UNIT 2 ( PAGE 5)
QUATUM MECHANICAL MODEL OF ATOM
Classical mechanics, based on Newton
Quantum mechanics is a theoretical science which deals with the study of the motions of the microscopic objects that have both observable wave like and particle like properties. It specifies the laws of motion that these objects obey. When quantum mechanics is applied to macroscopic objects (for which wave properties are insignificant) the results are same as those from the classical mechanics.
Erwin Schrodinger , in 1926, gave a wave equation to describe the behaviour of electron waves in atoms and molecules. In Schrodinger wave model of atom, the discrete energy levels or orbits proposed by Bohr are replaced by mathematical function Y, which are related with probability of finding electrons at various places around the nucleus.
Hydrogen Atom and the Schrodinger Equation
When Schrodinger equation was solved for hydrogen atom, the solution gives the possible energy states the electron can occupy and the corresponding wave function(s) (Y) (also called atomic orbitals) of the electron associated with each energy state. These quantised energy states and corresponding wave functions which are characterized by a set of three quantum numbers ( principal quantum number n, azimuthal quantum number ℓ and magnetic quantum number mℓ ) arise as a natural consequence in solution of Schrodinger equation. When an electron is in an energy state, the wave function corresponding to that energy state contains all informations about the electron. The wave function is a mathematical function whose value depends upon the co-ordinates of the electron in the atom. The quantum mechanical results of hydrogen atom successfully predict all aspects of the hydrogen atom spectrum including some phenomena that could not be explained by Bohr model.
Application of Schrodinger equation to multielectron atoms presents a difficulty ; the Schrodinger equation cannot be solved exactly for multi-electron atom. This difficulty can be overcome by using approximate methods. Such calculations with the aid of modern computers shows that orbitals in atoms other than hydrogen do not differ in any radical way from the hydrogen orbitals discussed above. The principal difference lies in the consequence of increased nuclear charge - all the orbitals are somewhat contracted. Unlike hydrogen orbitals whose energies depend only on the quantum number n, the energies of orbitals in multi-electron atoms depend on n and ℓ .
Important Features of the Quantum Mechanical Model of Atom
Quantum mechanical model of atom is the picture of the structure of the atom, which emerges from the application of the Schrodinger equation to atoms. The following are the important features of quantum mechanical model of atom.
1. The energy of electrons in atoms is quantised ( i.e., can have certain specific values).
2. The existence of quantised electronic energy levels is a direct result of the wave like properties of electrons.
3. Both exact position and exact velocity of an electron in an atom cannot be determined simultaneously (Heisenberg uncertainty principle). The path of the electron in an atom can, therefore, be never determined or known.
4. An atomic orbital is the wave function Y for an electron in an atom. Whenever an electron is described by a wave function, we say that the electron occupies that orbital. Since many such wave functions are possible for an electron, there are many atomic orbitals in an atom. These orbitals form the basis of the electronic structure of atoms. In each orbital electron has a definite energy. An orbital cannot contain more than two electrons. In multi-electron atom, the electrons are filled in various orbitals in the order of increasing energy For each electron of multi-electron atom, the electrons are filled in various orbitals in the order of increasing energy. For each electron of multi-electron atom, there shall, therefore, be an orbital wave function characteristic of the orbital it occupies. All the information about the electron in an atom is stored in its orbital wave function Y and quantum mechanics makes it possible to extract this information out of Y.
5. The probability of finding an electron at a point within an atom is proportional to the square of the orbital wave function i.e., ½Y½2 at that point. ½Y½2 is known as probability density and is always positive. From the value of ½Y½2 at different points within an atom , it is possible to predict the region around the nucleus where electron will most probably be found.
ORBIT AND ORBITAL
Orbit and orbital are synonymous. An orbit , as proposed by Bohr , is a circular path around the nucleus in which an electron moves. A precise description of this path of electron is impossible according to Heisenberg’s uncertainty principle. Bohr’s orbits, therefore, have no real meaning and their existence can never be demonstrated experimentally. An orbital on the other hand , is a quantum mechanical concept and refers to the one electron wave function Y in an atom. It is characterized by three quantum numbers (n, ℓ and m) and its values depends upon the three coordinates of the electron. Y has , by itself, no physical meaning. Its square of the wave function ½Y½2 which has a physical meaning. ½Y½2 at any point in the atom gives the value of probability density at that point. Probability density (½Y½2) is the probability per unit volume and a small volume (called a volume element) yields the probability of finding the electron in that volume (the reason for specifying a small volume element is that ½Y½2 varies from one region to another in space but its value can be assumed to be constant within a small volume element). The total probability of finding electron in a given volume can be calculated by the sum of all the products of ½Y½2 and the corresponding volume elements. It is thus possible to the probable distribution of electron in an orbital.
