The Hydrogen AtomAccording to Bohr

Physics 9 4

The atom

WeWe’’ve already talked about how tinyve already talked about how tinysystems behave in strange ways.systems behave in strange ways.

Now letNow let’’s talks talkaboutabout how a morehow a morecomplicatedcomplicatedsystem behaves.system behaves.

The atom!The atom!

Physics 9 5

Early models of the atom

The basic concept of atoms goes back to theThe basic concept of atoms goes back to theGreeks.Greeks.

John Dalton wasJohn Dalton waspretty much thepretty much thefirst tofirst to formulateformulatea modern concepta modern conceptof the atom.of the atom.

He formulated aHe formulated abasic modelbasic model ofofthe elements.the elements.

Physics 9 6

Early models of the atom Dmitri MendeleevDmitri Mendeleev

Physics 9 7

Early models of the atom A more sophisticatedA more sophisticated

model was developedmodel was developedby J.J. Thompson,by J.J. Thompson,who also discoveredwho also discoveredthe electron.the electron.

His modelHis model involved ainvolved aneutral atom.neutral atom.

Physics 9 8

Early models of the atom Thompson imaginedThompson imagined

the atom as being athe atom as being apositive sphere, withpositive sphere, withelectrons embeddedelectrons embeddedin the sphere. Thisin the sphere. Thisgave an overallgave an overallneutral atom.neutral atom.

Physics 9 9

Early models of the atom This is theThis is the ““plum pudding model.plum pudding model.””

A moreA moremodernmoderninterpretationinterpretation

Physics 9 10

Ernest Rutherford Of course, itOf course, it’’ss

important to testimportant to testtheories in science.theories in science.

Rutherford wanted toRutherford wanted toconfirm or discreditconfirm or discreditThompsonThompson’’s models modelvia direct experiment.via direct experiment.

Physics 9 11

Rutherford’s experiment The The Geiger-Marsden Geiger-Marsden experimentexperiment

Physics 9 12

Rutherford’s model RutherfordRutherford formulatedformulated

the familiar solar systemthe familiar solar systemmodel of the atom.model of the atom.

He imagined a centralHe imagined a centralnucleus with thenucleus with theelectrons orbiting it likeelectrons orbiting it liketiny planets.tiny planets.

This is a very pretty and intuitive model ofThis is a very pretty and intuitive model ofthe atom.the atom.

ThereThere’’s only one problems only one problem……

Physics 9 13

This atom is unstable!! RutherfordRutherford’’ss

model hasmodel haselectronselectrons orbitingorbitingaround thearound thenucleus. Theynucleus. They’’rereacceleratingaccelerating..

AcceleratingAcceleratingcharges radiatecharges radiateenergy!energy!

Physics 9 14

Remember de Broglie… WeWe’’ve alreadyve already

met de met de BroglieBroglie..

!

" =h

p

The de The de BroglieBrogliewavelength.wavelength.

WhatWhat does thisdoes thistell us?tell us?

Physics 9 15

Particle in a box A particle is confinedA particle is confined

to a small region ofto a small region ofspace byspace by BIG forcesBIG forces

Particles inParticles inconfined spacesconfined spacesform standingform standingwaves.waves.

Physics 9 16

Particle in a box The energy of thisThe energy of this

particle is particle is quantizedquantized..

Particles inParticles inconfined spacesconfined spacesform standingform standingwaves.waves.

!

En

=h2

8mL2n2

Physics 9 17

This idea might be useful… After all, the electronsAfter all, the electrons

are confined to orbits.are confined to orbits. WeWe’’ll come back to thisll come back to this

idea. But, thereidea. But, there’’s anothers anotherinteresting problem withinteresting problem withthe hydrogen atom thatthe hydrogen atom thatalso needed solving.also needed solving.

Physics 9 18

Atomic spectra Pass an electric current through an atomicPass an electric current through an atomic

gas.gas. Look at it through aLook at it through a

diffraction grating. Wediffraction grating. Weget a discrete set of lines.get a discrete set of lines.

This is theThis is the emission spectrum foremission spectrum for iron.iron.

Physics 9 19

The hydrogen spectrum Hydrogen is Hydrogen is muchmuch simpler. simpler.

