UNIT 2 ( PAGE 4)
BOHR’S MODEL FOR HYDROGEN ATOM
Niel’s Bohr , a Danish physicist put forward his model of the atom in 1913. He retained Rutherford’s model of very small positively charged nucleus at the centre. He also accepted the view that all the protons and neutrons ( and hence most of the mass of the atom ) are contained in the nucleus. He also agreed that negatively charged electrons are revolving around the nucleus in the same way as the planets are revolving round the sun. But he applied Planck’s Quantum Theory to the electron revolving round the nucleus. The important postulates of Bohr’s theory are :-
i) The electrons in an atom revolve around the nucleus only in certain selected circular orbits. These orbits are associated with definite energies and are called energy shells or energy Levels. These are numbered as 1,2,3...or designated as K,L, M, N, ..... etc. shells.
ii) Only those orbits are permitted in which the angular momentum of the electron is whole number multiple of (h/2), where ‘h’ is Planck’s constant. That is :
Angular momentum of the electron , m v r = n ( h / 2)
where n = 1, 2, 3, .....n.
m = mass of electron
v = the velocity of the electron.
r = the radius of the orbit.
In other words angular momentum is quantised.
iii) As long as the electron remains in a particular orbit, it does not lose or gain energy. This means that energy of electron in a particular orbit remains constant. That is why, these orbits are also called Stationary states.
iv) When energy from external source is supplied to the electron, it may jump to some higher energy level by absorbing a definite amount of energy (equal to the difference in energy between the two energy levels). When the electron jumps back to the lower energy level it radiates same amount of energy in the form of photon of radiation.
where is the frequency of the radiation emitted when
electron jumps from energy level E2 to E1.
Fig 19. Energy changes in electron jump.
Success of Bohr's Model
The main success of Bohr’s model are :
i) Bohr’s model could explain the stability of an atom. According to Bohr’s model, an electron revolving in a particular orbit cannot lose energy. The electron can lose energy only if it jumps to some lower energy level. If no lower energy level is vacant then the electron will keep on moving in the same orbit without losing energy and hence it explains the stability of the atom.
ii) Bohr’s theory helped in calculating energy of electron in a particular orbit of hydrogen . On the basis of various postulates of Bohr’s theory, it is possible to derive a mathematical relation for energy of an electron in the nth orbit of hydrogen. The expression is :
En = 2 2 m e4 Z2
n2 h2
where , Z = nuclear charge
m = mass of electron
e = charge on the electron
h = Planck’s constant.
Substituting the values of Z , m, e, h and in the above equation, we get :
En = 1312 kJ mol1 = 2.179 x 1021 kJ atom1
n2 n2
= (13.595 eV) atom1
n2
For single electron species such as He+, Li2+, etc. the energy of the electron in the nth orbit is given by the expression :
En = 2 2 m Z2 e4
n2 h2
where Z = Atomic number of the element :
Z = 2 for He+ and Z = 3 for Li2+.
iii) Bohr Theory can explain the atomic spectrum of hydrogen.
Radius of Bohr orbit
Radius of each circular orbit from the expression :
rn = 0.529 x 1010 (n2) m
One can see that as n increases , rn also increases indicating greater separation between the orbit and the nucleus. The radius of first orbit r1, is called Bohr radius ( n= 1) is 52.9 pm.
The radius for ions such as He+, Li2+ the radius is given as :
where Z is the atomic number and has values 2 and 3 for He+, Li2+ respectively.
Bohr's Theory and Emission of Line Spectra
Bohr’s theory successfully explained the emission of line spectra of hydrogen atom. Consider a hydrogen atom in the ground state. The electron is then in the first K-shell. The atom therefore has minimum energy. Now, suppose it absorbs energy from the surroundings. This causes the electron jump into one of the higher energy levels ( n = 2, 3, 4, etc. ) depending upon the energy absorbed by it.
The atom is in an excited state. This is an unstable situation. The electron therefore falls back almost immediately to one of the lower energy levels or even to its ground state. The difference in the energy possessed by the electron in the higher level and lower level is emitted by the atom in the form of Line spectrum.
