Q1:
The Bohr model
A Hydrogen-like ion is an ion that
A) is chemically very similar to Hydrogen ions
B) has the same optical spectrum as Hydrogen
C) has the same number of protons as Hydrogen
D) has the same number of electrons as a Hydrogen ion
E) is an atom that has lost all but one of its electrons
Summary of important Ideas
Announcements:
1) Electrons in atoms are found at specific energy levels
2) Different set of energy levels for different atoms
3) One photon emitted per electron jump down between
energy levels. Photon color determined by energy difference
between the two levels.
4) If electron not bound to an atom: Can have any energy. (For
instance kinetic energy of free electrons in the PE effect.)
Hydrogen
Lithium
Reading for Friday: 11.9 & 11.10
Exam 2 is next week (Tue. 7:30-9pm)
Practice exam is posted on CUlearn.
Electron energy levels in 2
different atoms: Levels have
different spacing (explains
unique colors for each type of
atom.
Energy
Topics: TZD, Chapters 3-5, 11.9, 11.10
(not to scale)
Atoms with more than one
electron: lowest levels filled.
Q2:
The Rydberg formula describes
Now we know about the energy
levels in atoms. But how can we
calculate/predict them?
a. the colors of light emitted from all types of atoms.
b. the angular distribution of particles scattering off
of atoms that Rutherford observed in his famous
experiment.
c. the colors of light emitted by hydrogen atoms.
d. the energy distribution of cathode rays
e. the frequencies of light emitted by helium atoms.
Need a model
Step 1: Make precise, quantitative observations!
Step 2: Be creative & come up with a model.
Step 3: Put your model to the test.
Balmer series:
A closer look at the
spectrum of hydrogen
Balmer series:
A closer look at the
spectrum of hydrogen
656.3 nm
410.3 486.1
434.0
Balmer (1885) noticed wavelengths followed a progression
λ=
91.19nm
1 1
−
22 n 2
656.3 nm
410.3 486.1
434.0
Balmer (1885) noticed wavelengths followed a progression
So this gets smaller
where n = 3,4,5, 6, .
Balmer correctly
predicted yet
undiscovered
spectral lines.
As n gets larger, what happens to wavelengths of
emitted light?
λ gets smaller and smaller, but it approaches a limit.
Hydrogen atom – Lyman Series
Rydberg’s formula
91.19nm
λ=
1
1
− 2
2
m n
Hydrogen energy levels
91.19nm
1
1
− 2
2
2
m2 n
Hydrogen
energy
levels
Predicts λ of nm transition:
Predicts λ of nm transition:
n
gets larger as n increase,
but no larger than 1/4
λ gets smaller and smaller, but it approaches a limit
Hydrogen atom – Rydberg formula
λ=
91.19nm
where n = 3,4,5,6, .
1 1
−
2 2 n 2 gets smaller as n increase
λlimit = 4 * 91.19nm = 364.7nm
Does generalizing Balmer’s formula work?
Yes! It correctly predicts additional lines in HYDROGEN.
Rydberg’s general formula
λ=
n
(n>m)
m
(m=1,2,3..)
0eV
Can Rydberg’s formula
tell us what ground
state energy is?
(n>m)
m
(m=1,2,3..)
-?? eV
m=1
Hydrogen energy levels
Balmer-Rydberg formula
λ=
0eV
A more general case: What is
the energy of each level
(‘m’) in hydrogen?
Look at energy for a transition
between n=infinity and m=1
1
≡ 0eV
Einitial − E final =
hc
λ
− E final =
=
0
hc  1
1 
 2− 2
91.19nm  m n 
hc
91.19nm
≡ 0eV
-?? eV
E1 = −13.6eV
Einitial − E final =
− E final =
0eV
91.19nm
1
1
−
m2 n2
Hydrogen energy levels
0
hc
λ
=
hc  1
1 
 2− 2
91.19nm  m n 
1
hc
91.19nm m 2
Em = −13.6eV
-?? eV
1
m2
The Balmer/Rydberg formula correctly
describes the hydrogen spectrum!
The Balmer/Rydberg formula is a
mathematical representation of an
empirical observation.
It doesn’t really explain anything.
Is it a good model?
How can we explain (not only calculate)
the energy levels in the hydrogen atom?
Next step: A semi-classical explanation
of the atomic spectra (Bohr model)
Rutherford shot alpha particles at atoms and he figured
out that a tiny, positive, hard core is surrounded by
negative charge very far away from the core.
• 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:
Q3:
Why don't planets spiral into the
sun?
A. Well, they do, but very, very slowly.
B. Because planets obey quantum mechanics,
not classical mechanics.
C. Because planets obey classical mechanics,
not quantum mechanics.
D. Because planets are much bigger than
electrons they don’t emit anything.
