Chapter 27 Early Quantum Theory and Models of the Atom

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.


blackbody radiation curves for three different temperatures. Note that frequency increases to the left.

KmTP 31090.2

The frequency of peak intensity increases linearly with temperature.


This spectrum could not be reproduced using 19th-century physics.

The constant h is now called Planck’s constant.

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:


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.


The photoelectric effect:

If light strikes a metal, electrons are emitted.

0max WKEhf

W0 : work function

The effect does not occur if the frequency of the light is too low

the kinetic energy of the electrons increases with frequency


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?

KEmax 12 mv 2;

0.93 eV 1.60 10 19 J eV 12 9.11 10 31 kg v 2 ,

v 5.7 105 m s.

We find the speed from

KEmax hf W 0 3.41 eV 2.48 eV 0.93 eV.

The maximum kinetic energy of the photoelectrons isE hf

hc

6.63 10 34 Jgs 3.00 108 m / s 1.60 10 19 J/eV 365 10 9 m 3.41 eV.

The energy of the photon is

J.s

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 wavelength6.00x10-7 m.

The momentum of the photon is

ph

6.6310 34 J s 6.0010 7 m 1.110 27kg 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.

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.

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.

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

is the de Broglie wavelength of the particle

mv

h

p

h

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

Hydrogen

Mercury

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.

27.11 Atomic Spectra: Key to the Structure of the Atom

The wavelengths of electrons emitted from hydrogen have a regular pattern:

(27-9)

This is called the Balmer series. R is the Rydberg constant:

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

n = 3 (red light)

n = 4 (blue light)

n = 5 (violet light)

and n = 6 (violet light)


Other series include the Lyman series for the UV-light:

And the Paschen series for the infrared light:

“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

Niels BohrPhysicist

1885 - 1962

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

Bohr radius

For Hydrogen

12 rnrn Higher orbit radii

Energy levels

For Hydrogen, Z = 1

...,3,2,16.132

nn

eVEn

eVE

eVE

eVE

51.1

4.3

6.13

3

2

1

Ground state (lowest energy level)

First excited state

Second excited state

...,3,2,1)6.13(

...,3,2,112

2

2

22

2422

nn

ZeVE

nnh

mkeZE

n

n

Each orbit has an energy of

n =3

n =2

n =1

27.12 The Bohr AtomBohr 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 photonsof light which will cause the electron to jump froma lower level to a higher level. Thus the energy ofthe photon must equal the difference in the energybetween 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 ?

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:

2 2

2 2 2

13.6 eV 3 13.6 eV 122 eV.n

ZE

n n n

1 2

122 eV0 0 122 eV.

1E E

3.4 eV

Extra slides


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


1. Number of electrons and their energy should increase with intensity

If light is a wave, theory predicts:

2. Frequency would not matter

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: 0.02160 J/s=1.3 J/s

The energy of a single photon is:E =hf=hc/λ

E =(6.6310-34 Js)(3108 m/s)/(55510-9 m)E =3.5810-19 J/photon

Number of emitted photons per second:(1.3 J/s)/(3.5810-19 J/photon)=3.61018 photons/s

The Compton Effect• The experiment was performed

by Arthur H. Compton (American Scientist, 1892-1962). An x-ray photon collides with a stationary electron. The scattered photon and the recoil electron depart the collision in different directions.

0

>0

Ccos)

0 is the incoming wavelength and is the emitted wavelength

C=h/(mec)=2.4310-12 m is the Compton wavelength

C-cos

(a) =2.4310-12 m (1-cos180)

=4.8610-12 m

(b) =2.4310-12 m (1-cos30)

=(0.134)(2.4310-12 m)

=3.2610-13 m

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.

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.

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) wherern = (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 differencebetween 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).

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.


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

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)

Balmer 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

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

infrared

visible

ultraviolet

Bohr Model

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Chapter 27 Early Quantum Theory and Models of the Atom