An orbital may be defined as a region in space around the nucleus where the probability of finding the electron is maximum.
Fig. 20. Illustration of Orbit and Orbital.
If the boundary is drawn which encloses a region where there is high probability of finding the electron (90-95%) of finding the electron, the figure obtained gives the general picture of the orbital. For the sake of simplicity it may be drawn as shown in Fig . The orbital is shown by dotted figure representing electron cloud. The intensity of the dots gives the relative electron probability of finding the electron in that region. It may be noted that there are some chances of finding the electron even outside the figure. The above figure provides a comparison with the circular orbit proposed by Bohr. The main difference between orbit and orbital are summed up below.
Orbit | Orbital |
1. It is a circular path around the nucleus in which an electron revolves. | 1. It is a region of space around the nucleus, where the electron is most likely to be found. |
2. It represents planar motion of an electron. | 2. It represents three-dimensional motion of an electron around the nucleus. |
3. The maximum number of electrons in an orbit is 2 n2, where 'n' stands for the number of orbit. | 3. An orbital cannot accommodate more than two electrons. |
4. Orbits are circular in shape. | 4. Orbitals have different shapes, e.g. s-orbitals are spherically symmetrical, whereas, p-orbitals are dumb-bell shaped. |
5. Orbits are non-directional in character and hence they cannot explain shapes of molecules. | 5.Orbitals(except s-orbitals) have directional character and hence they can account for shapes of molecules. |
6. Concept of well-defined orbit is against Heisenberg's uncertainty principle. | 6. Concept of orbitals is in accordance with Heisenberg's uncertainty principle. |
QUANTUM NUMBERS
A large number of orbitals are possible in an atom. Qualitatively, these orbitals can be distinguished by their size, shape and orientation. An orbital of smaller size means that there is more chance of finding the electron near the nucleus. Similarly, shape and orientation mean that there is more probability of finding the electron along certain directions than along others.
Atomic orbitals are precisely distinguished by quantum numbers. Each orbital is distinguished by three quantum numbers labelled as n, ℓ and mℓ .
Principal quantum number (n)
The principal quantum number n is a positive integer with values of 1, 2, 3 ….. etc. The principal quantum number determines the size and to a large extent the energy of the orbital. For hydrogen and hydrogen-like systems (e.g. He+ , Li2+), it alone determines the energy and size of the orbital. The larger the value of n, the larger the energy of the orbital. The principal quantum number also identifies the shell. There are n2 orbitals in the shell. All the orbitals of a given value of n constitute a single shell of atom and are represented by following letters.
n | 1 | 2 | 3 | 4 |
Shell | K | L | M | N |
Azimuthal quantum number (ℓ)
Each shell consists of one or more sub-shells or sub-levels. The number of sub-shells in a principal shell is equal to the value of n. There is only one sub-shell in the n = 1 shell. There are two sub-shells in the n = 2 shell, there are three in the n = 3 shell and so on. Each sub-shell in a shell is assigned an azimuthal or subsidiary quantum number , ℓ.
For a given value of n, ℓ can have n values ranging from 0 to ( n - 1) . For example, when n= 1 , only value of ℓ is 0 and there is only one sub-shell. When n = 2, there are two sub-shells that have ℓ values of 0 and 1, respectively. When n = 3, the three sub-shells have ℓ values of 0, 1 and 2.
ℓ | 0 | 1 | 2 | 3 | 4 | 5 |
notation | s | p | d | f | g | h |
The following table shows the permissible values of ℓ for a given principal quantum number and the corresponding subshell notations.
Sub-shell notations
n | ℓ | Sub-shell notation |
1 | 0 | 1s |
2 | 0 | 2s |
2 | 1 | 2p |
3 | 0 | 3s |
3 | 1 | 3p |
3 | 2 | 3d |
4 | 0 | 4s |
4 | 1 | 4p |
4 | 2 | 4d |
4 | 3 | 4f |
Magnetic quantum number (mℓ )
Each sub-shell consists of one or more orbitals. The number of orbitals in a sub-shell is given by (2 ℓ + 1) . In any ℓ = 0 sub-shell for example , there are 2(0) + 1 = 1 orbital. In any ℓ = 1 sub-shell, there are 2(1) + 1 = 3 orbitals. In any ℓ = 2 sub-shell there are 2(2) + 1 = 5 orbitals. In other words,
Sub-shell notation | s | p | d | f | g |
Value of ℓ | 0 | 1 | 2 | 3 | 4 |
Number of orbitals | 1 | 3 | 5 | 7 | 9 |
An s-subshell consists of one orbital, a p-subshell consists of three orbitals, a d-subshell consists of five orbitals, a f-shell consists of seven orbitals and so on. The quantum number ℓ also gives the shape of orbitals in the sub-shell.
