Dual Nature of Matter and Radiation
Light has been regarded as waves to explain the phenomena of reflection, refraction, diffraction etc. However in 1905, Albert Einstein suggested that light has a dual character, which means, it can behave as a particle as well as a wave. It was observed that some of the experimental facts regarding light could be explained only by assuming light to have particle like character while some other experimental facts could only be explained by assuming wave like character for light radiations. Since light is a kind of radiation, it may be concluded that all radiations behave like waves as well as particles. Such a wave like and particle like nature of radiation is known as dual nature of radiation.
Derivation of de-Broglie Relationship
The relationship may be derived by combining the mass-energy relationship proposed by Max Planck and Einstein. According to Planck, photon of light having energy E is associated with a wave of frequency n as:
E = hn
Distinction between Electromagnetic Waves and Matter Waves
| Electromagnetic waves | Matter waves |
|---|---|
| Electromagnetic are associated with electric and magnetic fields perpendicular to each other and to the direction of propagation of radiation | Matter waves may not be associated with electric and magnetic fields |
| Electromagnetic waves can be radiated into space or emitted | Matter waves are neither radiated into space nor emitted by the particles. These are simply associated with the particles |
Distinction between Wave and Particle
A particle occupies a well-defined position in space, which cannot be simultaneously occupied by another particle. If there is more than one particle in a given region of space, then their sum is equal to the number of individual particles. The sum can neither be more nor less. On the other hand, a wave is spread out in space. Two or more waves can co-exist in the same region. When two waves are present together, the resultant wave can be larger or smaller than the individual waves.
Uncertainty Principle
All moving objects in our daily life have well defined paths or trajectories. The path or trajectory of an object can be determined by knowing its position and velocity at various intervals of time. However, Werner Heisenberg, in 1927, pointed out that we can never measure accurately both the position and velocity (or momentum) of a microscopic particle as small as an electron. Consequently, it is not possible to talk of the trajectory of an electron. On this basis, Heisenberg put forward a principle known as uncertainty principle. According to Heisenberg's uncertainty principle, it is not possible to measure simultaneously both the position and velocity (or momentum) of a microscopic particle with absolute accuracy or certainty.
Physical Concept of Uncertainty Principle
To determine the position of an object, we must be able to see the object. This can be done with the help of light of suitable wavelength. When a beam of light falls on an object, the photons of this incident light are scattered by the object and the reflected light enters our eye. Now, if the object is large, its position and velocity will not change by the impact of the striking photons. Thus, it will be possible to determine both the position and velocity of the object simultaneously.
Significance of Uncertainty Principle in Daily Life
It should be borne in mind that the uncertainty is not due to lack of sufficiently refined techniques but it is due to the fact that one cannot observe microscopic things without disturbing them. No instrument can observe the position of an electron without affecting its motion. In other words, uncertainty principle is the fundamental limitation of nature. Thus, one cannot design an experiment to obtain an accurate value of both the position and momentum for microscopic objects. However, in daily life, this principle has no significance. This is because one comes across only large objects, (the objects which can be observed with naked eye), without altering their motion. The position and velocity of these objects can be determined accurately because in these cases, during the interaction between the object and the measuring device, the changes in position and velocity are negligible.
Probability Picture of Electrons and Concept of Atomic Orbital (Wave Mechanical Treatment)
The wave character of an electron and uncertainty in its position and momentum gave a serious blow to Bohr's model of an atom. According to Bohr, the electrons revolve around the nucleus in certain well-defined circular orbits. But the idea of uncertainty in position and velocity overruled the Bohr's picture of fixed orbits. Thus, the classical mechanics could not describe the behavior of electrons in atoms correctly. Therefore, the scientists started looking for a model, which could incorporate the dual character of matter and uncertainty principle. This resulted in a new approach called Wave Mechanics.
Why electron cannot exist in the nucleus?
On the basis of Heisenberg's uncertainty principle, it can be shown as to why electron cannot exist within the atomic nucleus. The radius of the atomic nucleus is of the order of 10-15 m. Now, if the electron were to exist within the nucleus, then the maximum uncertainty in its position would have been 10-15 m.
