1. Early Quantum Theory and Models of the Atom
Chapter 27
Early Quantum Theory and Models of the Atom

2. 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.

3. 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.

4. 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.

5. 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.

6. 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.

7. 27.3 Photon Theory of Light and 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
W0 : work function

8. 27.3 Photon Theory of Light and the Photoelectric Effect
kinetic energy vs. frequency:
f0 is the threshold
frequency

9. Photocells Photocells are an application of the photoelectric effect
When light of sufficiently high frequency falls on the cell, a current is produced

10. The energy of the photon is
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
J.s
The maximum kinetic energy of the photoelectrons is
We find the speed from

11. 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.

12. The momentum of the photon is
Example: Calculate the momentum of a photon of yellow light of wavelength
6.00x10-7 m.
The momentum of the photon is

13. 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.

14. 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.

15. 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
 is the de Broglie wavelength of the particle

16. 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.

17. 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

18. 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)
Hydrogen
Mercury

19. 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:

20. Niels Bohr Physicist
“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

21. Bohr Model
1. The electron travels in circular orbits about the positively charged nucleus. However, only certain orbits are allowed.

23. 27.12 The Bohr Atom
Bohr found that the angular momentum was quantized:

24. 27.12 The Bohr Atom Z : # of protons Bohr radius
Using the Coulomb force, we can calculate the radii of the orbits:
Z : # of protons
For Hydrogen
Bohr radius
Higher orbit radii

25. Energy levels Each orbit has an energy of For Hydrogen, Z = 1
Ground state (lowest energy level)
First excited state
Second excited state

26. n =3
n =2
n =1

27. 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.

28. 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.

29. infrared
visible
ultraviolet

30. 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.

31. 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:

32. Extra slides

33. 27.3 Photon Theory of Light and 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

34. 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

35. 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

36. Example: A 60 W light bulb operates at about 2. 1% efficiency
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

37. The Compton Effect Dl=l-l0=lC(1-cosq) l>l0
The experiment was performed by Arthur H. Compton (American Scientist, ). An x-ray photon collides with a stationary electron. The scattered photon and the recoil electron depart the collision in different directions.
Dl=l-l0=lC(1-cosq)
l0 is the incoming wavelength and l is the emitted wavelength
lC=h/(mec)=2.4310-12 m is the Compton wavelength

38. Dl=lC(1-cosq) Dl=4.8610-12 m (b) Dl=2.4310-12 m (1-cos30)
Example: Determine the change in the photon’s wavelength that occurs when an electron scatters an x-ray photon (a) at  =180 and (b)  =30.
Dl=lC(1-cosq) 
(a) Dl=2.4310-12 m (1-cos180)
Dl=4.8610-12 m
(b) Dl=2.4310-12 m (1-cos30)
Dl=(0.134)(2.4310-12 m)
Dl =3.2610-13 m

39. 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.

40. 27.12 The Bohr Atom
An electron is held in orbit by the Coulomb force:

41. 27.12 The Bohr Atom
The lowest energy level is called the ground state; the others are excited states.

42. 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.
Atomic hydrogen emission
Helium emission
Solar absorption

43. 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)

44. 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, …
Lyman series
And the wavelengths in the infrared region are given by the so-called Paschen series is used and is given by
1/ = R(1/32 – 1/n2), n= 4, 5,..
Paschen series

45. infrared
visible
ultraviolet

46. Bohr Model