Suppose the electron in an excited atom returns from a higher energy level associated with energy E2 to a lower level associated with energy E1. The difference in energy ( E2 - E1 ), is then emitted as a quantum of radiation in the form of a spectral line of frequency such that :
where c is the velocity of radiation and is the wave length. Since ‘c’ and ‘h’ are constants (E2 E1) has a definite value, each jump from one energy level to another will give rise to a spectral line of specific wave length in spectrum of hydrogen.
Bohr's theory and Absorption spectrum
If hydrogen is exposed continuously to a strong source of light ( e.g. an arc light ), then according to Bohr’s theory, the electron present in the atom of the element may pass from a lower energy to a higher energy level. Suppose the electron pass from a lower energy associated with energy E1 to a level associated with a higher energy E2. Then an energy equal to (E2 E1 ) will be absorbed from the light falling on it. As a result of this the spectral line of frequency corresponding to energy( E2 E1) will be missing from the spectrum of the light falling on the element. This gives rise to absorption spectrum.
To sum up
If the atom loses energy so that the electron passes from a higher to lower energy level, energy is evolved and a spectral line of specific wave length is emitted. This line constitutes Emission spectrum. If on the other hand, the atom gains energy so that the electron passes from a lower to a higher energy level, energy is absorbed and a spectral line of specific wave length will be absorbed. Consequently, a dark line will appear in the spectrum. This dark line constitutes Absorption Spectrum.
Sodium flame, for example, when examined in a spectroscope reveals two yellow lines D1 and D2 corresponding to wave lengths 589 nm and 589.6 nm respectively. These two lines constitute the emission spectrum of sodium. However, when sodium vapour was placed in the path of the arc light, the spectroscope reveals the presence of all the spectral lines of white light except for two lines corresponding to wave lengths of D1 and D2 which are now found missing. Two dark lines appear in their place. These two dark lines constitute the absorption spectrum of sodium.
Bohr's Theory and the origin of Hydrogen Spectrum
Hydrogen atom contains only one electron. But its spectrum consists of a large number of lines. Bohr supplied the reason. A given sample of hydrogen contains a very large number of atoms, of the order of several millions. Suppose energy is supplied to the sample of the gas. Now different atoms will absorb different amounts of energy. The solitary electron in different atoms will shift to different energy levels depending upon the energy absorbed by atoms. Some of the hydrogen atoms, for example will absorb sufficient energy to shift their electrons to second energy level only while some others may absorb more energy and may be able to shift their electrons to third or even to one of the higher energy levels. The electrons then tend to fall back, almost immediately , to one or other of lower energy levels. The various possibilities by which the electrons fall back from various excited states to lower energy levels are depicted in Fig 20.
During each transition, energy is released which appears in the form of radiation of a photon of light of specific wave length.
Suppose an electron lies in the energy level n2, where the energy is E2 and jumps to a lower energy level n1, where the energy is E1. The energies in these two states can be represented as :
Then the difference in energy E = E2 E1 should be emitted in the form of spectral line. If is the frequency of the line emitted then :
The emission of various series of spectral lines observed in the emission spectra of hydrogen can be easily explained with the help of Equation (1). The Lyman series are produced when electron jumps from second, third, fourth or a higher energy level to the first energy level. The frequencies of the various lines of this series can be obtained by substituting n1 = 1 and n2 = 2,3,4, etc in equation (1). The Balmer series result when the electron jumps from third, fourth, fifth etc. , energy levels to second energy level. The frequencies of various lines of the Balmer series can thus be obtained by substituting n1 = 2 and n2 = 3, 4, 5, 6 etc. In the same way the Paschen series originate from the electron jumps from fourth, fifth etc energy levels to the third energy level. The frequencies of these lines can be obtained by substituting n1 = 3 and n2 = 4, 5, 6 etc. in Equation (1) and so on. Brackett series originate from electron jumps from fifth, sixth etc energy levels to the fourth energy level. The frequencies of these lines can be obtained by substituting n1 = 4 and n2 = 5, 6, 7...etc. in Equation (1) . Pfund series originate from electron jumps from sixth , seventh, eighth etc., energy levels to fifth energy level. The frequencies of these lines can be obtained by substituting n1 = 5 and n2 = 6, 7, 8........etc. in Equation (1).
