Transcript Slide 1

Chapter 27
Early Quantum Theory and
Models of the Atom
27.2 Planck’s Quantum Hypothesis;
Blackbody Radiation
All objects emit radiation whose total intensity is
proportional to the fourth power of their temperature.
This is called thermal radiation; a blackbody is one
that emits thermal radiation only.
27.2 Planck’s Quantum Hypothesis; Blackbody Radiation
blackbody radiation curves for
three different temperatures.
Note that frequency increases
to the left.
The frequency of peak intensity
increases linearly with
temperature.
PT  2.9010 m  K
3
27.2 Planck’s Quantum Hypothesis;
Blackbody Radiation
This spectrum could not be reproduced using
19th-century physics.
A solution was proposed by Max Planck in
1900:
The energy of atomic oscillations within atoms
cannot have an arbitrary value; it is related to
the frequency:
The constant h is now called Planck’s constant.
27.2 Planck’s Quantum Hypothesis;
Blackbody Radiation
Planck found the value of his constant by fitting
blackbody curves:
Planck’s proposal was that the energy of an
oscillation had to be an integral multiple of
hf. This is called the quantization of energy.
27.3 Photon Theory of Light and the
Photoelectric Effect
Einstein suggested that, given the success of
Planck’s theory, light must be emitted in small
energy packets:
These tiny packets, or particles, are called
photons.
27.3 Photon Theory of Light and the Photoelectric Effect
The photoelectric effect:
If light strikes a metal,
electrons are emitted.
The effect does not occur if
the frequency of the light is
too low
the kinetic energy of the
electrons increases with
frequency
hf  KEmax W0
W0 : work function
27.3 Photon Theory of Light and the
Photoelectric Effect
kinetic energy vs. frequency:
f0 is the threshold
frequency
Photocells
• Photocells are an application of the
photoelectric effect
• When light of sufficiently high frequency
falls on the cell, a current is produced
Example: Barium has a work function of 2.48 eV. What is the
maximum kinetic energy of electrons if the metal is illuminated by
UV light of wavelength 365 nm? What is their speed?
The energy of the photon is
hc 6.63  10
E  hf 


1.60  10
34

 3.41 eV.
J/eV 365  10 m 
Jg
s 3.00  108 m / s
J.s
19
9
The maximum kinetic energy of the photoelectrons is
KEmax
 hf W 0  3.41 eV  2.48 eV  0.93 eV.
We find the speed from
KE max
 12 mv 2 ;
0.93 eV1.60  10
19
 9.11  10
J eV 
v  5.7  105 m s.
1
2
31

kg v 2 ,
27.4 Energy, Mass, and Momentum of a
Photon
Because a photon must travel at the speed of
light its momentum is given by:
Note mass of a photon is zero.
Example: Calculate the momentum of a photon of yellow light of wavelength
6.00x10-7 m.
The momentum of the photon is


p 

 6.00  10 m 
h
6.63  1034 J  s
7
1.1 1027 kg  m s.
Pair Production
The equation E = m c2 implies that it is possible to convert mass into
energy and vice versa.
One example of the conversion of energy to mass is pair production.
A high energy photon known as a gamma ray traveling near the nucleus of an
atom may disappear and an electron and a positron may appear in its place.
The electron and the positron have the same mass and carry the same
magnitude of electric charge; however, the electron is negatively charged and
the positron is positively charged.
The minimum energy of a gamma ray required for the pair production of
electron and positron is about 1.02 MeV (See EXAMPLE 27-9; p. 765).
If the energy of the gamma ray is above this amount, then excess energy is
shared equally between the particles in the form of kinetic energy.
Wave Particle Duality; the Principle of Complementarity
Young's interference experiment and single slit diffraction
indicate that light is a wave.
The photoelectric effect and the Compton effect indicate
that light is a particle.
Light is a phenomena that exhibits both the properties of
waves ad the properties of particles. This is known as
wave-particle duality.
Niels Bohr proposed the principle of complementarity
which says that for any particular experiment involving
light, we must either use the wave theory or the particle
theory , but not both. The two aspects of light
complement one another.
Wave Nature of Matter
Just as light exhibits properties of both particles and
waves, particles such as electrons, protons, and neutrons
also exhibit wave properties
In 1923, Louis de Broglie suggested that the wavelength
of a particle of mass m traveling at speed v is given by
h h
 
