Quantization of Energy
(A Thomson’s atomic model: Sir J.J Thomson was
first scientist to purpose structure of
Drawbacks of Thomson’s model:
1. Thomson’s
model could not explain the origin of spectral series in hydrogen atom.
2. It
could not explain scattering of α- particles.
Rutherford is α scattering:
In order to find out the structure of an atom and to
test the Thomson’s atomic model, Ernest Rutherford and his coworkers Geiger and
Mardsen experiment, they observe that:
i.
Most of the α particles were found to
cross the gold foil without deviation.
ii.
Some of the α particles were scattered
by small angles.
iii.
A few number of α particle were scattered
through very large angles.
From
that experimental observationRutherford with his coworkers concluded following
suggestions for structure of atom.
i.
The atom is mostly empty or contains
very light particles, which may not have significant effect on deflection.
ii.
Positive charge insides an atom is
concentrated in a small region and deflection is due to the force of repulsion
between positive charge of atom and α particle.
iii.
Large deflection also suggests that
almost all mass of an atom is concentrated in a small region.
(B) Rutherford’s atomic model: On
the basis of observation observed in α-scattering experiment, Rutherford and
Coworkers in 1911 A.D suggested the structure of an atom as follows;
i.
An atom consists of a nucleus in which
entire positive charge and almost whole mass of atom is concentrated. The size
of the nucleus is of the order of 10-15m which is small compared to
the size of atom 10-10m.
ii.
Electrons are spread over the outer
parts of nucleus in an atom thereby leaving lot of empty space in the atom.
iii.
The total positive charge concentrated
in nuclusieequl to the total charge carried by electrons around the nucleus and
hence an atom is electrically neutral.
Drawbacks of Rutherford’s Atomic Model:
i.
Rutherford’s atomic model failed to
explain the stability of atom.
ii.
Rutherford’s atomic model could not
explain the origin of line spectra of hydrogen atom.
Bohr’s
Atomic Model: Bohr modified Rutherford’s atom model based on quantum theory of
radiation in 1913 A.D and he was awarded Nobel Prize in physics in 1922 A.D.His
model was based on following two postulates:
(i)
The electron can revolve round the
nucleus only in those permissible orbits for which the angular momentum of the
electron is equal to integral multiple of
If
m is mass and v be the velocity of an electron revolving in an orbit of radius
r, then according to Bohr’s postulate,
Angular
momentum (L) =
Or, mvr = n
The
angular momentum of an electron is quantized. This is known as Bohr’s
quantization condition. The electron in the permissible or allowed orbits are
called stationary orbits. It does not radiate or absorb energy in this orbit.
This postulate explaininstability of atom.
(ii)
An electron radiates or absorbs energy
only it jumps from one stationary orbit to other. Electron emits energy when it
jumps from higher to lower energy level and it absorbs energy when it jumps
from lower to higher energy level.When an electron jumps from outer orbit n2
of energy En2 to inner orbit n1 of energy En1the
is radiated. The frequency f of radiation is given as:hf = En2 - En1.
This
is called Bohr’s frequency condition. This postulate not only explain the
origin of spectral series of hydrogen atom but also explain the quantization of
energy.
Bohr’s Theory of Hydrogen Atom:
Hydrogen
atom consists of nucleus having positive charge +e. Let m be the mass and –e be
the charge of an electron revolving around the nucleus in nth
stationary orbit of radiusrn with orbital velocity Vn.
The
electrostatic force of attraction between nucleus and electron is given by:
Fe
=
Where
The
centripetal force acting on the circular orbit is given by
Fc
=
Here,
the electrostatic force between nucleusand electron provides necessary
centripetal force. Therefore, we can have,
Fe
= Fc
Vn2
=
From Bohr’s
quantization condition,
m
Where n = 1,2,3…….. is principal
quantum number.
From equation (iii) and
(iv), we have
(
Or, rn =
This equation gives the radius of nth stationary orbit of
hydrogen atom and from the equation, it is clear that radii are proportional to
n2.
The radius of inner most. i.efirst orbit of hydrogen atom is
called Bohr’s radius a0.