Physical Significance of Wave Function
In the physical sense, y gives the amplitude of the wave associated with the electron. It is known that in case of light waves, the square of the amplitude of the wave at a point is proportional to the intensity of light. Extending the same concept to electron wave motion, the square of the wave function, y2 may be taken as intensity of electron at any point. In other words, y2 determines the probability of finding the moving electron in a given region or it gives the probability density. Thus, y2 has been called the probability density and y the probability amplitude. Thus, the solutions of Schrodinger wave equation replace the discrete energy levels or orbits proposed by Bohr and led to the concept of most probable regions in space in terms of y2. A large value of y2 means a high probability of finding the electron at that place and a small value of y2 means low probability. If y2 is almost zero at a particular point, it means that the probability of finding the electron at that point is negligible. Therefore, the wave mechanics approach gives meaningful wave functions, which describe the position and energy levels of electrons in an atom.
Energy Expression for Hydrogen Like Ions
Hydrogen like ions are those which contain only one electron. For e.g., He+, Li2+ etc. The energy expression for these ions may be written as:
Concept of Orbital
According to wave mechanics, we cannot simply say that the electron exists at a particular point but we talk about certain regions in space around the nucleus where the probability of finding the electron is maximum. Such regions are expressed by mathematical expressions and are called orbital wave functions or commonly known as orbitals. Therefore, the wave equation leads to the concept of orbitals instead of well-defined circular orbits.
Quantum Numbers
Schrodinger wave equation, when solved for hydrogen atom gives mathematical Y functions, which describe the allowed energy states in an atom. It is observed that physical meaningful solutions required three constants n, l and m, which are called quantum numbers. Each electron in an atom is characterized by a set of definite values of these numbers. In addition to these three numbers, another quantum number is also needed which specifies the spin of the electron. These four numbers, which are required to specify the position and energy of the electron in an atom, are called quantum numbers. These are Principal quantum number (n), angular quantum number (l), magnetic quantum number (m) and spin quantum number (s). These quantum numbers combined together give a complete address of the electron in an atom.
Principal Quantum Number (n)
This quantum number determines the main energy level or shell in which an electron is present. It is usually denoted by n and can have any whole number value such as: n = 1, 2, 3, 4 ….. etc. The various values of n as 1, 2, 3, 4 ….etc., are also called K, L, M, N etc. If n = 1, the energy level is closest to the nucleus and as the value of n increases, the distance of the energy level from the nucleus increases.
Azimuthal or Angular Quantum Number (l)
This quantum number denotes the sub-level or subshell in a given principal energy shell to which an electron belongs. This is also called secondary quantum number. It is denoted by l and corresponding to each value of n, there are n possible values of l ranging from zero to one less than n for that state i.e., l = 0, 1, 2...(n - 1).
Magnetic Quantum Number (m)
This quantum number describes the behavior of the electrons in the magnetic field. We know that the movement of electrical charge is always associated with magnetic field.
Spin Quantum Number (s)
It is observed that the electron is not only revolving around the nucleus but it is also spinning on its own axis. The spin of the electron produces a small magnetic field as a result of which the electron behaves as a 'tiny magnet'. This quantum number describes the spin orientation of the electron. It is designated by s. Since the electron can spin only in two ways: clockwise and anti-clockwise and, therefore, the spin quantum number can take only two values: + ½ or - ½ .
Pauli's Exclusion Principle
This principle was proposed by Pauli in 1952. It states that no two electrons in an atom can have same values for all the four quantum numbers. Thus, in the same atom, two electrons may have the same values for three quantum numbers but the fourth must be different. Electrons having the same value of n, l and m are said to belong to the same orbital. For instance, consider K shell, i.e. n =1. The electron will have only one value of (l) which is l = 0 and one value of m, which is m = 0 but it can have two values of s, either + ½ or - ½. This means that although n, l and m are the same for the two electrons but their spin quantum numbers are different. Thus, an orbital can have maximum of two electrons. Moreover, if an orbital has two electrons, they must be of opposite spin.
Orbital Wave Functions and Shapes of Orbitals
According to wave mechanics, orbitals are described by wave functions known as orbital wave functions. These orbital wave functions can be represented as a product of two functions: (i) Radial wave function and (ii) Angular wave function.
Shapes of s orbitals
s orbitals are non-directional and spherically symmetrical, This means that the probability of finding the electron is same in all directions at a particular distance from the nucleus, The 1s orbital is shown in the figure 1.3.