Any good theory of the hydrogen atomAny good theory of the hydrogen atomshould explain where these lines comeshould explain where these lines comefrom.from.

No one had a good theory, but there was aNo one had a good theory, but there was aformula that workedformula that worked……

Physics 9 20

The Rydberg formula… Johannes Johannes RydbergRydberg Examining the hydrogenExamining the hydrogen

atom he found a formulaatom he found a formulathatthat predicted thepredicted thespectra.spectra.

!

1

"= R

H

1

n1

2#1

n2

2

$

% &

'

( )

!

RH

=1.097 "107m

#1

This This IS NOTIS NOT a theory!a theory!

Physics 9 21

Niels Bohr In 1913,In 1913, BohrBohr

proposed a model toproposed a model toexplain theexplain thehydrogen atom,hydrogen atom,which explained thewhich explained theemission spectra.emission spectra.

The model wasThe model wasformulated in termsformulated in termsof of ““classicalclassical””quantum mechanics.quantum mechanics.

Physics 9 22

The main ideas… Bohr suggested that theBohr suggested that the electrons couldelectrons could

havehave only certain only certain classicalclassical properties. properties. The electrons can only travel in certainThe electrons can only travel in certain

discrete orbits. Their orbits dondiscrete orbits. Their orbits don’’t decay!t decay! The electrons can only change orbits byThe electrons can only change orbits by

absorbing or emittingabsorbing or emitting light. This changeslight. This changestheir energy discretely and the change istheir energy discretely and the change iscarried away (added) bycarried away (added) by radiatingradiating(absorbing)(absorbing) photons.photons.

Physics 9 23

The main ideas… How can this happen?How can this happen? BohrBohr said that thesaid that the

angular momentumangular momentumof the electron wasof the electron wasquantizedquantized..

This wasThis wasreinterpreted by dereinterpreted by deBroglieBroglie!!

!

L = mvr = nh

!

h "h

2#=1.05 $10

%34Js

Physics 9 24

Let’s see how this works BohrBohr

!

L = mvr = nh

de de BroglieBroglie

!

" =h

p

!

mvr = pr =hr

"= nh =

nh

2#

$2#r = n"

This is part of what gave de This is part of what gave de Broglie Broglie the ideathe idea

Physics 9 25

The mathematics The electron is heldThe electron is held

to the proton byto the proton byelectrical forces.electrical forces.

!

FE

=ke

2

r2

But, itBut, it’’s goings goingaround in a circle,around in a circle,too!too!

!

F =mv

2

r

Physics 9 26

The mathematics If theIf the electron is in aelectron is in a

stable orbit, thenstable orbit, then

!

ke2

r2

=mv

2

r

This gives the speedThis gives the speedof the electronof the electron

!

v =ke

2

mr

Physics 9 27

The mathematics Recall BohrRecall Bohr’’ss

quantizationquantizationconditioncondition

!

mvr = nh

Plug in thePlug in theexpression we foundexpression we foundfor the velocityfor the velocity

!

v =ke

2

mr

Physics 9 28

The mathematics This gives the allowedThis gives the allowed orbital radii.orbital radii.

TheyThey’’re quantized!re quantized!

!

rn

=h2

meke

2n2

This form can be re-expressed in terms ofThis form can be re-expressed in terms ofthe the Bohr Bohr radius:radius:

!

rn

= aBn2

!

aB

=h2

meke

2" 0.0528nm

Physics 9 29

The mathematics So, now weSo, now we know the quantized radii. Nowknow the quantized radii. Now

we can get the velocities!we can get the velocities!

!

vn

=ke

2

mrn

ThenThen

!

vn

=ke

2

h

1

n

This is oftenThis is oftenexpressed as aexpressed as afraction of fraction of cc..

!

vn

c=ke

2

hc

1

n"#

n

!

" #1

137.03599

Physics 9 30

The mathematics So, we now have the velocities, and theSo, we now have the velocities, and the

radii. What about the energy?radii. What about the energy? The energy is made up of two pieces,The energy is made up of two pieces,

kinetic and potential.kinetic and potential.

!

E =1

2mv

2"ke

2

r

Both the velocity and radius are quantized.Both the velocity and radius are quantized.Plug in these values!Plug in these values!

