Chapter 38C - Atomic Chapter 38C - Atomic PhysicsPhysics

A PowerPoint Presentation byA PowerPoint Presentation by

Paul E. Tippens, Professor of Paul E. Tippens, Professor of PhysicsPhysics

Southern Polytechnic State Southern Polytechnic State UniversityUniversity

A PowerPoint Presentation byA PowerPoint Presentation by

Paul E. Tippens, Professor of Paul E. Tippens, Professor of PhysicsPhysics

Southern Polytechnic State Southern Polytechnic State UniversityUniversity© 2007

Objectives: Objectives: After completing After completing this module, you should be this module, you should be

able to:able to:• Discuss the early models of the atom Discuss the early models of the atom

leading to the leading to the Bohr theoryBohr theory of the of the atom.atom.

• Demonstrate your understanding of Demonstrate your understanding of emission emission and and absorption spectraabsorption spectra and and predict the wavelengths or frequencies predict the wavelengths or frequencies of the of the BalmerBalmer, , LymanLyman, and , and PashenPashen spectral series.spectral series.

• Calculate the Calculate the energyenergy emitted or emitted or absorbedabsorbed by the hydrogen atom when by the hydrogen atom when the electron moves to a higher or lower the electron moves to a higher or lower energy level.energy level.

Properties of AtomsProperties of Atoms

• Atoms are stable and electrically Atoms are stable and electrically neutral.neutral.

• Atoms have chemical properties which Atoms have chemical properties which allow them to combine with other allow them to combine with other atoms.atoms.

• Atoms emit and absorb Atoms emit and absorb electromagnetic radiation with electromagnetic radiation with discrete energy and momentum.discrete energy and momentum.

• Early experiments showed that most Early experiments showed that most of the mass of an atom was of the mass of an atom was associated with positive charge.associated with positive charge.

• Atoms have angular momentum and Atoms have angular momentum and magnetism.magnetism.

• Atoms are stable and electrically Atoms are stable and electrically neutral.neutral.

• Atoms have chemical properties which Atoms have chemical properties which allow them to combine with other allow them to combine with other atoms.atoms.

• Atoms emit and absorb Atoms emit and absorb electromagnetic radiation with electromagnetic radiation with discrete energy and momentum.discrete energy and momentum.

• Early experiments showed that most Early experiments showed that most of the mass of an atom was of the mass of an atom was associated with positive charge.associated with positive charge.

• Atoms have angular momentum and Atoms have angular momentum and magnetism.magnetism.

Thompson’s Model for the Thompson’s Model for the AtomAtom

ElectronPositive

pudding

Thompson’s plum pudding

J.J. Thompson’s J.J. Thompson’s plum pudding modelplum pudding model consists of a sphere consists of a sphere of positive charge of positive charge with electrons with electrons embedded inside.embedded inside.This model would This model would explain that most of explain that most of the mass was the mass was positive charge and positive charge and that the atom was that the atom was electrically neutral.electrically neutral.

The size of the The size of the atom (atom (1010-10-10 m) m) prevented direct prevented direct

confirmation.confirmation.

Rutherford’s ExperimentRutherford’s Experiment

Rutherford Scattering Exp.

Gold foil Screen

Alpha source

The Thompson model was abandoned The Thompson model was abandoned in 1911 when Rutherford bombarded in 1911 when Rutherford bombarded a thin metal foil with a stream of a thin metal foil with a stream of positively charged alpha particles.positively charged alpha particles.

The Thompson model was abandoned The Thompson model was abandoned in 1911 when Rutherford bombarded in 1911 when Rutherford bombarded a thin metal foil with a stream of a thin metal foil with a stream of positively charged alpha particles.positively charged alpha particles.

Most Most particles particles pass right pass right through the through the foil, but a foil, but a few are few are scattered in scattered in a a backwardbackward direction.direction.

The Nucleus of an AtomThe Nucleus of an AtomIf electrons were distributed uniformly, If electrons were distributed uniformly, particles would pass straight through an particles would pass straight through an atom. Rutherford proposed an atom that is atom. Rutherford proposed an atom that is open space with positive charge open space with positive charge concentrated in a very dense nucleus.concentrated in a very dense nucleus.

Gold foil

Screen

Alpha scattering

+

-

-

Electrons must orbit at a distance in Electrons must orbit at a distance in order not to be attracted into the order not to be attracted into the nucleus of atom.nucleus of atom.

Electron OrbitsElectron Orbits

Consider the planetary model for electrons Consider the planetary model for electrons which move in a circle around the positive which move in a circle around the positive nucleus. The figure below is for the nucleus. The figure below is for the hydrogen atom.hydrogen atom.

Consider the planetary model for electrons Consider the planetary model for electrons which move in a circle around the positive which move in a circle around the positive nucleus. The figure below is for the nucleus. The figure below is for the hydrogen atom.hydrogen atom.