Note
Negative value of Energy of Electron
When the electron is at infinite distance from the nucleus, there is no force of interaction between the electron and nucleus. The energy of such a system is arbitrarily assumed to be zero. This is known as zero energy state. When unlike charges are brought together, energy is released by the system due to force of attraction between opposite charges. Thus when electron moves closer to the nucleus under its attractive influence energy is released. Consequently, its energy decreases and it becomes less than zero i.e., negative.
Shortcomings of Bohr Theory
Bohr’s theory of hydrogen atom enabled calculations on the radii and energies of these possible orbits in the hydrogen atom. The calculated values were in agreement with the experimental values. Bohr’s theory could also explain successfully the positions of various series of lines in the hydrogen spectrum. The theory was largely accepted and Bohr was awarded Nobel Prize in recognition of this work.
Even though Bohr’s theory could explain atomic spectra, this model could not explain a number of significant observations.
The main short comings of Bohr’s theory are :
i) Bohr’s theory failed to explain the spectra of multi-electron atoms. The spectroscopic measurements with high precision revealed the fine structure ( i.e., groups of very fine lines ) in the line spectrum of atoms. Bohr’s theory could not explain the occurrence of these fine spectral lines.
ii) Bohr’s atomic model failed to account for the effect of magnetic field on the spectra of atoms or ions. It was observed that when the source of spectrum is placed in a strong magnetic field, each spectral line is further split into a number of lines. This is called Zeeman’s Effect. This observation could not be explained on the basis of Bohr’s model.
iii) In 1923 de Broglie suggested that electron like light has a dual character. It has particle as well as wave character. Bohr treated the electron only as a particle.
iv) Another objection to Bohr’s theory came from Heisenberg’s Uncertainty Principle . According to this principle , it is impossible to determine simultaneously the exact position and momentum of small moving particle like an electron. The postulate of Bohr that electrons revolve in well defined orbits around the nucleus is thus not tenable.
The dual nature of matter
Einstein had suggested ( 1905 ) that light has dual nature, that is wave nature as well as particle nature. de Broglie ( 1923 ) proposed that like light, matter also has dual character. It exhibits wave as well as particle nature. He derived a relationship for the calculation of wave length ( ) of the wave associated with a particle of mass ‘m’ moving with a velocity ‘v’ as given below:
= h / m v
or, = h / p
where ‘h’ is Planck’s constant and p the momentum of the particle.
It may be noted that wave character has significance only in case of sub-atomic particles such as electrons. For objects of ordinary size wave length of wave associated with the object is too small to be detected.
The wave nature of electron was verified experimentally by Davison and Germer , by carrying out diffraction experiments with a beam of fast moving electrons. This wave nature of electrons is utilised in construction of electron microscope.
Heisenberg's Uncertainty Principle
According to this principle, it is not possible to determine precisely both the position and momentum ( or velocity ) of a microscopic moving particle. This principle is mathematically expressed as :
(x ) (p ) ( h / 4 )
where x is uncertainty with regard to the position and p is uncertainty with regard to momentum of the particle. Evidently, if x is very small , i.e., the position of the particle is known more or less exactly, p would be large, i.e., uncertainty with regard to momentum will be large. Similarly, if an attempt is made to measure exactly the velocity ( or momentum) of the particle, the uncertainty with regard to position will become large.
Significance of 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.
Illustration
If uncertainty principle is applied to an object of mass , say about a milligram (106 kg) , then
The value of V . X obtained is extremely small and is insignificant. Therefore one may say that in dealing with milligram-sized or heavier objects, the associated uncertainties are hardly of any real consequence.
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.
Illustration
In the case of a microscopic object like an electron on the on the other hand V . X obtained is much larger and such uncertainties are of real consequence. For example, for an electron , whose mass is 9.11 x 1031 kg. , according to Heisenberg uncertainty principle:
It, therefore, means that if one tries to find the exact location of the electron, say to an uncertainty of only 108 m, then the uncertainty V in velocity would be
which is so large so that the classical picture of electrons moving in Bohr’s orbits (fixed) cannot hold good. It therefore means that the precise statements of the position and momentum of electrons have to be replaced by the statements of probability that the electron has at a given position and momentum.