p mv
 is the de Broglie wavelength of the particle
Example:
Calculate the wavelength of a 0.21 kg ball traveling at 0.10 m/s.
We find the wavelength from
 = h/p = h/mv = (6.63 x 10–34 J · s)/(0.21 kg)(0.10 m/s)
= 3.2 x10–32 m.
Atomic Spectra
Emission spectra are produced by a high voltage placed across the
electrodes of a tube containing a gas under low pressure. The light
produced can be separated into its component colors by a diffraction
grating. Such analysis reveals a spectra of discrete lines and not a
continuous spectrum.
Hydrogen
Mercury
27.11 Atomic Spectra: Key to the Structure of the Atom
In 1885, J. Balmer developed a mathematical equation which could be
used to predict the wavelengths of the four visible lines in the hydrogen
spectrum. Balmer's formula states
The wavelengths of electrons emitted from hydrogen
have a regular pattern:
(27-9)
This is called the Balmer series. R is the Rydberg constant:
n = 3 (red light)
n = 4 (blue light)
n = 5 (violet light)
and n = 6 (violet light)
27.11 Atomic Spectra: Key to the Structure
of the Atom
Other series include the Lyman series for the UV-light:
And the Paschen series for the infrared light:
Niels Bohr
Physicist
1885 - 1962
“The opposite of a correct statement is a false statement. But
the opposite of a profound truth may well be another
profound truth.”
—Niels Bohr
Bohr Model
1. The electron travels in circular orbits about the
positively charged nucleus. However, only certain orbits
are allowed.
27.12 The Bohr Atom
Bohr found that the angular momentum
was quantized:
27.12 The Bohr Atom
Using the Coulomb force, we can calculate the
radii of the orbits:
Z : # of protons
For Hydrogen
rn  n r1
Bohr radius
2
Higher orbit radii
Energy levels
Each orbit has an energy of
2 Z e m k 1
En  
n  1, 2, 3, ...
2
2
h
n
Z2
En   (13.6 eV ) 2 n  1, 2, 3, ...
n
2
2 4
2
For Hydrogen, Z = 1
 13.6 eV
En 
n2
n  1, 2, 3, ...
E1   13.6 eV
Ground state (lowest energy level)
E2   3.4 eV
First excited state
E3   1.51eV
Second excited state
n =3
n =2
n =1
27.12 The Bohr Atom
Bohr proposed that values energy states were quantized.
Then the spectrum could be explained as transitions
from one level to another.
If an electron falls from one orbit,
also known as energy level, to
another, it loses energy in the form
of a photon of light. The energy of
the photon equals the difference
between the energy of the orbits.
A hydrogen atom can absorb only those photons
of light which will cause the electron to jump from
a lower level to a higher level. Thus the energy of
the photon must equal the difference in the energy
between the two levels.
infrared
visible
ultraviolet
Binding energy or ionization energy: minimum
energy required to remove and electron from
the ground state.
The ionization energy for hydrogen is 13.6 eV.
Example: How much energy is needed to ionize a
hydrogen atom in the n = 2 sate ?
3.4 eV
Example: Calculate the ionization energy of doubly
ionized lithium, Li2+ , which has Z = 3.
Doubly ionized lithium is like hydrogen, except that there are three
positive charges (Z = 3) in the nucleus. The square of the product of
the positive and negative charges appears in the energy term for the
energy levels. We can use the results for hydrogen, if we replace e2 by
Ze2:
En  
Z 2 13.6 eV 
n
2

