BSc 1 Chemistry
BSc 1 Chemistry
RELATION BETWEEN ENRGY,
WAVELENGHT, FREQUENCY AND WAVE NUMBER
(I) E=h v
Where, E= Energy, h= Planck’s constant and v= Frequency of radiation.
(ii) V= v̶ .c
E = h v̶ c
Where, v̶
Wave number, c = Velocity of light
(iii) v̶ 1/λ
Therefore E = h c/ λ
Where, λ =
Wavelength
Value of h = 6.626×10—27
ergs-sec = 6.626×10—34 joule-sec
C =
3.0×108 m/sec = 3.0×1010 cm/sec.
Hence, as E increases, v and v ̶ increase where λ decreases
Idea Of de—Broglie Matter Waves
Louis de-Broglie,
a French Physicist in 1924 proposed that matter has a dual character; as wave
and as particle. According to de-Broglie small particles of matter like, proton
electron and atom are in motion, they show particle as well as wave nature.
This dual character of matter is known as dual nature of matter. De-Broglie pointed out that the electron like
light, behave both as material particle and as a wave. The electron has dual
character, i. e. , it has both particle and wave nature.
De-Broglie’s Equation
This equation can be
derived by combining the mass- energy relationship introduces by Planck and
Einstein. Since this equation was derived by de-Broglie. Hence, on his name it
is called de-Broglie equation. With the help of this equation, Wavelength (λ)
of particle (electron) can be calculated.
Let an electron of mass
m move with a velocity c round the nucleus and be associated with a wavelength λ.
According to Max Planck
theory,
E = h
v ……………………. (1)
According to Einstein’s mass-energy relationship;
E = mc2
…………………………. (2)
Where, E = Energy of particles or an electron, h = Planck’s
constant
V = frequency,
m = Mass of electron, c = Velocity of light.
From equation (1) and
(2),
mc2
= h v = h c/λ (since, v = c/λ)
mc
= h/ λ
λ =
h/mc
………………………………. (3)
Equation (3) is for a photo. According to de-Broglie by
Substituting mass of particle m and
its velocity v in
Place
of c, the resultant equation can be applied to
Material
particle. Hence, equation (3) is given as follows
λ = h/mv = h/p …………… (4)
Where, p = mv = Momentum.
Equation (4) is called de-Broglie equation
and is Fig. 3: Electron wave
Applicable
to all material particles in motion. Around the nucleolus.
Wavelength,
λ= _h (Planck’s constant)/ Momentum of electron (p)
(λ) ∞ 1/Momentum (p)
Therefore
Momentum, (p) ∞ 1/ Wavelength
Hence, momentum of a moving particle is inversely proportional to its
wavelength.
(I)
Bohr’s assumption for the quantization of
orbit was arbitray one. But de-Broglie wave theory of electron explains this
fact as follows :
According to de-Broglie, the electron is not a
solid particle moving around the nucleus in a circular path but is a standing
wave extending around the nucleus in a circular orbit.
De-Broglie’s
equation plays an important role in Bohr’s atomic model. According to Bohr,
angular momentum of an electron is an integral multiple of h/.2π
Therefore
mvr = n . h/2π , where n = 1,2,3 …… = whole number.
mvr × 2π = n.h
Therefore 2πr = n.h/mv = n .
h/mv
From
de-Broglie’s equation substituting λ
= h/mv
2πr = n.λ ……………………….. (5)
Therefore
Circumference = n. Wavelength,
Hence, circumference of an orbit is an mtegral
multiple of wavelength.
The
above equation (5) clearly explain Bohr’s postulate that the electron can move
only in those orbits for which the angular momentum is an integral multiple of
h/2π.
(ii) This equation is also applicable in
calculation of wavelength (λ) of a moving particle (electron).
Experiment Verification of de-Broglie
Equation
(i)
Wave nature
: Davisson, Germier and
Thomson provided the experimental proof and showed that a beam of electrons
when reflected from a metal crystal (Ni crystal) produce a diffraction pattern.
Similar diffraction patterns have also been shown for neutrons, protons and
hydrogen. Since diffraction is a wave property, this experiment clearly proves
wave nature of particles.
(ii)
Particle nature :
When
an electron collides on zinc sulphide (ZnS) screen then its spot is formed. The
number of spot formed are equal to the number of electrons collide.
These spots are localized and do not extend in
whole region similar like wave. This proves particle nature of matter
(electron).
This behavior of waves as particles and
particles as waves is called dual nature of matter, i.e., wave-particle
duality. In this way, dual character of electron and quantitative nature of
de-Broglie equation is verified.
Limitation of de-Broglie Equation
De-Broglie
matter-wave nature, i.e., dual character has only significance when applied to
microscopic (tiny) particles. It had no significance for macroscopic bodies.
This is due to the fact that the matter waves associated with macroscopic
bodies are too small to be measured by any suitable method. This may be
explained as follows :
A ball having mass 100 gram moving with a
velocity 100/cm/sec, wavelength associated with the ball is given as :
λ = h/mv = 6.626×10-27/100×100
=
6.626×10-31 cm = 6.626×10-23 Å.
The above obtained wavelength is very
small even shorter than the electromagnetic radiation. Hence, it can be
measured by any suitable method. So, it has no significance. This proves that
de-Broglie equation has only significance for microscopic particles not for
macroscopic particles.
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