An single electron bashes into an atom in a discharge lamp If atom ...

An single electron bashes into an atom in a discharge lamp
Electron leaves hot filament with
nearly zero initial kinetic energy
10 V
-10eV
If atom fixed at the center of the tube,
list all the possible photon energies (colors) that you might see?
A. 1eV, 2eV, 3eV, 4eV, 7eV, 8eV
B. 4eV, 7eV, 8eV
C. 1eV, 3eV, 4eV
D. 4eV
E. Impossible to tell.
-
+
-2 eV
-3 eV
-6 eV

An single electron bashes into an atom in a
discharge lamp
d
-
10 V
+
D
-2 eV
-3 eV
-6 eV
If atom fixed at this point in tube,
-10eV
list all the possible photon energies (colors) that you might see?
A. 1eV, 2eV, 3eV, 4eV, 7eV, 8eV
B. 4eV, 7eV, 8eV
C. 1eV, 3eV, 4eV
D. 4eV Answer is D. Electron only gains about 5eV!
E. Impossible to tell.
Electron energy = qV = e(Ed),
where E is the electric field = (battery V)/(total distance D),
and d is the distance it goes before a collision.

Announcements
• Reading next class Finish chapter 5.
• Homework 7 assigned today.
• Exam 2 next week (Thursday)
– Old Exam up today on CULearn.
– List of topics up today on CULearn

Bohr, Moseley, and
Franck/Hertz

Balmer/Rydberg had a mathematical formula to describe
hydrogen spectrum, but no mechanism for why it worked!
Why does it work?
410.3 486.1 Balmer’s formula 656.3 nm
434.0
91.
19nm
1 1
2 2
m n
where m=1,2,3
and where n = m+1, m+2
Hydrogen energy levels
m=1, n=2

The Balmer/Rydberg formula is a
mathematical representation of an
empirical observation.
It doesn’t explain anything, really.
How can we calculate the energy levels
in the hydrogen atom?
A semi-classical explanation of the
atomic spectra (Bohr model)

Bohr Model
• When Bohr saw Balmer’s formula, he came up
with a new model that would predict it and
'solve' the problem of electrons spiraling into
the nucleus.
• The Bohr model has some problems, but it's
still useful.
• Why doesn’t the electron fall into the nucleus?
– According to classical physics, It should!
– According to Bohr, It just doesn’t.
– Modern QM will give a satisfying answer, but you’ll
have to wait a little longer.
Original paper: Niels Bohr: On the Constitution of Atoms and Molecules,
Philosophical Magazine, Series 6, Volume 26, p. 1-25, July 1913.)

Bohr's approach:
#1: Treat the mechanics classical (electron spinning around
a proton):
- Newton's laws assumed to be valid
- Coulomb forces provide centripetal acceleration.

Electrostatic potential energy
Potential energy of a single electron in an atom
PE of an electron at distance r from the proton is
, ke2 2
ke
PE
= 1.440eV·nm
r
+ + PE = -ke(Ze)
+ r
+ (For Z protons)
potential
energy
0
distance from proton

Bohr Model. # 1: Classical mechanics
The centripetal acceleration
a = v 2 / r is provided by the coulomb
force F = k·Ze 2 /r 2 .
(k = 1/( 4πε 0 ): Coulomb force const.)
Newton's second law mv 2 /r = k·e 2 /r 2
or: mv 2 = k·Ze 2 /r
The electron's kinetic energy is KE = ½ m v 2
The electron's potential energy is PE = - kZe 2 /r
E= KE + PE = -½ kZe 2 /r = ½ PE
Therefore: If we know r, we know E and v, etc…
F=k e 2 /r 2
v
+ E

Angular Momentum…
An electron is orbiting a proton at a radius r.
What is its angular momentum?
a) m enr into the board
b) m en out of the board
c) m en o the right
d) m en into the board
e) m enr pointed out of the board
Angular momentum

Bohr's approach:
#1: Treat the mechanics classical (electron spinning around
a proton):
- Newton's laws assumed to be valid
- Coulomb forces provide centripetal acceleration.
#2: Bohr's hypothesis (Bohr had no proof for this; he just
assumed it – leads to correct results!):
- The angular momentum of the electrons is quantized in
multiples of ћ.
- The lowest angular momentum is ћ.
ћ = h / 2π

