UNIT 01 ATOMIC STRUCTURE AND CHEMICAL BONDING
SYLLABUS
· Dual nature of matter and radiation.
· Heisenberg’s Uncertainty principle
· Quantum mechanics
· Quantum Mechanical Treatment of the Hydrogen atom.
· Atomic orbitals and their pictorial Representation.
· Electronic configuration of Elements.
· Chemical Bonding.
· Molecular Orbital Theory.
· Bonding in some Homonuclear Diatomic molecules.
· Metallic Bonding.
· Hybridisation.
· Intermolecular forces.
Albert Einstein in 1905 , suggested that light has a dual character, i.e., it can behave as a particle as well as wave. It was realized that some of the experimental facts regarding light could be explained only by assuming light to have wave like character while some other experimental facts could only be explained by assuming particle like character for light radiations.
Wave like character of radiation
Wave like character of light was proposed by Huygens. With the help of wave theory of light, Huygens explained the important phenomena of light such as reflection, refraction, diffraction and interference. In 1856, James Maxwell proposed that light and other forms of radiation propagate through the space in the form of waves and these waves have electric and magnetic fields associated with them. Therefore, these radiations are known as electromagnetic radiations or electromagnetic waves. Thus electromagnetic radiation is a form of energy which propagates through the space in the form of waves having electric and magnetic fields associated with them.
Limitations of wave character of radiation
The wave character of radiation failed to explain many phenomenon such as :
(i) Black body radiation
(ii) Photoelectric effect
Black body radiation
An ideal black body is defined as a perfect emitter of radiations. This means that a black body absorbs all the radiations falling on it. When such a body is heated, it emits radiations and it has been observed that no other body can emit radiant energy more than a black body. Hence a black body is the most efficient absorber radiant energy and it is also the most perfect emitter. This behaviour of black body
could not be explained on the basis of wave character of radiation.
Photoelectric effect
When a beam of light falls on certain metal plates in vaccum , the plates emit electrons. This effect was discovered by Hertz and is known as Photoelectric Effect. Photoelectric effect may be defined as the phenomenon of ejection of electrons from the surface of a metal when light of suitable frequency strikes it. The electrons, thus ejected are called photoelectrons.
Fig 1 Photoelectric Effect
It must be noted that the photoelectric effect is shown by any metal in a photoelectric cell only. The apparatus showing photoelectric effect is given in Fig 1. It comprises an evacuated tube with metal having low ionisation energy and constituting the negative electrode. Now light of sufficiently high energy strikes the metal, electrons are knocked off from its surface and move towards the positive electrode. The electrons constitute a current flowing through the circuit. Only a few metals show this effect under the action of visible light. Caesium with lowest ionisation energy emits electrons very easily.
Some important observations about photoelectric effects are :
(i) For a particular metal, there exists a threshold frequency such that, at frequencies below threshold, no electrons are emitted, no matter how large, the intensity is or how long the radiation occurs.
ii) The maximum kinetic energy of photoelectrons is directly proportional to the frequency of the incident radiation, but is independent of its intensity.
iii) The number of photoelectrons emitted per second is directly proportional to the intensity of the incident radiation but does not depend upon its frequency.
All these observations could not be explained on the basis of classical wave theory of light. According to classical theory, radiant energy is continuous and therefore , radiation of any frequency should have been able to bring about the ejection of the electrons simply by increasing the intensity of radiation. But all the experimental observations were contrary to what we expected.
Particle like character
The failure of classical electromagnetic theory to explain photoelectric effect led to the postulation of Planck’s quantum theory of radiation.
Planck’s Quantum Theory of Radiation
Planck in 1901 , gave a new theory of radiation known as quantum theory of radiation. The main features of the theory are given below :
(i) Radiant energy is not emitted or absorbed continously but discontinuously in the form of small packets of energy called quanta. Each wave packet or quantum is associated with definite amount of energy. In the case of light, the quantum of energy is often called photon.
