Today: Bohr Model
1. Model for discrete electron orbits in atoms.
2. Prediction of allowed radii from new
assumptions.
3. Discrete electronic energies calculated.
4. ‘Hydrogen-like’ ions
• HWK 7 due Wed. 10AM.
• Week 8 online participation available today
• Reading for Friday.: TZ&D Chap. 5.4,5.9, 6.1-6.4.
1
Balmer had a mathematical formula to describe
hydrogen spectrum, but no mechanism for why it
worked.
Hydrogen energy levels
Balmer’s formula
656.3 nm
410.3 486.1
434.0
91.19nm
λ=
1
1
− 2
2
m n
where m=1,2,3
and where n = m+1, m+2
m=1, n=2
2
Rutherford shot alpha particles at atoms
figured out: a tiny positive hard core
negative charge very far away.
• One possible model:
atom is like a solar system:
electrons circling the nucleus
like planets circling the sun…
• The problem is that accelerating
electrons should radiate light
and
• spiral into the nucleus:
3
*Elapsed time: ~10-11 seconds
Hydrogen atom sim…?
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
http://phet.colorado.edu
4
Clicker Question
Why don’t planets emit radiation and spiral into
the sun?
A. They do, but very, very slowly.
B. Because planets obey quantum mechanics,
not classical mechanics.
C. Because gravitational forces work differently
than electrical forces.
D. Because planets are much bigger than
electrons.
Answer is A. Gravitational radiation is much, much weaker
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than electromagnetic radiation.
Bohr Model
• Why Bohr model,
– A new model that would predict Balmer lines
– & solve the problem of electrons spiraling into
the nucleus.
• Bohr model has some problems, but still
useful.
6
know from
experiments
Bohr reasoning
1. 1/λnm = R (1/m2 - 1/n2) - Balmer
2. Gravity -1/r2 force gives orbits.
Coulomb -ke2/r2 force between electron and proton,
So would expect orbits.
3. Classical EM says electron going in circle should
radiate energy, spiral in. (accelerating charge radiates)
proton
+
-
Bohr hypothesisa. New mix of classical and QM
b. Fixed orbits (quant.) and fixed energies
c. Classical model (planet analog)
7
If going around in little orbits, important implications from
classical physics (review of phys I- planets etc)
-
+
v
Basic connections between
r, v, and energy!
r
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If going around in little orbits, important implications from
classical physics (review of phys I- planets etc)
Basic connections between
v
r, v, and energy!
Fcent
F = ma= Fcent = ?
r
(quick memory check)
a. -mv
+
b. -mv2/r
c. -v2/r2
d. -mvr
e. don’t remember learning
anything related to this
Ans b) Fcent = -mv2/r
Equate to Coulomb force, = kq+ q-/r2,
mv2/r =ke2/r2
mv2 = ke2/r
k =1/4πε0 (textbook)
9
Nucleus
Electron
++
++
-
Higher
Energy
-
Energy
levels
When electron moves to location further from the nucleus,
a. energy of electron decreases because energy is released as
positive and negative charges are separated, and there is a
decrease in electrostatic potential energy of electron since it is
now further away
b. energy of electron increases because it takes energy input to
separate positive and negative charges, and there is an
increase in the electrostatic potential energy of the electron.
c. energy of electron increases because it takes energy input to
separate positive and negative charges, and there is a
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decrease in the electrostatic potential energy of the electron.
Nucleus
Electron
++
++
-
Higher
Energy
-
Energy
levels
When electron moves to location further from the nucleus,
a. energy of electron decreases because energy is released as
positive and negative charges are separated, and there is a
decrease in electrostatic potential energy of electron since it is
now further away
b. energy of electron increases because it takes energy input to
separate positive and negative charges, and there is an
increase in the electrostatic potential energy of the electron.
c. energy of electron increases because it takes energy input to
separate positive and negative charges, and there is a
decrease in the electrostatic potential energy of the electron.
(Force on electron is less, but Potential Energy is higher!)
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Nucleus
Electron
++
++
F-
Higher
Energy
-
Energy
levels
Electron feels force toward nucleus.
Must work against that force to move
electron farther away, so increase in PE.
When electron moves to location further from the nucleus,
Answer is b. energy of electron increases because it takes
energy input to separate positive and negative charges, and
there is an increase in the electrostatic potential energy of the
electron. It’s like pushing a boulder out of a ditch (steep at first
and shallow later on).
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So electrons at higher energy levels are further from the nucleus!