Each orbital with in the given sub-shell is identified by a magnetic orbital quantum number mℓ which gives information about the orientation of the orbital.
For any sub-shell ( defined by ℓ values) , (2 ℓ + 1 ) values of mℓare are possible and these values are given by :
mℓ= - ℓ, -(ℓ-1), ….. 0……, (ℓ - 1), ℓ
Thus, for ℓ = 0 , the only permitted value of mℓ is 0 ( one s-orbital) For ℓ = 1, m can be - 1, 0 and +1 (three p-orbitals). For ℓ = 2 , m can be -2, -1, 0, +1, +2 ( five d-orbitals). It is noted that the values of m are derived from ℓ and that of ℓ are derived from n.
Each orbital in an atom is identified by a set of values for n, ℓ and m . An orbital described by the quantum numbers n=2, ℓ = 1 and m = 0 is an orbital in the p-subshell of the second shell , a 2p orbital.
Spin quantum number (ms)
The three quantum numbers labelling an atomic orbital can be used to label the electron in the orbital. However, a fourth quantum number, the spin quantum number , ms is necessary to describe an electron completely (George Uhlenbeck and Samuel Goudsmit proposed this quantum number). An electron has, besides charge and mass, intrinsic spin angular momentum commonly called spin. Spin angular momentum of the electron, a vector quantity, can have two orientations relative to a chosen axis. These two orientations are distinguished by the spin quantum number ms which can take the values of + ½ and - ½ . These are called the two spin states of the electron and are normally represented by two arrows (spin up) and ¯(spin down) respectively. Two electrons that have different ms values ( one + ½ and other - ½ ) are said to have opposite spins. An orbital cannot hold more than two electrons and these two electrons should have opposite spins.
To sum up, the four quantum numbers provide the following information :
1. n identifies the shell, determines the size of the orbital and also to a large extent the energy of the orbital.
2. There are n sub-shells in which the nth shell , ℓ identifies the sub-shell and determines the shape of the orbital. There are (2 ℓ + 1 ) orbitals of each type in a sub-shell, i.e., one s-orbital (ℓ = 0 ), three p-orbitals (ℓ = 1) and five d-orbitals (ℓ = 2 ) per sub-shell. To some extent ℓ also determines the energy of the orbital in multi-electron atom.
3. mℓ designates the orientation of the orbital. For a given value of ℓ, ml has ( 2 ℓ + 1 ) values, the same as the number of orbitals per sub-shell. It means that the number of orbitals is equal to the number of ways in which they are oriented.
4. ms refers to the orientation of the spin of the electron.
Relationship among values of n, ℓ and mℓ
Problems57. Which of the following are isoelectronic species ?
Na+, K+, Mg2+, Ca2+, S2- . Ar
58. What is the total number of orbitals associated with the principal quantum number n = 3 ?
59. Using s,p,d, f notations , describe the orbital with the following quantum numbers :
(a) n = 2 : ℓ= 1 (b) n = 4 : ℓ = 0 (c) n = 5 : ℓ = 3
(d) n = 3 : ℓ = 2.
60. What sub-shell are possible in n = 4 energy level ? How many orbitals are possible for this level ?
61. An electron is in 3d orbital. What possible values for quantum
numbers n, ℓ , m and s can it have ?
62. Can we have 5g subshell ? How many orbitals are possible for
this subshell ?
63. a) What subshells are possible in n=3 energy level ?
b) How many orbitals are possible in this level ?
64. Write the designations for orbitals with the following quantum
number :
a) n = 3 ; ℓ = 0 b) n = 5 ; ℓ = 2
65. What are the possible values of l for an electron in :
a) third energy level b) 3d-subshell
66. (a) An atomic orbital has n = 3. What are the possible values of ℓ ? (b) An atomic orbital has ℓ = 3 . What are the possible values of m ?
67. Using s, p, d notations, describe the orbital with the following quantum numbers :
a) n = 1 ; ℓ = 0 b) n = 2 ; ℓ = 0 c) n = 3 ; ℓ = 0
d) n = 4 ; ℓ = 0 e) n = 4 ; ℓ = 3
68. How many electrons in an atom may have the following quantum numbers ?
(a) n = 4, ms = - ½ (b) n = 3, ℓ = 0
SHAPES OF ATOMIC ORBITALS
The orbital wave function or y for an electron in an atom has no physical meaning. It is simply a mathematical function of the coordinates of the electron. However, for different orbitals the plots of corresponding wave functionsas a function of r (the distance from the nucleus) are different. Figure (a) gives such plots for 1s ( n = 1 , ℓ = 0) and 2s ( n = 2, ℓ = 0) orbitals.