32 13.6 eV 
n2
122 eV 


.
 122 eV  
E  0  E1  0   
  122 eV.
2

1 
n2
Extra slides
27.3 Photon Theory of Light and the Photoelectric Effect
The photoelectric effect:
If light strikes a metal,
electrons are emitted.
The effect does not occur if
the frequency of the light is
too low
the kinetic energy of the
electrons increases with
frequency
27.3 Photon Theory of Light and the
Photoelectric Effect
If light is a wave, theory predicts:
1. Number of electrons and their energy should
increase with intensity
2. Frequency would not matter
27.3 Photon Theory of Light and the
Photoelectric Effect
If light is particles, theory predicts:
• Increasing intensity increases number of
electrons but not energy
• Above a minimum energy required to break
atomic bond, kinetic energy will increase
linearly with frequency
• There is a cutoff frequency below which no
electrons will be emitted, regardless of intensity
• Example: A 60 W light bulb operates at about
2.1% efficiency. Assuming that all the light is
green light (λ =555 nm) determine the number
of photons per second given off by the bulb.
Light energy emitted per second:
0.02160 J/s=1.3 J/s
The energy of a single photon is:
E =hf=hc/λ
E =(6.6310-34 Js)(3108 m/s)/(55510-9 m)
E =3.5810-19 J/photon
Number of emitted photons per second:
(1.3 J/s)/(3.5810-19 J/photon)=3.61018 photons/s
The Compton Effect
• The experiment was performed
by
Arthur
H.
Compton
(American Scientist, 18921962). An x-ray photon collides
with a stationary electron. The
scattered photon and the recoil
electron depart the collision in
different directions.
 D0C1cosq)
>0
0

 0 is the incoming wavelength and  is the emitted
wavelength
 C=h/(mec)=2.4310-12 m is the Compton wavelength
Example: Determine the change in the photon’s wavelength that occurs
when an electron scatters an x-ray photon (a) at q =180 and (b) q =30.
DC1-cosq
(a) D=2.4310-12 m (1-cos180)
D=4.8610-12 m
(b) D=2.4310-12 m (1-cos30)
D=(0.134)(2.4310-12 m)
D =3.2610-13 m
Bohr’s Model of the Hydrogen Atom
1. The electron travels in circular orbits about the positively charged nucleus.
However, only certain orbits are allowed.
2. The allowed orbits have radii (rn) where
rn = (0.53 nm) n2 and n = 1, 2, 3, etc.
3. The orbits have angular momentum (L) given by L = mv rn = n h/2 where n = 1,2,3,
4. If an electron falls from one orbit, also known as energy level, to another, it loses
energy in the form of a photon of light. The energy of the photon equals the difference
between the energy of the orbits.
5. The energy level of a particular orbit is given by E = -13.6eV/n2 where n = 1,2,3,
If n = 1, the electron is in its lowest energy level and it would take 13.6 eV to remove
it from the atom (ionization energy).
6. A hydrogen atom can absorb only those photons of light which will cause the
electron to jump from a lower level to a higher level. Thus the energy of the photon
must equal the difference in the energy between the two levels.
27.12 The Bohr Atom
An electron is held in orbit by the Coulomb
force:
27.12 The Bohr Atom
The lowest energy level
is called the ground
state; the others are
excited states.
27.11 Atomic Spectra: Key to the Structure of the Atom
An atomic spectrum is a line spectrum – only certain
frequencies appear. If white light passes through such a
gas, it absorbs at those same frequencies.
(a)Atomic hydrogen emission
(b)Helium emission
(c)Solar absorption
Balmer series
In 1885, J. Balmer developed a mathematical equation which could be
used to predict the wavelengths of the four visible lines in the hydrogen
spectrum. Balmer's formula states
1/ = R(1/22 – 1/n2), n=3, 4, 5, …
n = 3 (red light)
n = 4 (blue light)
n = 5 (violet light)
and n = 6 (violet light)
Lyman series
For the spectral lines in the ultaraviolet (UV) region , the so-called
Lyman series is used and is given by
1/ = R(1/12 – 1/n2), n=2,3, 4, …
Paschen series
And the wavelengths in the infrared region are given by the socalled Paschen series is used and is given by
1/ = R(1/32 – 1/n2), n= 4, 5,..
infrared
visible
ultraviolet
Bohr Model