PHY 102: Waves & Quanta

Topic 12The Bohr Model

John Cockburn (j.cockburn@... Room E15)

•More on line spectra

•Orbital model of the hydrogen atom

•Failure of classical model

•Quantisation of orbital angular momentum: stationary states

•Successes and failures of the Bohr Model

Line Spectrum of hydrogen

•Hydrogen has line spectrum ranging in wavelength from the UV to the infrared

•Balmer (1885) found that the wavelengths of the spectral lines in the visible region of the spectrum could be EMPIRICALLY fitted to the relationship:

5......... 4, 3, m :4

6.364(nm)2

2

m

m

(The group of hydrogen spectral lines in the visible region still known as the Balmer Series)

Line Spectrum of hydrogen•Rydberg and Ritz subsequently obtained a more general expression which applies to ALL hydrogen spectral lines (not just visible), and also to certain elements (eg alkaline metals):

21

22

111

nnR

n2, n1 integers, n2 < n1

•R is called the Rydberg constant, which changes slightly from element to element.

•For hydrogen, RH = 1.097776 x 107 m-1

•Can a model of the atom be developed that’s consistent with this nice, elegant formula??

Rutherford Scattering

To explain results of the Rutherford scattering :

1) Atom must be mostly empty space

2) Positive charge must be concentrated in a small volume occupying a very small fraction of the total volume of the atom…………

Christmas pudding model doesn’t work

Nuclear model does work

Atomic radius ~ 10-10mNuclear radius ~ 10-14m

The Bohr Model (1912-13)Bohr suggested that the electrons in an atom orbit the positively-charged nucleus, in a similar way to planets orbiting the Sun

(but centripetal force provided by electrostatic attraction rather that gravitation)

Hydrogen atom: single electron orbiting positive nucleus of charge +Ze, where Z =1:

r

v

F+Ze

-e

r

v

F+Ze

-e

Bohr Model: electron energy

From electrostatics, the potential energy of the electron is given by:

r

Ze

r

eZe

r

qqU

0

2

00

21

44

)()(

4

r

v

F+Ze

-e

Bohr Model: electron energy

Centripetal force equation:

20

22

4 r

Ze

r

vme

Kinetic energy of electron:

r

Zevme

0

22

82

Bohr Model: electron energy

r

Zevme

0

22

82

Total energy of electron = P.E. + K.E:

r

Ze

r

eZe

r

qqU

0

2

00

21

44

)()(

4

r

Ze

r

Ze

r

Ze

0

2

0

2

0

2

884energy Total

But this classical treatment leaves us with a big problem………

Failure of the Classical model

The orbiting electron is an accelerating charge.

Accelerating charges emit electromagnetic waves and therefore lose energy

Classical physics predicts electron should “spiral in” to the nucleus emitting continuous spectrum of radiation as the atom “collapses”

CLASSICAL PHYSICS CAN’T GIVE US STABLE ATOMS………………..

Bohr’s postulates

• Only certain electron orbits are allowed, in which the electron does not emit em radiation (STATIONARY STATES)

•An atom emits radiation only when an electron makes a transition from one stationary state to another.

•The frequency of the radiation emitted when an electron makes a transition from a stationary state with energy E2 to one with energy E1 is given by:

h

EEf 12

Transition energies

Suppose an electron is initially in stationary state with energy E1, orbital radius r1. It then undergoes a transition to a lower energy state E2, with (smaller) radius r2:

120

2

20

2

10

2

21

11

888 rr

Ze

r

Ze

r

ZeEE

If Bohr’s postulates are correct, then the frequency of the radiation emitted in the transition is given by:

120

221 11

8 rrh

Ze

h

EEf

Rydberg-Ritz Revisited

21

22

111

nnR

21

22

11

nncRf

c = fλ

120

2 11

8 rrh

Zef

Bohr result:

Looks promising, if we can make the connection that r is somehow proportional to “integer squared”……………….

Quantisation of angular momentum

Bohr now makes the bold assumption that the orbital angular momentum of the electron is quantised………

Since v is perpendicular to r, the orbital angular momentum is just given by L = mvr.

Bohr suggested that this is quantised, so that:

nnh

mvr 2

IMPLICATIONS???..........................................................................

r

Zevme

0

22

82

Kinetic energy (earlier slide)

rm

Zev

e0

22

4

nnh

mvr 2

quantisation of A.M. (last slide)

22

222

mr

nv

22

22

0

2

4 mr

n

rm

Ze

e

2

022 4

Zem

nr

e

Bohr radius

So, introduction of the idea that angular momentum is quantised has the desired effect: rn2. Simplifying the expression for r a bit (Z=1 for hydrogen):

02

20

22

anrem

hnr n

e

a0, the radius of the n=1 orbit, is called the BOHR RADIUS

20

2

0 em

ha

e

We conclude that in the Bohr model only certain orbital radii (and electron velocities) are allowed.

120

2 11

8 rrh

ef

21

2200

2 11

8 nnah

ef

21

22

20

3

4 11

8 nnh

mef e

21

22

11

nncRf

Rydberg-Ritz

R=1.07 x 107 m-1

How nice.

Origins of hydrogen spectral lines:

Bohr Model: Shortcomings

•The Bohr model does an excellent job of explaining the “gross” features of hydrogen line spectrum

BUT•Doesn’t work well for many-electron atoms (even helium)

•Can’t explain fine structure of spectral lines observed at high resolution, or relative intensities of spectral lines

•Can’t explain effect of magnetic field on spectral lines (Zeeman effects), although Sommerfeld’s modifications (elliptical orbits, varying orientations) help to some extent

•Is fundamentally inconsistent with Heisenberg’s uncertainty principle

THE BOHR MODEL IS WRONG