Physics 9 31

The mathematics Plugging in thePlugging in the

values:values:

!

En

=1

2mv

n

2"ke

2

rn

= "ke

2

2rn

= "m

ek2e4

2h2

1

n2

The first term is aThe first term is aconstant!constant!

!

En

= "E0

n2

!

E0

=m

ek2e4

2h2

"13.6eV

1eV =1.602 #10$19J

The energy isThe energy isquantized!quantized!

Physics 9 32

The mathematics So, whatSo, what’’s the picture?s the picture?

!

vn

="c

n

!

En

= "E0

n2

!

rn

= aBn2

Physics 9 33

Bohr tackles the H spectrum Electrons jumpElectrons jump

from one orbit tofrom one orbit toanother.another.

The energyThe energychanges viachanges viaphotons.photons.

Physics 9 34

Bohr tackles the H spectrum The energy changes by a discrete amount.The energy changes by a discrete amount.

!

"En1#n

2

= $m

ek2e4

2h2

1

n1

2$1

n2

2

%

& '

(

) *

The energy is carried away by aThe energy is carried away by a photon.photon.

!

"En1#n

2

= $mek

2e4

2h2

1

n1

2$1

n2

2

%

& '

(

) * = E+ = hf

Physics 9 35

Bohr tackles the H spectrum The wavelength of a photonThe wavelength of a photon

!

" =c

f

The energy of a photonThe energy of a photon

!

E" = hf #hc

$

Plug this back in!Plug this back in!

Physics 9 36

Bohr tackles the H spectrum Plugging it back inPlugging it back in

!

1

"= #

$mek2e4

h3

1

n1

2#1

n2

2

%

& '

(

) *

Recall Recall RydbergRydberg::

!

1

"= R

H

1

n1

2#1

n2

2

$

% &

'

( )

!

RH

=1.097 "107m

#1

ButBut

!

"mek2e4

h3

=1.097 #107m

$1

Physics 9 37

Bohr tackles the H spectrum So, we understand the hydrogen spectrum!So, we understand the hydrogen spectrum!

!

1

"= #

$mek2e4

h3

1

n1

2#1

n2

2

%

& '

(

) *

Physics 9 38

It’s pretty - This can be extended to anyThis can be extended to any nucleus with anucleus with a

single electron (i.e., singly ionized He,single electron (i.e., singly ionized He,doubly ionized Li, etc.).doubly ionized Li, etc.).

The formulae change slightly.The formulae change slightly. For a nucleus with chargeFor a nucleus with charge q = q = ZeZe,, justjust

change change keke22 to to kZekZe22!!

!

rn

=h2

mekZe

2n2

!

vn

c=kZe

2

hc

1

n"#Z

n

!

En

= "m

ek2Z2e4

2h2

1

n2

Physics 9 39

It’s pretty - but it’s wrong! BohrBohr’’s model fails miserably when applieds model fails miserably when applied

to atoms with more than one electron!to atoms with more than one electron! People tried to fix it up, for example byPeople tried to fix it up, for example by

letting the electrons have elliptical orbits,letting the electrons have elliptical orbits,etc., but nothing worked.etc., but nothing worked.

It doesnIt doesn’’t matter how pretty the theory is - ift matter how pretty the theory is - ifit disagrees with experiment, it has to beit disagrees with experiment, it has to bethrown away!thrown away!

The basic model is wrong - it has too muchThe basic model is wrong - it has too muchclassical physics in it!classical physics in it!

Physics 9 40

We need quantum mechanics! The atom justThe atom just

doesndoesn’’t behave int behave ina classical way!a classical way!

The electronsThe electronsdondon’’t have well-t have well-defined orbits.defined orbits.

It would beIt would beanother 13 yearsanother 13 yearsbefore the correctbefore the correctanswer was found.answer was found.

Physics 9 41

We need quantum mechanics! TheThe electron orbitselectron orbits

for the s, p, dfor the s, p, dorbitalsorbitals..

The electron isThe electron isreally described byreally described bythethe SchrodingerSchrodingerequation (equation (non-non-relativisticallyrelativistically).).

!

"h2

2m#2$+V$ = ih

%$

%t