Coulomb’s law:2

204C

eF

r

Centripetal FC: 2

2C

mvF

r

2 2

204

mv e

r r Radius of Radius of

Hydrogen Hydrogen atomatom

2

204

er

mv

2

204

er

mv

FC

+

-

Nucleus

e-

r

Failure of Classical ModelFailure of Classical Model

v

+

-

Nucleus

e-

2

204

er

mv

2

204

er

mv

When an electron is When an electron is acceler-ated by the acceler-ated by the central force, it must central force, it must radiate radiate energyenergy..The loss of energy should The loss of energy should cause the velocity cause the velocity vv to to de-crease, sending the de-crease, sending the electron crashing into the electron crashing into the nucleus.nucleus.This does This does NOTNOT happen happen and the Rutherford atom and the Rutherford atom fails.fails.

Atomic SpectraAtomic SpectraEarlier, we learned that objects Earlier, we learned that objects continually emit and absorb continually emit and absorb electromagnetic radiation.electromagnetic radiation.In an emission spectrum, light is In an emission spectrum, light is separated into characteristic separated into characteristic wavelengths.wavelengths.

In an absorption spectrum, a gas absorbs In an absorption spectrum, a gas absorbs certain wave lengths, which identify the certain wave lengths, which identify the element.element.

Emission SpectrumEmission SpectrumGasGas

Absorption Absorption SpectrumSpectrum

Emission Spectrum for H Emission Spectrum for H AtomAtom

653 nm

486 nm 410 nm

434 nmCharacteristic wavelengths

nn = = 33

nn = = 44

nn = = 55

nn66

Balmer worked out a mathematical Balmer worked out a mathematical formula, called the formula, called the Balmer seriesBalmer series for for predicting the absorbed wavelengths from predicting the absorbed wavelengths from hydrogen gas.hydrogen gas.

2 2

1 1 1; 3, 4, 5, . . .

2R n

n

2 2

1 1 1; 3, 4, 5, . . .

2R n

n

R 1.097 x 107 m-1

R 1.097 x 107 m-1

Example 1:Example 1: Use the Balmer equation to Use the Balmer equation to find the wavelength of the first line (n find the wavelength of the first line (n = 3) in the Balmer series. How can you = 3) in the Balmer series. How can you find the energy?find the energy?

2 2

1 1 1; 3

2R n

n

R = R = 1.097 x 101.097 x 1077 m m-1-1

2 2

1 1 1 1(0.361);

2 3 0.361R R

R

7 -1

1

0.361(1.097 x 10 m ) = 656 nm

The frequency and the energy are found The frequency and the energy are found from:from:

c = fand E = hfc = fand E = hf

The Bohr AtomThe Bohr AtomAtomic spectra indicate that atoms emit or Atomic spectra indicate that atoms emit or absorb energy in discrete amounts. In absorb energy in discrete amounts. In 1913, Neils Bohr explained that classical 1913, Neils Bohr explained that classical theory did not apply to the Rutherford theory did not apply to the Rutherford atom.atom.

++

Electron orbits

e-

An electron can only An electron can only have certain orbits have certain orbits and the atom must and the atom must have definite energy have definite energy levels which are levels which are analogous to standing analogous to standing waves.waves.

An electron can only An electron can only have certain orbits have certain orbits and the atom must and the atom must have definite energy have definite energy levels which are levels which are analogous to standing analogous to standing waves.waves.

Wave Analysis of OrbitsWave Analysis of Orbits

++

Electron orbits

e- Stable orbits exist for Stable orbits exist for integral multiples of integral multiples of de Broglie de Broglie wavelengths.wavelengths.

22r = nr = nn = n = 1,2,3, 1,2,3, ……

2h

r nmv

Recalling that angular momentum is Recalling that angular momentum is mvrmvr, , we we write:write:

; 1, 2,3, . . . 2

hL mvr n n

; 1, 2,3, . . .

2

hL mvr n n

n n = 4= 4

The Bohr AtomThe Bohr Atom

++

The Bohr atom

Energy levels, n

An electron can have An electron can have only those orbits in only those orbits in which its angular which its angular momentum is:momentum is:

An electron can have An electron can have only those orbits in only those orbits in which its angular which its angular momentum is:momentum is:

; 1, 2,3, . . . 2

hL n n

; 1, 2,3, . . .

2

hL n n

Bohr’s postulateBohr’s postulate: When an electron : When an electron changes from one orbit to another, it changes from one orbit to another, it gains or loses energy equal to the gains or loses energy equal to the difference in energy between initial and difference in energy between initial and final levels.final levels.