For first orbit, n=1
Then r1 = a0 =
Substituting the known values, we have
Bohr’s radius (a0) =
Velocity of electron in the nth orbit of hydrogen
atom
Substituting the value of rn in equation (iv) we
have,
Or,
This gives the velocity of an electron in nth
stationary orbit of hydrogen atom.
Frequency of electron in the nth stationary orbit of
hydrogen atom
f=
This is the expression for orbital frequency of an electron in
its nth orbit of hydrogen atom.
Energy of electron in nth
orbit
An electron revolving around the nucleus possesses kinetic
energy as well as potential energy. The kinetic energy of electron is due to
its motion whereas the potential energy is due to the electrostatic force of
attraction between electron and nucleus. Thus,the total energy of an electron
revolving round the nucleus in certain orbit is sum of its kinetic and
potential energy.
The kinetic energy of electron in nth orbit is,
K.E =
Since we have
Using this equation in equation (1)
K.E =
The potential energy of electron in nth orbit is given
by
P.E = (electrostatic potential) ×(Charge of an electron) =
P.E =
Substituting the value of rn in above equation, we
get
P.E =
Now, the total energy of an electron in nth orbit is
E = K.E + P.E
E =
E =
This expression gives the total energy of electron in nth
orbit of hydrogen atom. The negative sign indicates that the electron is bound
to the nucleus.
Bohr’s Interpretation of
Hydrogen Atom:
When an electron in hydrogen atom jumps from higher energy state
n2 of energy En2 to lower energy state n1 of
energy En1, then the energy equal to the difference between the
energies of these two levels is radiated in the form of a photon. The frequency
of emitted radiation is given by
hf = En2 - En1
As we know,
En2 =
hf= En2 - En1
hf=
f=
Wave number of a radiation is the number of complete waves in
unit length in vacuum. It is the reciprocal of wavelength of radiation. It is denotedby
Wave number (
Or,
Or,
Where, R =
Spectral series of Hydrogen
Atom:
When an electron in a hydrogen atom jumps from higher energy
state to a lower energy state, it emits radiation of certain frequency or wavelength,
whichis called as spectral line. The wavelength of spectral line depends on the
two energy states between which transition of electron occurs. The transition
of electron between various orbits with different wavelength produce the spectral
series of hydrogen atom. Johann Balmer discovered the first such series in
1885. The different types of spectral series of hydrogen atom are shown in fig.
(A) Lyman Series: The spectral series obtained by the
transition of electron from higher energy level to lower energy level i.eground
state (n=1) is called Lyman series. This series lies in ultraviolet region.
The wave length of Lyman series are given
by
(
Here, n1= 1 and n2
= 2,3,4…. And R = 1.097×107 /m is Rydberg’sconstant
Here, n1= 2 and n2 = 3,4,5…. And R =
1.097×107 /m is Rydberg’sconstant
Here, n1= 3 and n2
= 4,5,6…. And R = 1.097×107 /m is Rydberg’sconstant
Here, n1= 4 and n2
= 5,6…. And R = 1.097×107 /m is Rydberg’sconstant
(E) P-fund series: The spectral series
obtained by the transition of electron from higher energy states to the fourth excited
state (n=5) is called P-fund series. This series lies in infrared region.
The wavelength of P-fund series are given
by
Here, n1= 5 and n2
= 6,7…. And R = 1.097×107 /m is Rydberg’sconstant
Energy Levels
in Hydrogen Atom:
Horizontal lines can represent the energy
of an electron in different stationary orbit, which is called energy level and
the diagram of such energy levels is called energy level diagram.
The energy of an electron revolving round
the nucleus in nth stationary orbit in hydrogen atomis given by,
En=
Substituting values of m,e,
En =-
This expression represents the energy of
an electron in nth orbit of hydrogen and negative sign signifies that
the electron is bounded to the nucleus.
For the first orbital hydrogen atom, n= 1
E1 = =-
It is the ground state of energy of
hydrogen atom.
For n=2,
E2 = =-
It is the energy of first excited state.
Energy of second and third excited state
can be given respectively as,
E3 = =-
E4 = =-
When the value of n increases, the value
of En also increases. Then energy levels are so close and they
constitute an energy continuum.