(ii) The amount of energy associated with a quantum of radiation is proportional to the frequency of radiation and is expressed as :
E a h n
E = h n
Where h is a fundamental constant known as Planck’s constant and n is the frequency of radiation. The numerical value of h is equal to 6.626 x 10-34 J s.
(iii) A body can emit or absorb energy only in terms of whole number multiples of quantum, i.e.,
E = n h n where n = 1, 2, 3. 4 , ……
This means that a body can emit or absorb energy equal to h n , 2 h n, 3 h n, ….or any other integral multiple of h n but it cannot emit or absorb energy equal to 1.6 h n or any other fractional value of , h n.
From Planck’s quantum theory of radiation, it may be concluded that radiations from any source do not represent a continuous flow of electromagnetic waves but a stream of individual tiny bundles of discrete energies called quanta or photons. This suggests that the light should have corpuscular or particle character.
Explanation of Photoelectric Effect
Einstein used the concept of quantum theory for explaining photoelectric effect and considered light as having a dual character- wave as well as particle. When light falls on the metallic surface , a photon of light is absorbed , which interacts with the most loosely bound electron in the metal surface, thereby parting-with all its energy. Part of this energy acquired by electron is just sufficient to separate itself from the metal surfaces ; while the remaining energy appears in the form of kinetic energy of the ejected electron.
Thus :
Energy of the photon
= ( Threshold energy ) + (K.E acquired by electron
ejected)
h n = h no + ½ m v2.
K.E = h n - h no
The validity of the above expression was tested by Millikan by calculating the value of h. It was found to be 6.57 x 10-34 J s.
Since for a particular metal no and h are constants, so kinetic energy of photoelectrons is directly proportional to frequency. Kinetic energy of photoelectrons is independent of intensity because increase in intensity does not effect the energy of photons rather it simply increases the number of photons falling on the surface of metal and hence increases the number of photoelectrons.
The photoelectric effect supports the particle nature of light because only a light of suitable frequency and not of any frequency can bring about the emission of photoelectrons. Also, increasing the frequency of radiation increases the velocity of electrons whereas increasing the intensity does not change it.
DUAL NATURE OF MATTER OR ELECTRON
When an electron was discovered , the various experiments carried out to characterize it (like Thomson’s experiments, Millikan’s oil drop experiments) showed that an electron behaves as a particle. In Bohr’s theory , electron is treated as a particle. However, in 1924 , Louis de Broglie suggested that just as light exhibits wave and particle properties, all microscopic material particles such as electrons, protons, atoms, molecules, etc. have also dual character. According to de Broglie, all material particles in motion also possess wave characteristics.
de Broglie Equation
According to de Broglie, the wave length associated with a particle of mass m, moving with velocity v , is given by the relation,
where h is Planck’s constant and mv is the momentum of the particle. The waves associated with material particles are called matter waves.
Derivation of de Broglie relationship
de Broglie using Planck's Quantum Theory and Einstein's theory derived a relationship between the magnitude of wave length (l) associated with the particle of mass m moving with a velocity c .
According to Planck, the photon energy is given by :
E = h n ……….(1)
and by Einstein's mass-energy relation , the energy is given by :
E = m c2 ………..(2)
Comparing (1) and (2) , we get :
( since n = c / l )
( where p = m c )
The equation (3) is called de-Broglie's equation.
de Broglie pointed out that the same equation might be applied to material particle by using m for the mass of the particle instead of the mass of photon and replacing c by the velocity of the particle v :
de Broglie wave length
According to de Broglie, every moving particle of mass 'm' and velocity 'v' , is associated with a wave length l is given by :
h = Planck's constant
p = m v
= momentum of the particle.
l is called de Broglie wave length .
Note
(i) The wave length associated a moving particle is
inversely proportional to its mass, i.e.,
(ii) Greater the mass of the moving object, shorter the wave length of the matter wave associated with it.
(iii) A lighter moving object has a larger wave length associated with it.