According to Max Born, the square of the wave function (i.e., y2 ) at a point gives the probability density of the electron at that point. The variation of y2 as a function of r for 1s and 2s orbitals is given in Fig (b)
Fig (a) The plots of the orbital wave function y(r)
The variation of probability density y2 (r) as a function of distance r of the electron from the nucleus for 1s and 2s orbitals
The curves for 1s and 2s orbitals are different.
It may be noted that for 1s orbital the probability density is maximum at the nucleus and it decreases sharply as we move away from it. On the other hand , for 2s orbital, the probability density first decreases sharply to zero and again starts increasing. After reaching a small maxima it decreases again and approaches zero as the value of r increases further. The region where this probability density function reduces to zero is called nodal surfaces or simply nodes. In general , it has been found that ns-orbital has (n-1) nodes, that is , number of nodes increases with increase in principal quantum number n. In other words, the number of nodes for 2s orbital is one, two for 3s and so on.
These probability density variations can be visualised in terms of charge cloud diagrams.
Probability density plots of 1s , 2s and 2p orbitals
In these diagrams , the density of dots in a region represents electron probability density in that region.
Boundary surface diagrams of constant probability density for different orbitals give a fairly good representation of the shapes the orbitals. In this representation, a boundary surface or contour surface is drawn in space for an orbital on which the value of probability density ½m½2 is constant. In principle any such boundary surfaces may be possible. However, for a given orbital, only that boundary surface diagram of constant probability density is taken to be good representation of the shape of the orbital which encloses a region or volume in which the probability of finding the electron is very high, say 90%. The boundary surface diagram for 1s and 2s orbitals are given in Fig.
Boundary surface diagram for 1s and 2s orbitals
Boundary surface diagram for a s-orbital is a sphere centred on the nucleus. In two dimensions, this sphere looks like a circle . It enclose a region probability of finding the electron is about 90%.
The 1s and 2s orbitals are spherical in shape. All s-orbitals are spherically symmetric, ie., the probability of finding the electron at a given distance is equal in all the directions. It is also observed that the size of the s-orbital increases with increase in n , that is , 4s > 3s > 2s > 1s and the electron is located further away from the nucleus as the principlal quantum number increases.
The boundary surface diagrams for three 2p-orbitals (ℓ =1) are shown in Figure.
Boundary surface diagrams of the three 2p-orbitals
In these diagrams , the nucleus is at the origin. Here unlike s-orbitals, the boundary surface diagrams are not spherical. Instead each of the p-orbital consists of two sections called lobes that are either side of the plane that passes through the nucleus. The probability density is zero on the plane where the two lobes touch each other. The size, shape and energy of the three orbitals are identical. They differ however, in the way the lobes are oriented. Since the lobes may be considered to lie along the X, Y and Z axis, they are given the designations 2Px, 2Py and 2 Pz. It should be understood , however, that there is no simple relation between the values of mℓ ( - 1, 0, and +1) and X, Y and Z directions. The three p-orbitals are mutually perpendicular. Like s-orbitals, p-orbitals increase in the principal quantum number and hence the order of the energy and size of various p-orbitals is 4p > 3p > 2p. Like s-orbitals, the probability density functions for p-orbitals also pass through value zero, besides at zero and infinite distance, as the distance from the nucleus increases. The number of nodes are given by the n - 2 , that is number of radial node is 1 for 3p orbital, two for 4p orbital and so on.
For ℓ = 2, the orbital is known as d-orbital and the minimum value of principal quantum number (n) has to be 3, as the value of ℓ cannot be greater than n - 1. There are five mℓ values ( - 2, - 1, 0, +1, +2) for ℓ = 2 and thus there are five d-orbitals shown in Fig.
Boundary surface diagrams of the five 3d-orbitals
The five d-orbitals are designated as dxy, dyz, dxz, d x2- y2 and dz2. The shapes of first four d-orbitals are similar to each other, whereas that of the fifth one dz2, is different from others, but all five d-orbitals are equivalent in energy. The d-orbitals for which n is greater than 3 ( 4d, 5d, …..) also have shapes similar to 3d-orbital, but differ in energy and size.
Besides the radial nodes( i.e., probability density function is zero) , the probability density functions for np and nd orbitals are zero at the plane(s), passing through the nucleus(origin). For example, in the case of Pz orbital, XY-plane is a nodal plane, in the case of dxy orbital, there are two nodal planes passing through the origin and bisecting the XY plane containing Z-axis. These are called angular nodes and the number of angular nodes are given by ‘ℓ’, i.e., one angular node for p-orbitals, two angular nodes for ‘d’ orbitals and so on. The total number of nodes are given by ( n - 1) , i.e., sum of ℓ angular nodes and ( n - ℓ - 1) radial nodes.