When n=∞, then
E∞ = =-
This state is called ionized state. The
energy level diagram of hydrogen atom is shown in figure below:
Limitation of Bohr’s theory of
hydrogen atom:
i.
It cannot explain the spectra of the atom.
ii.
It cannot explain about the intensity of
spectral lines.
iii.
The fine structure of spectral lines
cannot be explained based on this theory.
Excitation and Ionization Potential:
Excitation
potential: The minimum amount of energy required to excite an atom, i.e to move
electron from its ground state to higher energy state is known as excitation
energy and the corresponding potential is known as excitation potential.
The
amount of energy required to excite an electron from ground state( n=1) to its first
excitation state (n=2) is called first excitation energy. The energy required
to excite an electron from ground state to first excited state in hydrogen atom
is given by
E=
E2 – E1 = -3.4 – (-13.6) = 10.2eV
Similarly,
the energy required to excite electron to second excited state in hydrogen atom
is,
E=
E3-E1 = -1.51 –(-13.6) = 12.09eV
The
excitation potential for second excited state of hydrogen atom is 12.09eV and
so on.
Hence,
the minimum potential required to excite an electron from it ground state to
the given excited state is called excitation potential.
Ionization Potential:The minimum amount of
energy required to ionize an atom is called as ionization energy and the
corresponding potential is known as ionization potential.
The
energy required to excite an electron from ground state to ionized sate is
given by, E= E∞ - E1 = 0 –(-13.6) = 13.6eV
The
ionization energy is numerically equal to ground state energy. The ionization
energy required to knock out the electron from that atom.
Hence,
the minimum potential required to remove the electron from the ground state to
ionized state is called ionization potential.
Emission and Absorption Spectra:
Emission Spectra: when electron in excited state jumps to
lower energy state then the emission spectra of different wavelengths is obtained.
On the basis of nature of emitted spectral lines, spectra are classified into
three types,
(i)
Line spectra: A spectra in which discrete
spectral lines having particular wavelength or frequency can be observed is
called line spectra. This type of discontinuous spectra is produced by excite
atom or ions. Hydrogen, Mercury spectrum are some examples of line spectra.
(ii)
Band Spectra: The emission spectrum,
which consists a separate group of spectral lies, is known as band spectrum. In
general, band spectra are produced by gases which contain more than one atom.
(iii)
Continuous spectra: The spectra which
consist of unbroken or continuous range of wavelength are called continuous
spectra. Continuous spectra are produced by hot solids, liquid and highly
compressed gases.
Absorption Spectra: When electrons are
taken from lower energy state to the higher energy state, the absorption
spectra are obtained. In practices, when a light from a source having
continuous spectrum is passed through a medium in gaseous state, it is observed
that the resulting spectrum appears as a continuous spectrum in which some characteristics
wavelength are absent. The absent wavelength are seen as dark lines in the
spectrum, such a spectrum is called as absorption spectra.
A substance, which emits
lights certain wavelength as emission spectra at certain temperature, would
also absorb light of same wavelength at the same temperature. On absorbing the
light of the wavelength, atom will be excited into higher energy state from
lower energy state. Therefore, some radiations are absent in the spectrum. This
is the reason due to which there are some dark lines in the solar spectrum.
Absorption spectra are also of three types. They are line
spectra, band spectra and continuous spectra. The absorption spectra of
hydrogen atom as shown in fig.
De-Broglie’s Theory: Duality:A
major advance in understanding of atomic structure begins in 1924. A French
physicist Loui Victor de Broglie suggested the dual nature of matter i.e wave-
particle duality in his doctoral thesis in 1924 and awarded Nobel Prize in
physics in 1929 for this valuable work. His hypothesis was based on following
facts.
i.
In the universe, whole of the energy is
in the form of radiation and matter. Therefore, both forms of energy should possess
similar characteristics.
ii.
Nature loves symmetry: The matter and
radiation are symmetrical in many ways. As the radiation has dual nature, the
matter should also possess dual nature. According to de-Broglie, a moving
particle behaves sometimes as a wave and sometimes as a particle. The waves
associated with a moving particle are called matter waves or de-Broglie waves.
The wavelength associated with matter waves is called de-Broglie wavelength.
The
de- Broglie wavelength of a particle of mass (m) moving with speed (v) is
ℷ
=
Where
h is Planck’sconstant.
