1, 0, +1
(three orbitals)
2, 1, 0, +1, +2
(five orbitals)
2
6
10
18
4 0
(4s-orbital)
1
(4p-orbital)
2
(4d-orbital)
3
(4f-orbital) 0
(one orbital)
1, 0, +1
(three orbitals)
2, 1, 0, +1, +2
(five orbitals)
3, 2, 1, 0, +1, +2, +3
(seven orbitals)
2
6
10
14
32
3. HUND’S RULE OF MAXIMUM
MULTIPILICITY
This rule states that pairing of electrons in the orbitals of a particular sub-shell ( p, d, or f ) does not take place until all the orbitals of the sub-shell are singly occupied. Moreover, the singly occupied orbitals must have the electrons with parallel spins.
This is due to the fact that electrons being identical in charge , repel each other when present in the same orbital. This repulsion can , however, be minimised if two electrons move as far apart as possible by occupying different degenerate orbitals.
Further all the singly occupied orbitals will have parallel spins i.e., in the same direction viz., either clockwise or anticlockwise. This is due to the fact that two electrons with parallel spins (in different orbitals) will encounter less interelectronic repulsions in space than when they have opposite spins(while staying in different orbitals).
Consider for example, the following illustration :
For the element nitrogen, which contains 7 electrons, the following configurations can be written.
In accordance with Hund’s rule , the configuration (a) in which there are three unpaired electrons occupying 2Px, 2Py and 2Pz orbitals have parallel spins (either all clock-wise or anti-clockwise ) is correct, while configuration (b) in which the unpaired electrons do not have parallel spins is incorrect. The configuration (c) in which all the pairing of electrons has been in the 2Px-orbital without putting the third electron in 2Pz orbital is also not consistent with the Hund’s rule of maximum multiplicity.
By the term maximum multiplicity, we mean that total spin of unpaired electrons is maximum. For example, the total spin of the unpaired electrons in configuration (a) , (b) and (c) are 1½ , ½ and ½ . Thus in accordance with the Hund’s rule, configuration (a) with maximum multiplicity of 1½ is correct.