Energies of Orbitals
The energy of an electron in a hydrogen atom is determined solely by the principal quantum number. Thus the energy of the orbitals increases as follows:
1s < 2s = 2p < 3s = 3p = 3d < 4s = 4p = 4d = 4f < …
and is depicted in Fig. A diagram representing relative energies of various orbitals in an atom is called energy level diagram.
Although the shapes of 2s and 2p orbitals are different , an electron has the same energy when it is in the 2s orbital as when it is present in 2p orbital. The orbitals having the same energy are called degenerate. The 1s in hydrogen atom corresponds to the most stable condition and it is called the ground state and the electron residing in this orbital is most strongly held by the nucleus. An electron in 2s , 2p or higher orbitals in a hydrogen atom is in excited state.
Energy level diagram for multi-electron atoms
The energy of an electron in a multielectron atom , unlike that of hydrogen atom, depends not only on its principal quantum number(shell) , but also on its azimuthal quantum number(subshell). That is , for a given principal quantum number, s, p, d, f …. all have different energies. The main reason for having different energies of the subshells is the mutual repulsion among the electrons in a multi-electron atoms. The only electrical interaction present in hydrogen atom is attraction between the negatively charged electron and the positively charged nucleus. In multi-electron atoms, besides the presence of attraction between the electron and nucleus, there are repulsion terms between every electron and other electrons present in the atom. Thus stability of an electron in multi-electron atom is because total attractive interactions are more than the repulsive interactions. In general , the repulsive interaction of the electrons in the outer shell with electrons in the inner shell are more important. On the other hand, the attractive interactions of an electron increases with increase of positive charge ( Ze) on the nucleus. Due the presence of electrons in the inner shells, the electrons in the outer shell will not experience the full positive charge on the nucleus (Ze), but will be lowered due to partial screening of positive charge on the nucleus by inner shell electrons. This is known as shielding of the outer shell electrons from the nucleus by inner shell electrons, and the net positive charge experienced by the electron from the nucleus is known as effective nuclear charge (Zeffe). Despite the shielding of the outer electrons from the nucleus by inner shell electrons, the attractive force experienced by outer shell electrons increase with increase of nuclear charge. In other words, energy of interaction between the nucleus and electron (that is orbital energy) decreases (that is more negative) with increase of atomic number (Z).
Both the attractive and repulsive interactions depend upon the shell and shape of the orbital in which electron is present. For example, being spherical in shape, the s-orbital shields the electron from the nucleus more effectively as compared to p-orbital. Similarly because of the difference in their shapes, p-orbitals shield the electrons from the nucleus more than the d-orbitals, even though all these orbitals are present in the same shell. Further more due to spherical shape, s-orbital electron spends more time close to the nucleus in comparison to p-orbital and p-orbital spends more time in the vicinity of the nucleus in comparison to d-orbital. In other words, for a given shell (principal quantum number), the Zeff experienced by the orbital decreases with increase of azimuthal quantum number(ℓ), that is, the s-orbital will be more tightly bound to the nucleus than p-orbital and a p-orbital in turn will be better tightly bound than the d orbital. The energy of s orbital will be lower (more negative) than that of p orbital and that of p orbital will be less than that of d orbital and so on. Since the extent of shielding of the nucleus is different for different orbitals, it leads to the splitting of the energies of the orbitals with in the same shell (or same principal quantum number), that is , energy of the orbital depends upon the values of n and ℓ . The lower the ( n +ℓ ) for an orbital, the lower is its energy. If two orbitals have the same value of ( n +ℓ ) , the orbital with lower value of n will have the lower energy. The Fig depicts the energy levels of multi-electron atoms.
Energy level diagram for multielectron atom
The main features of energy level diagrams of multi-electron atoms are as follows :
i) The different sub-shells of a particular energy level may have different energies. For example , energy of 2s-subshell is different from the energy of 2p-subshell.
ii) In a particular energy level, the sub-shell having higher value of ℓ has higher energy. For example, energy of 2p-subshell (ℓ = 1) is higher than energy of 2s-subshell (ℓ = 0 ). In general, increasing order of energies of different types of sub-shells in a particular energy level is :
s < p < d < f
iii) As the value of n increases, some sub-shells of lower energy level may have energy higher than the energy of some sub-shells of higher energy level. For example, energy of 3d is higher than the energy of 4s although the latter belongs 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 < .......
In a multi-electron atom energy of the 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 or Bohr’s - Bury’s Rule. According to this rule :
(a) In neutral atoms a sub-shell with lower value of ( n + ℓ ) has lower energy. For example, 4s-orbital has lower energy than 3d orbital . For 4s-orbital ; n = 4 and ℓ = 0 . Hence ; ( n + ℓ) = 4 + 0 = 4 For 3d-orbital n = 3 and ℓ = 2 ; ( n + ℓ ) = 3 + 2 = 5.