What does this say about energy?
r
Fcent
v
Basic connections between
r, v, and energy!
mv2 =ke2/r
+
E = KE + PE = 1/2mv2 +PE
0 distance from proton
potential
energy
PE =?
PE = -ke2/r
so E = 1/2ke2/r -ke2/r = -1/2ke2/r
if know E, know r!
if know r, know E!
if know r or E, know v!
Bohr hypothesis- only certain E levels.
Hop down to lowest level, and then stable.
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Energy (total) levels
for electrons
3rd ex. lev.
2nd ex. lev.
1st excited
level
ground level
Bohr- “Electron in orbit with only certain particular energies”.
This implies that an electron in Bohr model of hydrogen atom:
a. is always at one particular distance from nucleus
b. can be at any distance from nucleus.
c. can be at certain distances from nucleus corresponding to
energy levels it can be in.
d. must always go into center where potential energy lowest
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0 distance from proton
potential
energy
Warning:
Bad mix of representations
potential energy (curve)
total energy (lines)
Energy levels
for electrons
3rd ex. lev.
2nd ex. lev.
1st excited
level
ground level
Bohr- “Electron in orbit with only certain particular energies”.
This implies that an electron in Bohr model of hydrogen atom:
a. is always at one particular distance from nucleus
b. can be at any distance from nucleus.
c. is at certain distances from nucleus corresponding to energy
levels it can be in.
d. must always go into center where potential energy lowest
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v
r
-
+
Fcent
so E = -1/2 ke2/r
if know E, know r!
if know r, know E!
if know r or E, know v!
Bohr hypothesis: only certain E levels.
e hop down to lowest level, giving off photons
when make jump, stable in lowest level.
Does not radiate more.
0 distance from proton
potential
energy
But what determines these “special” energies?
Complex argument based on idea that
at large sizes, electron should radiate
classically, differences only at small size.
(correspondence principle).
Quantized angular momentum L = mvr=nh
Predicted special E’s.
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Bohr calculated special energies.
label energy level with n (n = 1, 2, 3, …)
involved bunch of constants, h, m, e, c
that when combined (see book) give
En = -13.6 eV/n2
This then predicts size of jumps between
levels.
Agreed with observed spectra/Balmer
series to four decimal places!!
(since E and r, connected, also predicts
radius of each orbit. Lowest orbit is “Bohr
radius”, ab=0.053 nm, rn =abn2)
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Review Bohr Model – see book 5.6
Bohr started with 3 basic ideas:
Ordinary
Classical
Mechanics
1. Energy Cons.: E = KE + PE = ½mv2 - ke2/r
2. Centripetal Force: Fcent = mv2/r = ke2/r2
3. Angular Momentum Quantization L = n= Totally new idea:
Derived from
Correspondence
Solve 3 for v ⇒ mvr = n= ⇒ v = n=/mr
Principle
Sub 3 into 2, solve for r to get rn = n2=2/mke2 = n2aB
Hydrogen orbital radii
Sub 2 into 1 to get E = -ke2/2r
Hydrogen
Sub rn into E to get En = -mk2e4/2=2n2 = E1/n2 energies
where E1 = -13.6eV = ground state energy of H
& aB= =2/mke2 = Bohr radius = size of H in gnd state.
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Note: k =1/4πε0 (textbook)
Successes of Bohr Model
• Explains source of Balmer formula and predicts
empirical constant from fundamental constants:
1/λ12 = R(1/n22 - 1/n12) ⇔ Ephoton = E1(1/n22 - 1/n12)
R = 1/(91.2nm) = mk2e4/4πc=3
• Explains variations in R for different single
electron atoms.
• Predicts approximate size of hydrogen atom
• Explains (sort of) why atoms emit discrete
spectral lines
• Explains (sort of) why electron doesn’t spiral into
nucleus
19
Which of the following principles of classical
physics is violated in the Bohr model?
A.
Opposite charges attract with a force inversely
proportional to the square of the distance between
them.
B. The force on an object is equal to its mass times its
acceleration.
C. Accelerating charges radiate energy.
D. Particles always have a well-defined position and
momentum.
E. All of the above.
Note that both A & B are used in derivation of Bohr model.
20
Bohr model is a weird mix of classical
physics and arbitrary rules…
• Why is angular momentum quantized yet
Newton’s laws still work?
• Why don’t electrons radiate when they are
in fixed orbitals yet Coulomb’s law still
works?