Shapes of p orbitals
For p-orbitals (l=1), there are three possible orientations corresponding to m = -1, 0, +1 values. This means that there are three p - orbitals in each p-subshell. These are designated as px, py and pz; For e.g., 2px, 2py and 2pz.
Shapes of d orbitals
For d-orbital (l = 2), there are five possible orientations corresponding to m = - 2, -1, 0, + 1, +2. This means that there are five orbitals in each d-subshell. For 3d subshell, these are designated as 3dxy, 3dyz, 3dxz, 3dx2- y2 and 3dz2. These five orbitals are equal in energy but differ in their orientations. The shapes of 3d orbitals are shown in the figure 1.5.
Electronic Configurations of Atoms
The distribution of electrons in different orbitals is known as electronic configuration of the atom. The orbitals are indicated by a box (square or circular) and electrons are indicated by arrows. An empty square will mean an empty orbital, while a single electron will be indicated by drawing one arrow in the square and two electrons will be shown by a pair of arrows with their heads in the reverse direction. The direction of the arrow gives the orientation of its spin.
Aufbau Principle
The Aufbau principle states that in the ground state of an atom, an electron enters the orbital of lowest energy first, and then the subsequent electrons are fed in the order of increasing energies into the orbitals. The relative energies of various orbitals are given in fig.1.6. From the figure, the following sequence is observed for orbitals in the increasing energy:1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s.
Pauli's Exclusion Principle
According to this principle, an orbital can accommodate maximum of two electrons and these two must have opposite spins. This means that an orbital can have 0, 1 or 2 electrons. Moreover, if an orbital has two electrons, they must be of opposite spins.
Hund's Rule of Maximum Multiplicity
According to this rule, electron pairing will not take place in orbitals of same energy (same sub-shell) until each orbital is first singly filled with parallel spin. In other words in a set of orbitals having same energy (degenerate orbitals), the electrons distribute themselves to occupy separate orbitals with same spin as far as possible. This rule can be illustrated by considering the example of carbon.
First Row Elements
Electronic Configurations of Atoms
Based upon the rules and sequence of energy levels, one can write the electronic configurations of atoms of some rows of the periodic table.
Second Row Elements
In lithium (Z =3), the third electron goes to the next 2s orbital. In Beryllium (Z = 4), the fourth electron fills 2s orbital and in boron (Z = 5), the next electron goes to the next available 2p orbital. In carbon, the sixth electron is also to go into the 2p orbital as 2p subshell can accommodate six electrons. Here the Hund's rule of maximum multiplicity applies. In other words, the electrons enter the orbitals of same energy with parallel spin until all are singly filled. Thus, in accordance with Hund's rule, the differentiating electron must be placed in either 2py or 2pz,orbital.
Third Row Elements
The third row elements are from sodium to argon. The build up of these eight elements proceeds exactly in the same manner as discussed above. Here 3s- and 3p-orbitals are filled in place of 2s- and 2p-orbitals as has been done in case of Li to Ne.
Fourth Row Elements
In potassium (Z=19) and calcium (Z=20), electrons enter the 4s-orbitals in accordance with the fact that 4s-orbital is slightly lower in energy than 3d-orbital.
Exceptional Configurations of Chromium and Copper
The above configurations show that the electronic configurations of chromium and copper are slightly different from that of other elements.
Molecular Orbital Theory
Molecular orbital theory was proposed by F. Hund and R.S. Mulliken in 1932. The basic idea of molecular orbital theory is that atomic orbitals of individual atoms combine to form molecular orbitals. The electrons in molecules are present in the molecular orbitals, which are associated with several nuclei. These molecular orbitals are filled in the same way as the atomic orbitals in atoms are filled.
Linear Combination of Atomic Orbitals (LCAO) Method
According to LCAO method, the orbitals are formed by the linear combination (addition or subtraction) of atomic orbitals of the atoms, which form the molecule. This theory is applicable to hydrogen molecule as follows. Consider that hydrogen molecule consists of two atoms A and B. Each of these atoms has 1s-orbital of lowest energy. The atomic orbitals of these atoms may be represented by the wave functions yA and yB for hydrogen atoms A and B respectively. Now, when these atomic orbitals are brought closer, they combine to form molecular orbitals. According to LCAO method, the linear combination of atomic orbitals can take place by addition and by subtraction of wave functions of atomic orbitals.