(b) If the two sub-shells have equal values of ( n + ℓ ) , the sub-shell with lower value of n has lower energy. For example, consider 3p and 4s orbitals. For 4s-orbital n = 4 and ℓ = 0 . Hence (n + ℓ) = 4 + 0 = 4 For example consider 3p and 4s orbitals . For 4s-orbital n = 4 and ℓ= 0 . For 3p-orbital n = 3 and ℓ = 1 . Hence ( n + ℓ ) = 3 + 1 = 4 . Hence 3p-orbital has lower energy than 4s because it has lower value of n.
Note
The energies of the orbitals in the same subshell decrease with increase in the atomic number (Zeff). For example, energy of 2s orbital of hydrogen atom is greater than that of 2s orbital of lithium and that of lithium is greater than that of sodium and so on, that is ,
E2s(H) > E2s(Li) > E2s(Na) > E2s(K)
RULES FOR FILLING 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 electrons into 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 figure 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 accomodate. 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 ‘ms’ can be either +½ or -½. Thus there are two combinations of quantum numbers, as shown below:-
n = 1 ; ℓ = 0 ; mℓ = 0 ; ms = +½
n = 1 ; ℓ = 0 ; mℓ = 0 ; ms = - ½
This shows that in K-shell, there is only one sub-shell ℓ = 0 and only two electrons of opposite spins can be accomodated.
For L-shell, since n = 2 , ‘ℓ’ can have two values ( 0 and 1 ) , mℓ can have one value for ℓ = 0 and three values for
ℓ = 1 ( - 1, 0, +1 ) and s can have two values ( +1/2 and -1/2 ) for each value of m. These possibilities give rise to eight combinations of four quantum numbers, keeping in view of the Exclusion Principle.
n = 1 ; ℓ = 0 ; m ℓ = 0 ; m s = +½
n = 1 ; ℓ = 0 ; m ℓ = 0 ; m s = - ½
n = 2 ; ℓ= 1 ; m ℓ = - 1 ; m s = +½
n = 2 ; ℓ= 1 ; m ℓ = - 1 ; m s = - ½
n = 2 ; ℓ= 1 ; m ℓ = 0 ; m s = +½
n = 2 ; ℓ = 1 ; m ℓ = 0 ; m s = - ½
n = 2 ; ℓ = 1 ; m ℓ = +1 ; m s = +½
n = 2 ; ℓ = 1 ; m ℓ = +1 ; m 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 accomodate 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).
3. Hund's Rule of Maximum Multiplicity
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 subshell , there exists two possibilities I and I I.
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 subshell, this will go to Pz ( the third empty orbital) rather than to Px and Py. Pairing will be there if the fourth electron is added as shown in III.
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 , 2Py 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
Electronic Configurations
In order to represent electron population of an orbital, the principal quantum number ( n) is written before the orbital symbol, while the number of electrons in the orbital is written as superscript near the right hand top of the orbital symbol. For example , if we have two electrons in the s-orbital of first energy level then it is written as 1s2. Sometimes electronic configurations are represented in a different way by representing each orbital by square or a circle and electrons are represented by putting arrows as illustrated below.
Problem
69. Write the electronic configurations of the following elements:
i) H ii) He iii) Li iv) Be v) B
ii) C vii) N vii) O viii) F ix) Ne
x) Na xi) Mg
Stability of Filled and Half-filled orbitals
The filled and half-filled orbitals have extra stability due to the following reasons.
i) Symmetrical distribution of electrons
It is well-known that symmetry leads to stability. Thus if the addition or removal of an electron results in symmetrical distribution of the electrons in the orbitals, the electronic configuration becomes more stable. In such case there will be half-filled or completely filled orbitals. Electrons in the same subshell ( here 3d ) have equal energy but different spatial distribution. Consequently, their shielding of one another is relatively small and the electrons are strongly attracted by the nucleus.
ii) Exchange Energy
The stabilizing effect arises whenever two or more electrons with the same spin are present in the degenerate orbitals of a subshell. These electrons tend to exchange their positions and the energy released due to this exchange is called exchange energy. The number of exchanges that can take place is maximum when the subshell is either half-filled or completely filled(Fig).
Possible exchange for a d5 configuration
As a result the exchange energy is maximum and so is the stability.
The exchange energy is the basis of Hund’s rule that electrons which enter orbitals of equal energy have parallel spins as far as possible. In other words, the extra stability of half-filled and completely filled subshell is due to : (i) relatively small shielding , (ii) smaller coulombic reulsion energy , and (iii) larger exchange energy.