Bohr’s postulateBohr’s postulate: When an electron : When an electron changes from one orbit to another, it changes from one orbit to another, it gains or loses energy equal to the gains or loses energy equal to the difference in energy between initial and difference in energy between initial and final levels.final levels.

Bohr’s Atom and Bohr’s Atom and RadiationRadiation

EmissionEmission

AbsorptionAbsorption

When an electron drops to a lower level, radiation is emitted; when radiation is absorbed, the electron moves to a higher level.

When an electron drops to a lower level, radiation is emitted; when radiation is absorbed, the electron moves to a higher level.

Energy: hf = Ef - Ei

By combining the idea of energy levels By combining the idea of energy levels with classical theory, Bohr was able to with classical theory, Bohr was able to predict the radius of the hydrogen atom.predict the radius of the hydrogen atom.

Radius of the Hydrogen Radius of the Hydrogen AtomAtom

; 1, 2,3, . . . 2

hL mvr n n

; 1, 2,3, . . .

2

hL mvr n n

Radius as Radius as function of function of

energy energy level:level:

nhrmv

Bohr’s Bohr’s radiusradius

2

204

er

mvClassicClassic

al al radiusradius

By eliminating By eliminating rr from these equations, we from these equations, we find the velocity find the velocity vv; elimination of; elimination of v v gives gives

possible radii rpossible radii rnn::2

02n

ev

nh

2

02n

ev

nh

2 20

2n

n hr

me

2 2

02n

n hr

me

Example 2:Example 2: Find the radius of the Find the radius of the Hydrogen atom in its most stable state Hydrogen atom in its most stable state (n = 1).(n = 1).

2 20

2n

n hr

me

2 2

02n

n hr

me

m = 9.1 x 10-31 kg

e = 1.6 x 10-19 C

2

2

2 -12 34 2NmC

-31 -19 2

(1) (8.85 x 10 )(6.63 x 10 J s)

(9.1 x 10 kg)(1.6 x 10 C)r

r =r = 5.31 x 10 5.31 x 10-11 -11 mm r = 53.1 pm

Total Energy of an AtomTotal Energy of an AtomThe total energy at level The total energy at level nn is the sum of is the sum of the kinetic and potential energies at that the kinetic and potential energies at that

level.level. 221

20

; ; 4

eE K U K mv U

r

Substitution Substitution for for v v and and rr gives gives expression for expression for total energy.total energy.

2

02n

ev

nh

2 20

2n

n hr

me

But we recall that:

4

2 2 208n

meE

n h

4

2 2 208n

meE

n h

Total energyTotal energy of Hydrogen of Hydrogen atom for level atom for level nn..

Energy for a Particular Energy for a Particular StateState

It will be useful to simplify the energy It will be useful to simplify the energy formula for a particular state by substitution formula for a particular state by substitution

of constants.of constants.m = 9.1 x 10-31 kg

e = 1.6 x 10-19 C

o = 8.85 x 10--12 C2/Nm2

h = 6.63 x 10-34 J s

2

2

4 -31 -19 4

2 2 2 -12 2 2 -34 2C0 Nm

(9.1 x 10 kg)(1.6 x 10 C)

8 8(8.85 x 10 ) (6.63 x 10 Js)n

meE

n h n

-18

2

2.17 x 10 JnE n

-18

2

2.17 x 10 JnE n

2

13.6 eVnE n

2

13.6 eVnE n

OrOr

Balmer RevisitedBalmer Revisited

4

2 2 208n

meE

n h

4

2 2 208n

meE

n h

Total energyTotal energy of Hydrogen of Hydrogen atom for level atom for level nn..

Negative because Negative because outside energy to outside energy to raise raise n n level.level.

When an electron moves from an initial When an electron moves from an initial state state nnii to a final state to a final state nnff, energy , energy involved is:involved is: 4 4

0 2 2 2 2 2 20 0 0

1 1;

8 8ff

hc me meE E E

hc h n h n

4 4

2 3 2 2 2 2 3 20 0

1 1 1 ; If

8 8f f i f

me meR

h cn n n h cn

Balmer’s Equation:

7 -12 2

0

1 1 1; 1.097 x 10 m

f

R Rn n

Energy LevelsEnergy LevelsWe can now visualize the hydrogen atom We can now visualize the hydrogen atom with an electron at many possible energy with an electron at many possible energy levels. levels.

EmissionEmission

AbsorptionAbsorption

The energy of the atom The energy of the atom increases increases on absorption (on absorption (nnff > > nnii) and ) and de-creasesde-creases on on emission (emission (nnff < n < nii).).Energy of nth level: 2

13.6 eVE

n

The change in energy of the atom can be The change in energy of the atom can be given in terms of initial given in terms of initial nnii and final and final nnff levels: levels:

2 20

1 113.6 eV

f

En n

2 2

0

1 113.6 eV

f

En n

Spectral Series for an Spectral Series for an AtomAtom

The The Lyman seriesLyman series is for transitions to is for transitions to n = 1n = 1 level.level.The The Balmer seriesBalmer series is for transitions to is for transitions to n = 2n = 2 level.level.