Differences between Atomic and Molecular Orbitals
| Atomic Orbital | Molecular Orbital |
|---|---|
| An electron in atomic orbital is under the influence of only one positive nucleus of the atom | An electron in molecular orbital is under the influence of two or more nuclei depending upon the number of atoms present in the molecule |
Relative Energies of Bonding and Antibonding Molecular Orbitals
In the case of bonding molecular orbital, the attraction of both the nuclei for both the electrons is increased. This results in lowering of energy. In the case of antibonding molecular orbital, the electrons try to go away from the nuclei and this corresponds to repulsive state.
Differences between Bonding and Antibonding Molecular Orbitals
| Bonding MO | Antibonding MO |
|---|---|
| Bonding molecular orbital is formed by the addition of overlapping of atomic orbitals. The wave function of the bonding MO may be written as :ψ(MO)= ψA+ψB | Antibonding molecular orbital is formed by the subtraction of overlapping of atomic orbitals. The wave function for the antibonding MO may be written as: ψ(MO) = ψA-ψB |
| They are formed when the lobes of the combining atomic orbitals have same sign | They are formed when the lobes of the combining atomic orbitals have opposite sign |
Combination of 2s and 2p Atomic Orbitals to form Molecular Orbitals
Like 1s-orbitals, 2s-orbitals combine by addition and subtraction of overlappingwavefunctions to form bonding and antibonding molecular orbitals. These are labeled as s2s,and s*2s. These molecular orbitals have exactly the same shapes as s1s and s*1s MOs but they are slightly larger in size.
Differences between Sigma and Pi Molecular Orbitals
| σ Molecular orbitals | π Molecular orbitals |
|---|---|
| It is formed by the head-to-head overlap of atomic orbitals along internuclear axis | It is formed by sidewise overlap of atomic orbitals perpendicular to internuclear axis |
| The overlap region is maximum | The overlap region is minimum |
Conditions for the Combination of Atomic Orbitals
The combining atomic orbitals should not differ much in energies. For e.g., in case of homonuclear diatomic molecules of the type A2' 2s-orbital of one atom can combine with 2s-orbital of another atom but 1s-orbital of one atom cannot combine with 2s-orbital of another atom.
Energy Level Diagram for Molecular Orbitals
The relative energies of molecular orbitals depend upon the following two factors:
(i) the energies of the atomic orbitals combining to form molecular orbitals.
(ii) the extent of overlapping between the atomic orbitals.
Rules for Filling Molecular Orbitals
Each molecular orbital can accommodate maximum of two electrons having opposite spins. This is in accordance with Pauli's exclusion principle.
Electronic Configurations and Molecular Behavior
Stability of a Molecule
We have learnt that the electrons in bonding molecular orbitals contribute to attraction between the atoms and thereby help in the formation of bond. On the other hand the electrons in antibonding molecular orbitals contribute to repulsion between the atoms and, therefore oppose the formation of bond. Thus, the electrons in bonding molecular orbitals will give stability to the molecule whereas those in antibonding molecular orbitals will decrease the stability.
Bonding in Some Diatomic Molecules
Bonding in some homonuclear diatomic molecules of the elements of first and second rows of the periodic table:
1. Hydrogen molecule (H2)
2. Hydrogen molecule ion (H2+)
3. Hypothetical helium molecule (He2)
Hydrogen Molecule
It is formed by the combination of two hydrogen atoms. Each hydrogen atom has one electron in 1s-orbital and, therefore, there are two electrons in hydrogen molecule. Both these electrons are to be accommodated in the lowest energy molecular orbital. According to Pauli's exclusion principle, these two electrons should have opposite spins.
Hydrogen Molecule Ion
This is formed by the combination of hydrogen atom containing one electron and hydrogen ion having no electron. Therefore, this ion has only one electron.
Hypothetical Helium Molecule
Each helium atom has two electrons in 1s-orbital and therefore, there are four electrons in He2 molecule. These electrons will be accommodated in si1ls and s*1s MOs as shown below.
Helium Molecule Ion
This molecule contains three electrons, two from one helium atom and one from the other helium (He+) ion. The molecular orbital electronic configuration of the molecule is:
Nitrogen Molecule
Oxygen Molecule
Fluorine Molecule
The electronic configuration of fluorine atom is 1s2 2s2 2p5 and therefore, there are 14 electrons in the valence shell of F2 molecule.