The
de-Broglie wavelength is independent of nature and charge of the particle. A
particle in motion exhibits wave nature if its de-Broglie wavelength is of the
order of its size.
De- Broglie wavelength of Photon:
Consider
the photon to be a particle of mass (m) moving with velocity of light (c).
According to Einstein’s mass energy relation; the energy of photon is, E = mc2
According
to Planck’s quantum theory of radiation, the energy of photon having frequency
f is given by
E=hf
Comparing
above equation, we get
hf=
mc2
h
ℷ
=
Since
the matter also possesses dual nature. So, the wavelength of the wave associated
with a particle of mass (m) moving with velocity (v) is
ℷ
=
This
equation represents the expression for de-Broglie wavelength. From this
equation, it is clear that the greater will be the momentum, the shorter will
be the wavelength and vice-versa i.e ℷ α
De- Broglie
of Electron:
The wavelength associated with a moving
with a moving electron can be calculated by using de- Broglie wave theory.
Consider an electron having mass m and charge (e)be accelerated by a potential
difference of V volts. Let v be the velocity gained by the electron. According
to work energy theorem, the kinetic energy gained by the electron must be equal
to the work done (P.E) in moving electron,
i.e K.E = work done
ℷ
=
De- Broglie wave theory and Bohr’s atom model:
The
de-Broglie wave theory is the roof of atomic model. This theory can explain the
Bohr’s quantization condition. The electron revolving around the nucleus in an
orbit forms standing waves as shown in fig. The circumference of the orbit must
be equal to the integral multiple of wavelength of de-Broglie wave produced by
the electron.
i.e
2Πr = nℷ
where
n= 1,2,3…….. is integer and r = radius of the orbit.
According
to de-Broglie wave theory.
ℷ =
Substituting
the value of ℷ,
2Πr
=
mvr=
This
equation represents the Bohr’ quantization condition. Hence, the de-Broglie hypothesis
of matter wave is the agreement with Bohr’s theory.
Application of de-Broglie
waves:
i.
De- Broglie wavelength of a double- slit
interference pattern is produced by using electrons as the source.
Heisenberg’s Uncertainty Principle:This
principle states that, it is impossible to determine precisely and
simultaneously the values of both the members of pair of physical variables
which describe the motion of an atomic system. According to this principle, the
position and momentum of a particle cannot be determined simultaneously to any
desired degree of accuracy.
If
ΔP and ΔX be uncertainties in momentum and positon respectively, the
ΔP
×ΔX ≥
If
ΔX is very small then ΔP will be very large and vice versa. It means if one
variable is measured accurately, the measurement of other quantity become
uncertain. The uncertainty is a direct consequence of dual nature of matter.
The uncertainty relation is universal and holds for all canonically conjugate
physical quantities.
The
uncertainty relation for energy and time is
ΔE
×Δt≥
Where,
ΔE is uncertainty in energy and Δt is uncertainty in time.
Production of X-ray:
Coolidge
Tube: X-ray can be produces by bombardment of fast moving electron i.e cathode
ray on the metal target of high atomic weight and melting point.
X-ray
are produced in Coolidge tube. It is a modern X-rays tube designed by Coolidge
in 1993. It consists of an evacuated glass tube at the pressure about 10-5mm
of Hg having cathode and anode. The cathode consists of tungsten filament (F)
having high resistivity which is heated by low-tension battery and is placed in
a metal cap(C) to focus the electrons to the target. The metal target (T) i.e
anode of tungsten or molybdenum having a high melting point and high atomic
weight.
Working: The filament F is heated by low-tension
battery and the electrons are emitted from it. These electrons are accelerated
by applying very high potential( 50-100KV) between filament and target. These
energetic electrons are focused to a point on the target with the help of metal
cap(C). When the fast moving electrons strike the target, X-rays are produced.
While producing X-rays, about 98% of the energy of the incident electrons is
converted into heat and hence the target gets heated. So the target is cooled
by circulating cold water through the copper pipe continuously. About 2% of the
energy of the incident electrons is converted into X-rays.
Control of intensity and quality:
i.