The The Pashen Pashen seriesseries is for is for transitions to transitions to n n = 3= 3 level. level.The The Brackett Brackett seriesseries is for is for transitions to transitions to n n = 4= 4 level. level.

n n =2=2

n n =6=6

n n =1=1

n n =3=3n n =4=4

n n =5=5 2 20

1 113.6 eV

f

En n

2 2

0

1 113.6 eV

f

En n

Example 3:Example 3: What is the energy of an What is the energy of an emitted photon if an electron drops from emitted photon if an electron drops from the the nn = 3 = 3 level to the level to the nn = 1 = 1 level for the level for the hydrogen atom?hydrogen atom?

2 20

1 113.6 eV

f

En n

2 2

0

1 113.6 eV

f

En n

Change in Change in energy of the energy of the

atom.atom.

2 2

1 113.6 eV 12.1 eV

1 3E

E = -12.1

eVE = -12.1

eVThe energy of the atom decreases by 12.1 The energy of the atom decreases by 12.1 eV as a photon of that energy is emitted.eV as a photon of that energy is emitted.

You should show that You should show that 13.6 eV13.6 eV is required is required to move an electron from to move an electron from n = 1n = 1 to to n = n = ..

Modern Theory of the Modern Theory of the AtomAtom

The model of an electron as a point particle The model of an electron as a point particle moving in a circular orbit has undergone moving in a circular orbit has undergone significant change.significant change.

• TheThe quantum modelquantum model now presents the now presents the location of an electron as a probability location of an electron as a probability distribution - a distribution - a cloudcloud around the around the nucleus.nucleus.

• Additional quantum numbersAdditional quantum numbers have have been added to describe such things as been added to describe such things as shape, orientation, and magnetic spin.shape, orientation, and magnetic spin.

• Pauli’s exclusion principlePauli’s exclusion principle showed showed that no two electrons in an atom can that no two electrons in an atom can exist in the exact same state.exist in the exact same state.

Modern Atomic Theory Modern Atomic Theory (Cont.)(Cont.)

The Bohr atom for The Bohr atom for Beryllium suggests Beryllium suggests a planetary model a planetary model which is not strictly which is not strictly

correct.correct.

The n = 2 level of The n = 2 level of the Hydrogen atom the Hydrogen atom is shown here as a is shown here as a

probability probability distribution.distribution.

SummarySummary

Bohr’s model of the atom assumed the Bohr’s model of the atom assumed the electron to follow a circular orbit around a electron to follow a circular orbit around a positive nucleus.positive nucleus.

Bohr’s model of the atom assumed the Bohr’s model of the atom assumed the electron to follow a circular orbit around a electron to follow a circular orbit around a positive nucleus.positive nucleus.

FC

+

-

Nucleus

e-

r Radius of Radius of Hydrogen Hydrogen

AtomAtom

2

204

er

mv

2

204

er

mv

Summary (Cont.)Summary (Cont.)In an emission spectrum, In an emission spectrum, characteristic wavelengths appear on characteristic wavelengths appear on a screen. For an absorption spectrum, a screen. For an absorption spectrum, certain wavelengths are omitted due certain wavelengths are omitted due to absorption.to absorption.

Emission SpectrumEmission SpectrumGasGas

Absorption Absorption SpectrumSpectrum

Summary (Cont.)Summary (Cont.)

Balmer’s Equation:

7 -12 2

0

1 1 1; 1.097 x 10 m

f

R Rn n

653 nm

486 nm 410 nm

434 nmSpectrum for nf = 2 (Balmer)

nn = = 33

nn = = 44

nn = = 55

nn66

The general equation for a The general equation for a change from one level to change from one level to

another:another:

Emission spectrumEmission spectrum

Summary (Cont.)Summary (Cont.)Bohr’s model sees the hydrogen atom with Bohr’s model sees the hydrogen atom with an electron at many possible energy levels. an electron at many possible energy levels.

EmissionEmission

AbsorptionAbsorption

The energy of the atom The energy of the atom increases increases on absorption (on absorption (nnff > > nnii) and ) and de-creasesde-creases on on emission (emission (nnff < n < nii).).Energy of nth level: 2

13.6 eVE

n

The change in energy of the atom can be The change in energy of the atom can be given in terms of initial given in terms of initial nnii and final and final nnff levels: levels:

2 20

1 113.6 eV

f

En n

2 2

0

1 113.6 eV

f

En n

CONCLUSION: Chapter 38CCONCLUSION: Chapter 38CAtomic PhysicsAtomic Physics