Reasons for the failure of the Bohr model
In Bohr model, an electron is regarded as a charged particle moving in well defined circular orbits about the nucleus. The wave character of the electron is not considered in Bohr model. Further an orbit is a clearly defined path and this path can completely defined only if both the exact position and the exact velocity of the electron at the same time are known. This is not possible according to Heisenberg’s uncertainty principle. Bohr model of hydrogen atom , therefore , not only ignores dual behaviour of matter but also contradicts Hesenberg'’ uncertainty principle. In view of these inherent weakness in the Bohr model, there was no point in extending Bohr model to other atoms. What was needed was an insight into the structure of the atom, which took into account wave-particle duality of matter and was consistent with Heisenberg’s uncertainty principle. This came with the advent of quantum mechanics.
Problems
31. Find out the number of electrons, protons and neutrons in the following : a) fluorine b) Fluoride ion and fluorine molecule.
32. Calculate the :
i) Number of electrons which will together weigh one gram.
ii) Mass of one mole of electrons.
iii) Charge on one mole of electrons.
31. How many protons and neutrons are in the following nuclei ?
32. Write the complete symbol for :
i) the nucleus with atomic number 56 and mass number 138 .
ii) the nucleus with atomic number 26 and mass number 55.
iii) the nucleus with atomic number 4 and mass number 9.
iv) Z = 17 , A = 35
v) Z = 92 , A = 233
33. From the symbol 2He4 for element helium , write down the
mass number and atomic number of the element.
34. Write the electronic configurations of an element X whose
atomic number is 12.
35. What is the atomic number of an element whose atomic
nucleus has mass number 23 and neutron number 12. What is
the symbol of the element ?
36. The atom of an element X has 4 protons, 5 neutrons and 4 electrons. Write down its atomic number and atomic mass ( mass number ).
37. Complete the table given below:
Element Mass number Atomic number Number of protons Number of neutrons
39K 39 19 …….. ……..
37Cl ….. ….. ……. …….
35Cl ….. ….. ……. …….
38. Complete the following table :
Element Atomic
number Protons Electrons Neutrons Mass number
A 17 …….. …… 18 ……
B ….. ……. 14 14 ……..
C …… 9 ….. ……. 19
39. The atomic number of nitrogen is 7 and that of hydrogen is 1.
How many electrons are there in the ammonium ion NH4+ ion ?
40. Nuclear radius is of the order of 1013 cm ; while atomic radius is of the order of 108 cm. Assuming the nucleus and atom to be spherical, what fraction of atomic volume is occupied by the nucleus ?
41. What is the frequency of the wave length of a photon emitted during a transition from the n = 5 state to n = 2 state in hydrogen atom ?
42. Calculate the energy assoiciated with the first orbit of He+. What is the radius of the orbit ?
43. What will be the wave length of a ball of mass 0.1 kg moving with a velocity of 10 m s1 ?
44. Calculate the mass of a photon with wave length 3.6 A.
45. If the energy difference between two electronic states is 214.68 kJ mol1. Calculate the frequency of light emitted when electron drops from the higher to lower state.
46. What is the wave length of light emitted when the electron in hydrogen atom undergoes a transition from the energy level n = 4 to an energy level with n = 2 ? What is the colour corresponding to the wave length ?
47. Calculate the wave length of a line in the Balmer series which is associated with a drop of electron from 3rd to 2nd orbit.
48. What is the energy associated with one mole of radiation of wave length 103 m ?
49. Calculate the momentum of the particle which has a de Broglie
wave length of 2.5 x 1010 m.
50. The mass of an electron is 9.1 x 1031 kg. If its kinetic energy
is 3.0 x 1025 J, calculate its wave length.
51. Calculate the mass of a photon with wave length 360 pm.
52. What is the maximum number of emission lines when excited
electron of a H atom in n = 6 drops to the ground state ?
53. What is the energy in Joules , required to shift the electron from the hydrogen atom from first Bohr orbit to the fifth Bohr orbit and what is the wave length of the light emitted when the electron returns to the ground state ? The ground state electron energy is 2.18 x 1011 ergs.
54. A microscope using suitable photons is employed to locate an electron in an atom within a distance of 0.1 A. What is the uncertainty involved in the measurement of its velocity ?
55. A golf ball has a mass of 40 g , and a speed of 45 m s1. If the speed can be measured within accuracy of 2%, calculate the uncertainty in the position.
56. Calculate the wave length of an electron moving with a velocity of 2.05 x 107 m s1 .