(iv) Basically de Broglie's equation is true for material particles of all sizes and dimensions. However, in the case of large macro objects, the wave character is almost negligible and it cannot be measured properly. In the case of small or micro objects (like electrons), only wave character is of significance. Hence de-Broglie equation is more useful for small (or micro) particles.
Distinction between electromagnetic waves and matter waves
Electromagnetic radiation | Matter waves |
1. These waves are associated with electric and magnetic fields. | 1. These waves are not associated with electric and magnetic fields. |
2. Electromagnetic waves can be emitted or radiated in space. | 2. Matter waves are neither radiated into space nor emitted by the particles. These are simply associated with the particles. |
3. All electromagnetic waves travel with the same velocity. | 3. Matter waves travel with different velocities. |
4. The velocity of alll electromagnetic waves is equal to that of light ( = 3 x 108 ms-1) | 4. The velocity of matter waves is different from that of light. |
Significance of de Broglie equation
Although de Broglie equation is applicable to all material objects, but it has significance only in the case of microscopic particles. This can be illustrated with the following examples :
Consider a ball of mass 0.1 kg moving with a speed of 60 m s-1. From de Broglie equation, the wave length of the associated wave is :
It is apparent that this wave length is too small for ordinary observation.
On the other hand, an electron with a rest mass equal to 9.106 x 10-31 kg i.e., approximately 10-31 kg moving at the same speed would have a wave length :
which can be easily measured experimentally.
Since we come across macroscopic objects in our everyday life, therefore de Broglie relationship has no significance in our every day life. That is why we do not observe any wave nature associated with the motion of a running car or a cricket ball.
Problems
1. What will happen to the wave length associated with a moving particle if its velocity is doubled ?
2. Two particles A and B are moving with the same velocity but wave length of A is found to be double than that of B . What do you conclude ?
3. A molecule of O2 and that of SO2 travel with the same velocity. What is the ratio of their wave lengths ?
4. What is the main difference between the wave emitted by a bulb and that associated with a particle ?
5. Two particles A and B are in motion. If the wave length associated with particle A is 5 x 10-8 m, calculate the wavelength associated with particle B if its momentum is half of A.
6. Calculate the wave length of 1000 kg rocket moving with a velocity of 3000 km per hour.
7. The sodium flame test has a characteristic yellow colour due to emissions of wavelength 589 nm. What is the mass equivalence of one photon of wavelength ?
DUAL NATURE OF ELECTRON
The following experimental evidence confirm the dual particle and wave nature of electrons. The particle character of the electron is proved by the following different experiments.
Verification of particle character
(i) When an electron beam strikes a zinc sulphide screen , a spot of light known as scintillation is produced. Since scintillation is localized on the zinc sulphide screen, therefore the striking electron which produces it also must be localized and is not spread out on the screen. But the localized character is possessed by particles. Hence electron has particle character.
(ii) Thomson’s experiment for determination of the ratio of charge to mass (i.e., e/m) and Milliken’s oil drop experiment for determination of charge on the electron also show that electron has particle character.
(iii) The phenomenon of Black body radiation and photoelectric effect also prove the particle nature of radiation.
Verification of wave character
(i) Davison and Germer’s experiment
Davison and Germer (1927) observed that when a beam of electrons is allowed to fall on the surface of a nickel crystal and the scattered or reflected X-rays are received on a photographic plate, a diffraction pattern (consisting of a number of concentric rings), similar to that produced by X-rays is obtained.
Fig 2 X-ray diffraction rings produced by nickel crystal
Since X-rays are electromagnetic waves, i.e., they have confirmed to have wave character, electrons must also have wave character. Moreover, the wave length determined from diffraction is found to be very nearly the same as calculated from de Broglie equation.
(ii) Thomson’s experiment
G.P. Thomson (1928) performed experiments with thin gold foil in place of nickel crystal. He observed that if the beam of electrons after passing through the thin foil of gold is received on the photographic plate placed perpendicular to the direction of beam, a diffraction pattern is observed as before (Fig 3).