Hypothetical Neon Molecule
The electronic configuration of neon atom is 1s2 2s2 2p6 and the molecular orbital electronic configuration of neon molecule is:
Ne2:KK(s2s)2(s*2s)2(s2pz)2(p2px)2(p2py)2(p*2px)2(p*2py)2 (s*2pz)2
Metallic Bond (Band Theory)
Metals constitute about three fourths of all the known elements. They have characteristic properties such as bright luster, high electrical and thermal conductivity, malleability, ductility and high tensile strength. The attractive force, which binds various metal atoms, is called metallic bond. The metallic bond is neither a covalent nor an ionic bond. Molecular orbitals form between two atoms when atomic orbitals of the two atoms overlap. In some cases (such as benzene), the atomic orbitals of three or more atoms overlap to form molecular orbitals that are associated with all the atoms. Such molecular orbitals are said to be delocalized. The number of molecular orbitals formed is equal to the number of atomic orbitals participating in overlapping.
Conduction in Metals
Electrons become free to move throughout a crystal when they are excited to a higher energy level within the band. If the highest energy band that is occupied is not fully filled, electrons may be excited from lower energy level to higher energy level by supplying a very small amount of energy because vacant orbitals lie just above the occupied orbitals of highest energy.
Hybridization
The structures of different molecules can be explained on the basis of hybridization. For e.g., in case of carbon, the ground state electronic
Hybridization in Elements of Third Period Involving d-Orbitals
The elements of the third period contain d-orbitals also in addition to s- and p-orbitals. The 3d-orbitals are comparable in energy to the 3s- and 3p-orbitals. These d-orbitals are also involved in the hybridization and they explain the geometries of molecules of elements of third period. This results in covalencies of 5, 6 and 7, which are not known amongst the compounds of second period elements.
sp3d Hybridization
This Hybridization involves the mixing of one s, three p and one d-orbital. These five orbitals hybridize to form five sp3d-hybrid orbitals. The mixing of five orbitals is shown in figure 1.23. These hybrid orbitals point towards the corners of a trigonal bipyramidal geometry. In this case, the three orbitals forming a plane are directed towards the corners of an equilateral triangle while the other two are perpendicular to the plane of the triangle lying above and below it.
sp3d2 Hybridization
In this case, one s, three p and two d-orbitals get hybridized to form six sp3d2hybrid orbitals which adopt octahedral arrangement.
sp3d3 Hybridization
This involves the mixing of one s three p and three d-orbitals forming seven sp3d3 hybrid orbitals having pentagonal bipyramidal geometry. The geometry of IF7 molecule can be explained on the basis of sp3d3 Hybridization.
dsp2 Hybridization
In addition to above types of Hybridization, dsp2 type of hybridization is also known particularly in case of transition metal ions. The orbitals involved in this type of Hybridization are dx2- y2, s and two p. The four dsp2 hybrid orbitals adopt square planar geometry.
Hybrid Orbitals and Molecular Shapes of Molecules Involving d Orbitals
The Hybridization and molecular shapes of some molecules involving d-orbitals are summarized in the table given below.
Intermolecular Forces
The attractive forces between molecules are called intermolecular forces. Intermolecular attractions hold two or more molecules together. Intermolecular attractions can be of the following types:
1) Dipole-dipole interactions
2) London forces (Dispersion forces)
3) Hydrogen bonds.
London Forces
London forces are the interparticle forces among the non-polar molecules such as hydrogen (H2), oxygen (O2), chlorine (Cl2), etc., in solid or liquid states.
Hydrogen Bond
Hydrogen bond is an intermolecular force of attraction that exists in the molecules that have hydrogen atom bonded to nitrogen, oxygen, or fluorine. Hydrogen bond is a strong dipole-dipole attraction.
Conditions for Hydrogen Bonding
Since hydrogen bond comes into existence as a result of dipole-dipole interactions between the molecules, therefore, following conditions are required for effective hydrogen bonding.
Examples of Hydrogen Bonding
In hydrogen fluoride, hydrogen atom is bonded to highly electronegative atom, fluorine (electronegativity = 4). It has been found that in solid state hydrogen fluoride consists of long zig-zag chains of H-F molecules associated by H-bonds. On heating, progressively, the length of the chain shortens, and associated units become quite small.
Summary or Key concepts
It is not possible to measure simultaneously both the position and velocity (or momentum) of a microscopic particle with absolute accuracy or certainty.