Control of intensity: The intensity of
X-rays depends on the number of electrons striking the target metal. The number
of electrons depends on the current passing through the filament.
i.e
intensity of X-ray (I) α filament current (I).
Hence,
the intensity of x-rays can be control by adjusting filament or cathode
current.
ii.
Control of quality: The frequency of
X-ray resembles the quality of x-rays. The frequency of x-rays depends on the
voltage between anode and cathode.
Let
m be the mass of an electron, e the charge of an electron and V be the
potential difference between two electrodes. Then,
Maximum
kinetic energy gained = work done
Wherefmax is maximum frequency
of x-rays
from above equationfmax =
Also the minimum possible wavelength of
x-rays produced is given by:
ℷmin =
Properties of x-rays:
The properties of x-rays are as follows:
i.
X-ray are electromagneticradiations of
short wavelength ranging from 10-12 to 10-9m.
ii.
They are not deflected by electric field
as well as magnetic fields.
iii.
They travel with velocity of light in
vacuum.
iv.
X-rays have ionizing power.
v.
They exhibit phenomenon of reflection,
refraction, diffraction, interference. etc
Uses of x-rays
i.
X-rays are used for diagnosis. They used
to detect the fractures of bones, diseased organs.
ii.
X-rays are used in detective departments
to detect the smuggling of precious metal, explosive metal.
iii.
X-ray are also used om scientific research
to study the structure of crystal and their chemical composition.
X-ray diffraction and Laue Experiment:
The
wave nature of x-ray was confirmed by using Laue experiment. In this
experiment, Von Laue and his workers passed a fine beam of x-rays through a
crystal of zinc Sulphide, behind which a photographic plate was placed. After
exposing x-rays several hours and developing the photographicplate, it is
observing many faint but regularly spots around bright spot on the plate. So,
the regular pattern of spots in photographic plate due to diffraction of
x-rays, is called Laue pattern for diffraction.
Thus,
from this experiment, it is concluded that:
i.
X-rays are electromagnetic waves of very
short wavelength.
ii.
The atoms in a crystal are arranged in a
regular three –dimensional lattice.
Bragg’s Law:
Sir
William Henry Bragg and his son wir William Lawrence Bragg studied the
diffraction of X-ray.
Consider
a crystal containing a set of parallel planes having inter planner distance
(d). suppose parallel beam of monochromatic x-ray of wavelength ℷ is incident
on crystal lattice with glancing angle θ . Let AB be the incident ray and BC be
the reflected ray in XX’ plane. Similarly, A’B’ be the incident ray and B’C’ be
the reflected ray YY’ plane as shown in fig.
To
calculate the path difference between two reflected rays, let us drawn
perpendicular BD on A’B’ and BE on B’C’ respectively.
Then,
path difference = DB’ + B’E
In
right angled triangle B’BD
Sinθ
=
Or,
DB’ = BB’sinθ
Or,
DB’ = dsinθ
Similarly
in right angled triangle B’BE,
B’E
= dsinθ
Now,
the path difference between two reflected rays = DB’+ B’E
= dsinθ+dsinθ
= 2dsinθ
For
maximum intensity of reflected beam, path difference must be equal to an
integral multiple of ℷ.
i.e
path difference = nℷ
or,
2dsinθ= nℷ
where,
n=1,2,3,4….. is integer
This
equation is known as Bragg’s law or Bragg’ equation.
Applying
Bragg’s law, the wavelength of x-ray can be calculated as ℷ =
The crystal plane spacing can be determined
by using Bragg’s equation.
Here, d =
X-ray
Spectra:
X-ray are produced from x-ray tube and
they are of different wavelength. The intensity of x-ray at a specified
accelerating potential are called x-ray spectra. Urey and his co-workers
analyzed the intensity and wavelength of x-ray beam, obtained in the x-ray tube
, by using different potentials and same target and they plotted a graph
between them. The graph is obtained as shown in figure. X-ray spectrum are
basically of two types.
i.
Continuous spectra: Continuous x-ray
spectra consists of x-rays spectra of all possible wavelength within the range.
ii.
Characteristics x-ray spectra: when the
accelerating potential is increased, it is observed that for certainwavelength
of x-ray, the intensity of x-ray become maximum. These are called as characteristics
x-ray spectra.
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