Fig 3 Electron diffraction rings produced by a thin gold film.
This again confirmed the wave nature of electrons.
de Broglie's concept of dual nature of matter finds application in the construction of electron microscope and the study of surface structure of solids by electron diffraction.
Wave nature of electron and quantisation of angular momentum
Consider an electron moving in a circular orbit around the nucleus. The associated wave train would be as shown in Fig 4a .
Fig4 . The wave trains associated with electron moving in a circular path around a nucleus.
If the wave is to remain continually in phase (Fig a) , the circumference of the circle must be an integral multiple of wave length l , that is ,
Circumference = n l
….……(1)de-Broglie’s equation is :
………(2)Combining (1) and (2) , we get:
……….(3)If the angular momentum is bigger or smaller than the value ( nh / 2p) given in equation (3) , the wave will no longer remain in phase (Fig 4 b).
Since mvr represents the angular momentum of the electron, it follows that the electron can move only in those orbits for which angular momentum is an integral multiple of h/2p . In other words, the angular momentum is quantised.
HEISENBERG UNCERTAINTY PRINCIPLE
All moving objects have well defined paths or trajectories. The path or trajectory of an object can be determined by knowing the velocity at various intervals of time. Werner Heisenberg (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.
Mathematically, uncertainty principle may be put as :
where , Dx = uncertainty in position.
Dp = uncertainty in momentum.
h = Planck's constant.
The sign ³ means that the product of Dx and Dp can be greater than or equal to , but never smaller than h/4p. If Dx is made small , Dp increases and vice versa.
Since p = m v , therefore , Dp = m Dv . The above relation , therefore can also be written as :
which means that position and velocity of an object cannot be simultaneously known with certainty.
It must also be understood that the uncertainty principle applies to location and momentum along the same axis i.e., if Dx is the uncertainty in position along X-axis, then Dp must also be the uncertainty in momentum along X-axis. Here it may be emphasised that this principle is not due to any limitation of the measuring instrument.
Explanation
Suppose we attempt to measure both the position and momentum of an electron. To pin-point the position of the electron, we have to use light so that the photon of light strikes the electron and the reflected photon is seen in the microscope (Fig).
Fig 5 Change of momentum and position of electron on impact with a photon.
As a result of hitting , the position as well as the velocity of the electron are disturbed.
But according to the principle of optics , the accuracy with which the position of the particle can be measured depends upon the wavelength of light used. The uncertainty in the position is ±l. The shorter the wavelength , the greater is the accuracy. But shorter wavelength means higher frequency and hence higher energy. This high energy photon on striking the electron changes its speed as well as direction.
Alternatively, shorter wavelength implies higher momentum ( as l = h/p i.e., p = h/l ). Thus photon will have high momentum and a larger but indefinite amount of it will be transferred to the electron at the time of collision. This will result in greater uncertainty in the velocity of electron. On the other hand , decreasing the momentum means increasing the wave length which will lead to greater uncertainty in position.
Significance of Heisenberg uncertainty principle
One of the important implications of the Heisenberg uncertainty Principle is that it rules out existence of definite paths or trajectories. The trajectory of an object is determined by its location and velocity at various moments. If we know where a body is at a particular instant and if we know its velocity and force acting on it at that instant, we can tell where the body would be sometime later. We therefore conclude that the position of an object and its velocity fix its trajectory. Since for a sub-atomic object such as an electron, it is not possible to simultaneously determine the position and velocity at any given instant to an arbitrary degree of precision, it is not possible to talk of the trajectory of an electron.
Although Heisenberg’s uncertainty principle holds good for all objects , but it is of significance only for microscopic particles. The energy of the photon is insufficient to change the position and velocity of bigger bodies when it collides with them. Since in our everyday life we come across big objects only, the position and velocity can be measured accurately, Heisenberg’s uncertainty principle has no significance in everyday life.