Bohr Model. #2: Quantized angular momentum
Bohr postulated that the angular
momentum of the electron could only
have the quantized values of:
L= nћ
And therefore: mvr = nћ, (n=1,2,3…)
or: v = nћ/(mr)
Substituting this into mv 2 = k·e 2 /r :
m(nћ/(mr)) 2 =k·Ze 2 /r: Solve for r and get:
rn 2
rBn
, with rB
2
2
ke
m
2
2 m(
ke )
En ER
/ n , with ER
2
2
52.
9pm
F=k e 2 /r 2
, r B: Bohr radius Z=1
2
13.
6
eV
v
, E R: Rydberg
Energy

Bohr Model. Results
2
r rBn
, with rB
2
2
ke
, r B: Bohr radius
m
2
2 m(
ke )
En ER
/ n , with ER
2
2
52.
9pm
2
13.
6
eV
, E R: Rydberg
Energy
The Bohr model not only predicts a reasonable atomic radius
r B, but it also predicts the energy levels in hydrogen to 4 digits
accuracy!
Possible photon energies: 1 1
E En
Em
ER
2 2
m n
(n > m)
The Bohr model 'explains' the Rydberg formula!!

Only discrete energy levels possible.
Electrons hop down towards lowest level, giving off photons
during the jumps. Atoms are stable in lowest level.
potential
energy
0 distance from proton
Lmin
Bohr couldn't explain why the angular
momentum is quantized but his
model lead to the Rydberg-Balmer
formula, which matched to the
experimental observations very well!
He also predicted atomic radii reasonably well and
was able to calculate the Rydberg constant.

Successes of Bohr Model
• 'Explains' source of Balmer formula and predicts
empirical constant R (Rydberg constant) from
fundamental constants: R= 1 / 91.2 nm=mk 2 e 4 /(4c 3 )
Explains why R is different for different single
electron atoms (called hydrogen-like ions).
• 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

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.

Moseley.

Basic questions
1. Does the Bohr model tell us anything else? Is it anything
more than an oddity that explains Hydrogen?
2. All of the interactions we’ve seen in connection with
“quantization” involves light. Is “quantization” a property of
light alone?

Balmer-Rydberg and Bohr
v
91.
19nm
1 1
2 2
m n
L= nћ
mvr = nћ, (n=1,2,3…)
ke
rn 2
rBn
, with rB
2
2
m
2
2 m(
ke )
En ER
/ n , ER
2
2
2
52.
9
pm
13.
6eV
Hydrogen energy levels
m=1, n=2

Hydrogen like and Electron
2
/ Z,
with r 2
ke m
2
rn rBn
B
2
En Z
ER
R
2
2 m(
ke )
/ n , E 2
2
2
shells
52.
9
pm
13.
6eV
As more and more electrons are added
they fill up in a particular ordering (more
on this later)
Important fact: only 1 electron per
allowed state
The closest ones (core electrons) are
bound quite similarly to this equation.
For Hydrogen like atoms
(one electron)

How do we make x-rays?
High velocity
KE > 1 keV

How do we make x-rays?
Knocks out a core electron and then moves on.
Do we know anything about KE or KE?
No. We can but we don’t

How do we make x-rays?
Level 2 electron falls into empty state releasing photon
This process produces something called a K x-ray
What is the energy of such a photon?

Basic experimental setup

X-ray energies (K-a line)
The energy scales are
likely related to the
Bohr model.
What is the charge
seen by the bound
electrons?

Moseley Balmer and Rydberg
This electron sees what
charge from the nucleus?
Turns out we can ignore any
electrons outside this.
a) Ze b) (Ze) 2
c) e(Z-2) d) e(Z-1)

Moseley Balmer and Rydberg
This electron sees what
charge?
a) Ze b) (Ze) 2
c) e(Z-2) d) e(Z-1)
Nucleus has Z charge and one
electron is screening it. So
e(Z-1)

Implications
All atoms measured agree with Balmer, Rydberg and Bohr
with only slight modifications.
With only slight modifications this explains other x-ray
spectral lines
Lanthanide series: Many elements “discovered” but
chemistry wasn’t yet enough to sort them all out.
Moseley’s work ordered them, found how many there were,
and predicted missing elements (promethium).
"You see actually the Rutherford work [the nuclear atom] was not taken seriously.
We cannot understand today, but it was not taken seriously at all. There was no
mention of it any place. The great change came from Moseley.“ N. Bohr.