Note :- It may be noted that configurations of atoms can also be written in the condensed form by taking the configurations of noble gases as the core. The configurations of inert gases representing core are written as [He]2, [Ne]10, [Ar]18, [Kr]36, [xe]54 and [Rn]86. For example, electronic configurations of scandium( Z = 21 ) may be written as :
21Sc : [Ar]18,3d1, 4s2
Exceptional Configurations of Copper and Chromium
The electronic configurations of Cr ( Z = 24 ) and Cu ( Z = 29 ) do not follow the general trend. The electronic configuration of Cr and Cu are expected to be as follows:
Cr : 1s2, 2s2, 2p6, 3s2, 3p6, 3d4, 4s2 and
Cu : 1s2, 2s2, 2p6, 3s2, 3p6, 3d9, 4s2
But actually their configurations are :
Cr : 1s2, 2s2, 2p6, 3s2, 3p6, 3d5, 4s1 and
Cu : 1s2, 2s2, 2p6, 3s2, 3p6, 3d10, 4s1
These anomalies are attributed to the fact that exactly half-filled and completely filled orbitals ( ie., d5, d10, f7 and f14 ) have lower energy and hence extra stability . The cause of this extra stability has been attributed to the symmetry effects and Exchange energy effects. Thus to acquire stability , one of the 4s electrons goes to the nearby 3d-orbitals so that 3d-orbitals get half-filled in Cr and completely filled in copper.
Problems
70. Write the electronic configurations of Chromium and copper.
71. Arrange the following orbitals in the order in which electron may be normally expected to fill : 2p, 3s, 4d, 3p, 3d.
72. Give the electronic configurations of 19K, 25Mn, 20Ca.
73. Give the electronic configurations of :
i) H- ii) Na+ iii) F- iv) Mg2+
74. Write the electronic configurations of :
i) Li+ ii) O2- iii) N3-
iv) F+ v) Al3+
75. What are the atomic numbers of elements whose outermost electrons are represented by (a) 3s1 (b) 2p3 and (c) 3p5 ?
76. Which atoms are indicated by the following configurations ?
a) [He]2s1 (b) [Ne]3s23p3 and (c) [Ar]4s23d1.
77. What is the lowest value n that allows g orbitals to exist ?
78. An electron is one of the 3d orbitals. Give the possible values of n , ℓ , and mℓ for this electron.
79. An atom of element contains 29 electrons and 35 neutrons. Deduce (i) the number of protons and (ii) the electronic configuration of the element.
80. Give the number of electrons in the species H2+, H2 and O2+.
81. An atomic orbital has n = 3. What are the possible values of , ℓ , and mℓ ?
82. List the quantum numbers ( ℓ , and mℓ) of electrons for 3d orbital.
83. Which of the following orbitals are possible ? 1p, 2s, 2p and 3f.
84. Using s, p, d , d , f notations, describe the orbital with the following quantum numbers :
(a) n = 1 ; ℓ = 0 (b) n = 3 ; ℓ = 1
(c) n = 4 ; ℓ = 2 (d) n = 4 ; ℓ = 3
85. Explain giving reasons , which of the following sets of quantum numbers are not possible :
(a) n = 0 ; ℓ = 0 ; m ℓ = 0 ; m s = +½
(b) n = 1 ; ℓ = 0 ; m ℓ = 0 ; m s = - ½
(c) n = 1 ; ℓ = 1 ; m ℓ = 0 ; m s = +½
(d) n = 2 ; ℓ = 1 ; m ℓ = 0 ; m s = - ½
(e) n = 3 ; ℓ = 3 ; m ℓ = -3 ; m s = +½
(f) n = 3 ; ℓ = 1 ; m ℓ = 0 ; m s = + ½
86. How many electrons in an atom may have the following quantum numbers ?
(a) n = 4 ; m s = - ½ (b) n = 3 ; ℓ = 0
87. Show that circumference of the Bohr orbit for hydrogen is an integral multiple of de Broglie wave length associated with the electron revolving around the orbit.
88. Write the electronic configurations of Cu2+ and Cr3+ ions.
89. What atoms are indicated by the following electronic configurations :
i) 1s2, 2s2, 2px1, 2py1, 2pz1 and
ii) ii) 1s2, 2s1, 2px1, 2py1, 2pz1
Are these atoms in their ground state or in excited states ?
90. How many number of unpaired electrons are there in Cu, Br-and K+
91. What designations are given to orbitals having :
i) n = 2 ; ℓ = 1 (ii) n = 2 ; ℓ = 0
iii) n = 4 ; ℓ = 3 (iv) n = 4 ; ℓ = 2
v) n = 4 ; ℓ = 1
92. Show that the circumference of the Bohr’s orbit for hydrogen atom is an integral multiple of de Broglie wave length associated with the electron revolving around it.