Answer is A. “Gravitational radiation” (i.e. gravitational waves)
is much, much weaker than electromagnetic radiation.
*Elapsed time: ~10-11 seconds
Electrostatic potential energy
(‘Tidal forces’ likely dominate the energy loss of the planets).
Potential energy of the electron in hydrogen
D
F
+
Nucleus
Electron
++
++
-
Higher
Energy
-
Energy
levels
r
We define electron’s PE as 0 when far, far away from the proton!
Electron's PE = -work done by electric field from r1=∞r2=D
Coulomb’s constant
D
D
∫ F • dr = ∫
When an electron moves to location further away from the
nucleus its energy increases because energy is required to
separate positive and negative charges, and there is an
increase in the electrostatic potential energy of the electron.
Force on electron is less, but Potential Energy is higher!
Electrons at higher energy levels are further from the nucleus!
-
∞
∞
kqelect q prot
r2
D
dr
D
dr
1
ke 2
PE = kqelect q prot ∫ 2 = kqelect q prot
=−
r
r∞
D
∞
-e e
(k = 1/( 4πε0 ): Coulomb force const.)
(for hydrogen)
Potential energy of a single electron in an atom
PE of an electron at distance D from the proton is
PE = −
ke 2
D
How can we calculate the energy
levels in hydrogen?
λ=
, ke2 = 1.440eV—nm
0 distance from proton
potential
energy
PE = -ke(Ze)
+ +
+
D
+ (For Z protons)
91.19nm
1
1
−
m2 n2
Step 1: Make precise, quantitative observations!
Step 2: Be creative & come up with a model.
How to avoid the Ka-Boom?
*Elapsed time: ~10-11 seconds
Bohr Model
Bohr's approach:
• 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?
#1: Treat the mechanics classical (electron spinning around
a proton):
– According to classical physics, It should!
– According to Bohr, It just doesn’t.
– Modern QM will give a more satisfying answer, but
you’ll have to wait till next week.
- 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.
Original paper: Niels Bohr: On the Constitution of Atoms and Molecules,
Philosophical Magazine, Series 6, Volume 26, p. 1-25, July 1913.)
Bohr Model. # 1: Classical mechanics
The centripetal acceleration
a = v2 / r is provided by the coulomb
force F = k·e2/r2.
(k = 1/( 4πε0 ): Coulomb force const.)
or:
Bohr Model. #2: Quantized angular momentum
F=k e2/r2
v
r
Newton's second law mv2/r
= k·e2/r2
mv2 = k·e2/r
The electron's kinetic energy is KE = ½ m v2
The electron's potential energy is PE = - ke2/r
E= KE + PE = -½ ke2/r = ½ PE
Therefore: If we know r, we know E and v, etc
ћ = h/2π
circular
orbit of
electron
+ E
Bohr assumed 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 mv2 = k·e2/r leads to:
rn = rB n 2 , with rB =
h2
= 52.9 pm
ke 2m
En = ER / n 2 , with ER =
F=k e2/r2
v
r
circular
orbit of
electron
, rB: Bohr radius
m( ke 2 ) 2
= 13.6 eV , ER: Rydberg
Energy
2h 2
Bohr Model. Results
Successes of Bohr Model
2
r = rB n 2 , with rB =
h
= 52.9 pm , rB: Bohr radius
ke 2m
En = ER / n 2 , with ER =
• 'Explains' source of Balmer formula and predicts
empirical constant R (Rydberg constant) from
fundamental constants: R=1/91.2 nm=mk2e4/(4πch3)
• 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
m( ke 2 ) 2
= 13.6 eV , ER: Rydberg
Energy
2h 2
The Bohr model not only predicts a reasonable atomic radius
rB, but it also predicts the energy levels in hydrogen to 4 digits
accuracy!
Possible photon energies:
1
 1
Eγ = En − Em = ER  2 − 2 
n 
m
(n > m)
The Bohr model 'explains' the Rydberg formula!!
Q4:
Shortcomings of the Bohr model:
• 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
• Can't explain shapes of molecular orbits
and how bonds work
• Can’t explain doublet spectral lines
Which of the following principles of classical
physics is violated in the Bohr model?
A.
B.
C.
D.
E.
Coulomb’s law
Newton’s F = ma
Accelerating charges radiate energy.
Particles always have a well-defined position and
momentum (Heisenberg’s uncertainty principle)
All of the above.
Note that both A & B are used in derivation of Bohr model.
Questions?
Answers to clicker questions
Ideas for how to resolve these
problems?
Matter waves: TZD Chapter 6
Q1: E
Q2: C
Q3: A “Gravitational radiation” (i.e. gravitational waves) is much, much weaker
than electromagnetic radiation. (‘Tidal forces’ likely dominate the energy
loss of the planets).
Q4: C