Note : On the basis of Heisenberg’s uncertainty principle, it can be shown that why electrons cannot exist with in the atomic nucleus. This is because the diameter of the atomic nucleus is of the order of 10-14 m. Hence if electron were to exist within the nucleus, the maximum uncertainty in its position would have been 10-14 m.
Taking the mass of electron as 9.1 x 10-31 kg , the minimum uncertainty in velocity can be calculated by applying uncertainty principle as follows :
This value is much higher than the velocity of light (viz. 3 x 108 ms-1) and hence is not possible.
PROBLEMS
8. Calculate the momentum of a particle which has a de-Broglie wave length of 0.1 nm.
9. Calculate the wave length associated with a cricket ball of 100 g moving with a velocity of 30 m s-1. Comment your result.
10. A particle moving with a wave length 6.6 x 10-6 m is moving with a velocity of 10-4 m s-1. Find the mass of the particle.
11. What must be the velocity of a beam of electrons , if they are to display a de Broglie wave length of 100Å ?
12. The wave length of a moving body of mass 0.1 mg is
3.12 x 10-29 m. Calculate the kinetic energy of the
body.
13. Electromagnetic radiation of wave length 242 nm just sufficient to ionise the sodium atom. Calculate the ionisation energy of sodium in kJ mol-1.
14. When would the wave length associated with an electron be equal to the wave length associated with a proton. Mass of proton = 1.6725 x 10-27 kg.
15. Two particles A and B are in motion. If the wave length associated with particle A is 5 x10-8 m, calculate the wave length of particle B, if its momentum is half that of A.
16. Calculate the uncertainty in velocity of an electron if the uncertainty in its position is 100 pm.
17. If the uncertainty in its position is of the order of 100 pm, calculate the uncertainty in velocity of a cricket ball of mass 0.15 kg.
18. Calculate the uncertainty in velocity of a wagon of mass 2000 kg whose position is known to an accuracy of ±10 m.
19. Calculate the uncertainty in position of a dust particle with mass equal to 1 mg if the uncertainty in its velocity is 5.5 x 10-20 ms-1 .
20. On the basis of Heisenberg’s uncertainty principle, show that electron cannot exist within the atomic nucleus (radius 10-10 m).
OPERATOR
While studying the state of a system , we make various measurements of its properties such as mass, volume, momentum , position and energy. Each individual property is called observable. In order to determine the value of the observable property, we have to perform certain mathematical operations. This operation is reresented by an operator. Therefore, operator is a mathematical command or instruction which acts on a mathematical function. For example , in the equation 4 x 5 = 20, the operation is multiplication and the operator is ‘x’ . We can also express the multiplication operation with some symbol , designated as
QUANTUM MECHANICS
On the basis of dual nature of matter and Heisenberg Uncertainty Principle, Erwin Schrodinger (1926) developed a new branch of science called quantum mechanics ( also known as wave mechanics). Schrodinger equation, which for a system such as an atom or a molecule whose energy does not change with time (time independent) , is written as
H Y = E Y …………..(1)
Where H is the total energy operator called Hamilitonian. Hamilitonian operator is the sum of kinetic energy operator (T) and potential energy operator (V) i.e., H = T + V. The potential energy operator V for a system is normally equal to its expression for potential energy V. Y is the wave function of the system and E is the total energy of the system . Equation (1) may therefore be written as ( T + V ) Y = E Y. Writing Schrodinger equation for a system, therefore , means writing the explicit mathematical forms of T and V in the above equation. This equation is then solved to get E and Y for the system.
The quantum mechanical study of any system consists of :
1. Writing Schrodinger equation for the system.
2. Solving Schrodinger equation for the meanigful solutions of the wave functions and corresponding energies. The meanigful solutions of Y function must follow the conditions i.e., they have to be single valued, continuous and finite.
3. Calculation of all the observable properties of the system from Y.
The quantum mechanical approach has gained wide acceptance as the results obtained are in excellent agreement with the experimental findings. In particular, the Schrodinger equation is very well suited to interpret the experimental information about the structure of atoms and molecules.