93. Find energy of each photons which :
iii) corresponds to light of frequency 3 x 1015 Hz
iv) have a wave length of 0.50 A.
QUESTIONS.
1. What are sub-atomic particles of atoms ?
2. What are the properties of cathode rays ?
3. Compare the properties of electron proton and neutron.
4. Explain the part played by cathode rays in the elucidation of the structure of atom.
5. Explain the part played by positive rays in the elucidation of structure of atom.
6. In what respects cathode rays differ from positive rays ?
7. Explain the part played by alpha particle scattering in the elucidation of structure of atom.
8. In Rutherford 's experiment some of alpha particles, when bombarded against gold foil, were retraced their path. Give reason for this observation.
9. Explain Rutherford 's model of atom.
10. What is the composition of atom.
11. Give the drawbacks of Rutherford 's model of an atom.
12. What is the difference between a hydrogen atom and a proton ?
13. Define the terms:
i) Atomic number ii) Mass number iii) Electron.
14. Give an account of discovery of neutron.
15. How is atomic number related to atomic mass of an atom ?
16. Compare the characteristics of electron, proton and neutron.
17. Write a note on Moseley's work in understanding the importance of atomic number.
18. What is meant by electromagnetic nature of light ? Why is it called so ? Give its characteristics.
19. Explain the terms:
i) spectrum ii) Electromagnetic spectrum.
iii) continuous spectrum iv) Atomic spectrum.
20. Write briefly on quantum theory of electromagnetic radiation.
21. What is Bohr's atomic theory ?
22. How could Bohr's theory of atomic structure account for hydrogen spectrum ?
23. List the shortcomings of Bohr's theory.
24. What is photoelectric effect ? What are its main features ?
25. How is energy of 'quanta' of light related to frequency of light ?
26. How is photoelectric effect be used to show that light behaves as a stream of particles ?
27. How is particle nature of light explained on the basis of photoelectric effect ?
28. Explain the dual character of light radiations.
29. What is de Broglie's relation ?
30. What do you understand by quantum mechanical model of the atoms ?
31. State and explain Heisenberg's uncertainty principle.
32. How does wavelength of a moving particle vary with its mass ?
33. What is meant by an atomic orbital ?
34. What do you think of the matter wave, when a moving object comes to rest ?
35. What is meant by de Broglie wave length ?
36. Differentiate between an orbit and orbital.
37. What do you understand by quantum numbers ? What is the significance of these quantum numbers ?
38. Give the shape of the sub-shell for ℓ = 0, 1, 2, 3.
39. State and explain Pauli's exclusion principle.
40. How do four quantum numbers arise for the location of electrons in atoms ? What are the restrictions on their values ?
41. Which quantum number specify the orientation in space for an orbital ?
42. Why is there is no 2d or 3f orbital ?
43. Which of the following orbitals are not possible ?
44. Describe and draw the shapes of :
i) s-orbital ii) p-orbital iii) d-orbitals.
45. Which of the following orbitals are spherically symmetrical ?
i) Px ii) Py ii) s
46. What the maximum number of electrons that can be accomodated in s, p, d and f-subshells ?
47. How can atomic orbitals be specified ?
48. State and illustrate 'Hunds's rule ' of maximum multiplicity. What is its importance ?
49. Explain with appropriate reason why three p-electrons in nitrogen are unpaired and have parallel spins.
50. What is Pauli's exclusion principle ? What is its importance ?
51. What is Aufbau principle ?
52. What is (n + ℓ ) rule ?
53. What are the rules governing the probable electronic configurations of atoms ?
54. Explain why atoms with half-filled and completely filled orbitals have extra stability .
55. Write the electronic configuration of chromium and copper.
56. Arrange the following orbitals in the order in which electron be normally expected to fill:
3s, 2p, 4d, 3p, 3d.
57. The expected electronic configuration of chromium is [Ar]3d4 4s2, though actually it is [Ar]3d5 4s1. Comment.
58. How many electrons are there in the valence quantum level of copper atom ? Give reason.
59. Write the electronic configuration of K, Mn and Ca.
60. Give the electronic configurations of :
i) H- v) N-
ii) F- vi) F+
iii) Mg2+ vii) Al3+
iv) Li+ viii) O2-
Write the electronic configurations of Cu2+ and Cr3+.
61. The two extra-nuclear electrons in 1s orbital of helium have antiparallel spins. Why not they have parallel spins ?
62. The element A (atomic weight = 39) and B ( atomic weight = 80) contain 20 and 45 neutrons respectively in their nucleus. Give their electronic configurations.
63. What is e/m value of an electron ?
64. What is a photon ? How is the energy of a photon related to its (a) frequency (b) wave length ?