• No way to know a priori which rules to
keep and which to throw out…
21
What things CAN’T the Bohr model explain?
• WHY is angular momentum quantized?
• WHY don’t electrons radiate when they are in
fixed orbitals?
• How does electron know which level to jump to?
(i.e. how to predict intensities of spectral lines)
• Can’t be generalized to more complex (multielectron) atoms
• Shapes of molecular orbits and how bonds work
• Can’t explain doublet spectral lines
22
Ideas for how to resolve these
problems?
23
L
Waves
λ1=2L
• Physicists at this time
may have been
confused about atoms,
but they understood
waves really well.
• They understood that for
standing waves,
boundary conditions
mean that waves only
have discrete modes.
• E.g. guitar strings
= node = fixed point
that doesn’t move.
f1=c/2L
λ2=L
f2=c/L
λ3=2L/3
f3=3c/2L
λ4=L/2
f4=2c/L
λ5=2L/5
f5=5c/2L
λn=2L/n
f5=nc/2L
…
24
Standing Waves on a Ring
Just like standing wave
on a string, but now the
two ends of the string
are joined.
What are the restrictions on the wavelength?
A. r = λ
B. r = nλ
n = 1, 2, 3, …
C. πr = nλ
D. 2πr = nλ
E. 2πr = λ/n
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Standing Waves on a Ring
• Answer: D. 2πr = nλ
• Circumference = 2πr
• To get standing wave on ring:
Circumference = nλ
Must have integer number of wavelengths to
get constructive, not destructive, interference.
• n = number of wavelengths
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deBroglie Waves
• deBroglie (French grad student)
suggested: maybe electrons are actually
little waves going around the nucleus.
• This seems plausible because…
– Standing waves have quantized frequencies,
might be related to quantized energies.
– Einstein had shown that light, typically thought
of as waves, have particle properties. Might
not electrons, typically thought of as particles,
have wave properties?
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deBroglie Waves
What is n for electron wave in this picture?
4
3
A.
B.
C.
D.
E.
1
1
5
10
10
20
Cannot determine from picture
Answer: C. 10
5
2
6
7
9
8
n = number of wavelengths.
It is also the number of the energy level En = -13.6/n2.
So the wave above corresponds to
E10 = -13.6/102 = -0.136eV
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deBroglie Waves
n=1
n=2
n=3
…n=10
= node = fixed point
that doesn’t move.
29
deBroglie Waves
• If electron orbits are standing waves, there is a
relationship between orbital radius and
wavelength.
• But what is the wavelength of an electron?!
• For photons, it was known that photons have
(momentum)
momentum E= pc= hc/ λ
p
⇒ p=h/λ ⇒ λ=h/p
λ
• deBroglie proposed that this
(wavelength)
is also true for massive particles (particles w/mass)!
• λ=h/p=“deBroglie wavelength”
30
deBroglie Waves
Given the deBroglie wavelength (λ=h/p)
and the condition for standing waves on a ring
(2πr = nλ), what can you say about the angular
momentum L of an electron if it is a deBroglie
wave?
A.
B.
C.
D.
E.
L = n=/r
L = n=
L = n=/2
L = 2n=/r
L = n=/2r
L = angular momentum = pr
p = (linear) momentum = mv
(Recall: = = h/2π)
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deBroglie Waves
• Substituting the deBroglie wavelength (λ=h/p)
into the condition for standing waves (2πr = nλ),
gives:
2πr = nh/p
• Or, rearranging:
pr = nh/2π
L = n=
• deBroglie EXPLAINS quantization of angular
momentum, and therefore EXPLAINS
quantization of energy!
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deBroglie Waves
• This is a great story.
• But is it true?
• If so, why no observations of electron
waves?
• What would you need to see to believe
that this is actually true?
• Next: Electron interference!
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Models of the Atom
• Thomson – Plum Pudding
–
–
–
–
–
– Why? Known that negative charges can be removed from atom.
– Problem: just a random guess
• Rutherford – Solar System
– Why? Scattering showed hard core.
– Problem: electrons should spiral into nucleus in ~10-11 sec.
–
+
• Bohr – fixed energy levels
– Why? Explains spectral lines.
– Problem: No reason for fixed energy levels
+
• deBroglie – electron standing waves
– Why? Explains fixed energy levels
– Problem: still only works for Hydrogen.
+
• Schrodinger – quantum wave functions
– Why? Explains everything!
– Problem: None (except that it’s hard to understand)
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