QUANTUM MECHANICAL TREATMENT OF THE ATOM
On the basis of de Broglie's hypothesis and Heisenberg's uncertainty principle a new model of atom was developed during 1920's. In this model behaviour of the electron in an atom is described by an equation known as Schrodinger wave equation. Schrodinger equation for hydrogen atom can be written in terms of cartesian cordinates (x, y, z) or in terms of the spherical polar cordinates (r, q, f) of the electron with respect to the nucleus.
¶2 Y + ¶2 Y + ¶2 Y + 8 p2m ( E - V ) Y= 0
¶ x2 ¶y2 ¶z2 h2
where x, y and z are the three space co-ordinates. m is the mass of the electron, h is the Planck's constant,
E is the total energy and V is the potential energy of the electron,Y is amplitude of the electron wave and is called wave function and ¶2 Y refers to the second derivative of
¶ x2
Y with respect to x only and so on.
Since the atom has spherical symmetry, it is more convenient to write Schrodinger equation in terms of polar coordinate systems are shown in Fig.
The coordinates x, y and z of the electron with respect to nucleus in terms of polar coordinates are given by :
x = r sinq cos f
y = r sinq sinf
z = r cosq
x2 + y2 + z2 = r2
When Schrodinger equation in polar cordinates is solved for hydrogen atom, it gives possible energy states and corresponding wave function [y(r , q, f)] (called atomic orbitals or hydrogenic orbitals, which are the mathematical functions of the coordinates of the electron associated with each energy state). It can be shown that an atomic orbital is the product of two factors : (i) the radial part, dependent on r and (ii) the angular part , dependent on q and f. The quantized electronic states of the hydrogen atom are given by
….(1)where me is the mass of electron in kg , e is its charge in C , h is the Plank constant , n is the principal quantum number eo is vaccum permittivity . It is a fundamental constant and has the value 8.85419 x 10-12 J -1 C2 m-1. It may be noted that in actual calculations , the reduced mass (m) is used in place of me.
where mn is the mass of the nucleus. The mass of nucleus is very large in comparison to the mass of electron and therefore , except for precise work reduced mass can be taken equal to me.
The enrgy expression in CGS units may be expressed as :
En = - 2 p2 me e4
n2 h2 ……..(2)
Note
It may be noted that the above expression(Equ, 1) may be written as :
En = - 2 p2 me e4
n2 h2
The electronic energy of the hydrogen atom depends only on n and is independent of ℓ and m.
The above quantised energy states and corresponding wave functions which are characterized by set of three quantum numbers (principal quantum number n, azimuthal quantum number ℓ and magnetic quantum number mℓ )arise as a natural consequence in the solution of the Schrodinger equation. The restrictions on the values of these quantum numbers also come naturally from this solution. The quantum mechanical solution of the hydrogen atom successfully predicts all aspects of the hydrogen atom spectrum and other phenomenon that could not be explained by Bohr model.
The Schrodinger equation cannot be solved exactly for a multi-electron atom. However, solutions of resonable accuracy can be obtained using approximate methods. The total wave function for multi-electron atom can be constructed from atomic orbitals having different energies. These atomic orbitals are the function of coordinates of a single electron and their angular part has the same form as for hydrogen atom depending up on ℓ and mℓ values. However, their radial part is different and also takes into account the mutual repulsion between the electrons and depends on n, ℓ and charge Z on the nucleus. According to the quantum mechanical model of atom, these atomic orbitals form the basis of electronic structure of atoms. In multi-electron atom the electrons are filled in various orbitals in the order of increasing energy according to certain rules .
Significance of Y
As a moving electron is associated with a wave and a wave is completely defined by its amplitude , therefore Y refers to the amplitude of electron wave. It has no physical significance. However the square of Y i.e., Y2 has a physical significance. Just like light radiations where square of amplitude gives intensity of light , similarly , in electron wave Y2 gives the intensity of electrons. In other words, the knowledge of Y2 is helpful in assessing the probability of electron in a particular region.
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