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PHYS219 Fall semester 2014
Lecture 26: Electron Energy Levels in
the Hydrogen Atom
Dimitrios Giannios
Purdue University
Gas Discharge Tube – gases produce Discrete
Wavelengths (late 1800s)
Glass tube, evacuated
and back-filled with gas
Glass prism
light is dispersed
Focus on light emitted from Hydrogen
discharge tube
Balmer’s empirical formula (1884) explains the observed
visible wavelengths from hydrogen gas with high accuracy
n=7 n=6 n=5
n=4
é1
1ù
= R H ê - ú ; n = 3, 4,5...
êë22 n 2 úû
l
1
RH = 1.097 × 107 m-1
n=3
The Bohr Model for the electronic energies
in atomic Hydrogen
To better understand the origin of
Balmer’s empirical formula, Bohr
formulated a theory of the hydrogen
atom in 1911.
http://www.anonymousartofrevolution.com/2012/08/great-people.html
The Hydrogen Atom: 1e- and 1p
Attractive Force via Coulomb’s Law
? ? ?
e-
p
e-
Bohr’s prediction:
ao ~ 0.0529 nm
(52.9 pm)
(5.29×10-11 m)
Electrically
Neutral H
Atom
p
r=ao
symbol
charge
mass
neutron
n
0 (zero)
proton
P
+1.6 x 10-19 C
electron
e-1.6 x 10-19 C
1.675 x 10-27 kg 1.673 x 10-27 kg 9.11 x 10-31 kg
Note: 1 C = 1 Coulomb
Bohr Model (1911): Assumptions
(hybrid model combining classical and quantum physics)
• Electron moves in special circular orbits – stationary states
• Only certain orbits are allowed; quantization of angular momentum
determines radius of orbit: r=n2ao; n=positive non-zero integer
• Electron gives off no radiation when in stationary orbit
• Radiation only emitted when electron makes transition from one
stationary orbit to another
allowed
electron
orbits
Nucleus
(1 proton for H)
Bohr’s model for light emission from H
Quantized
electron
orbits
“ground state”
n=1
rn = n2 a0
ao = 0.0529nm
rn
n=2
n=3 .......
nucleus
Using this model, you can derive Balmer’s
empirical formula
Why Quantized Orbits?
rn
rm
Each orbit gives rise to a discrete energy level
Energy
Newtonian
Physics
n=∞
n=3
n=2
n=1
E=0
Quantum
Physics
ground state
same result
that Einstein
proposed
ΔE = Ef - Ei
emitted photon
bound
states
Bohr predicts the allowed electron
energy levels for H atom
Prediction of
Bohr’s Model:
Energy
E=0
n=3
n=2
n=1
E3=-1.51 eV
E2=-3.40 eV
E1=-13.6 eV
ground state
Example – Balmer Lines
Transition
from
n=7 n=6 n=5
n=4
ΔE = En - Em
where E n = DE =
E0
-
E0
E0
n
2
and E m = -
= hf = h
E0
m
c
m2 n 2
l
é 1
1 E0é 1
1ù
1ù
ê
=
- ú = RH ê
- ú
2
2
2
êë m
l hc êë m n úû
n 2 úû
2
; E 0 = 13.6eV
n=3
to
the n=2 level
Example
•An electron in the n = 3 orbit has an energy of
-13.6 eV/32= -1.51 eV
•An electron in the n = 2 orbit has an energy of
-13.6 eV/22= -3.40 eV
•An electron that drops from n=3 to n=2 gives up
3.40 eV - 1.51 eV=1.89 eV
of energy, the energy of the red-color photon.
•The 656 nm emission line from H has a frequency
f=4.57×1014 Hz. A photon of this color has an energy of
hf= 3.03×10−19 J (1.89 eV).
Atomic Structure
The results of many experiments collectively suggest
that all matter is made up of small, indivisible units
which have a unique identity.
Study of chemistry suggests a number of elementary
substances (elements) that show unique chemical
behavior.
These elements are made up of identical tiny particles
called atoms (Greek for “without division”).
Classify the Elements by Mass
- Dalton’s contribution (1804) light
• Hydrogen (H)
• Helium (He)
• Lithium (Li)
• Beryllium (Be)
• Boron (B)
• Calcium (Ca)
heavy
• etc.
Periodic Table by Sorted by Chemical “Activity”
Mendeleev 1869
by mass (atomic number):
atomic
number
by atomic number &
chemical activity:
1st shell
2nd shell
3rd shell
Why does this classification scheme make
sense?
Each chemical element has fixed number of
electrons and protons
Example:
C has 6 protons, 6 electrons.
Electronic energy states
in isolated atom are
QUANTIZED
E=0
p
s
p
Allowed
Energies
s
Chemistry of each element
determined by most weakly bound
electron energy states
Check out http://www.colorado.edu/physics/2000/applets/a2.html
Electronic Energy Shells
1st 20 Chemical Elements
Bohr’s Model (1911)
1st shell holds 2 e’s
Source: www.chemicool.com/elements/
2nd
shell
holds
8 e’s
Schrödinger’s Model
(1926)
3rd
shell
holds
8 e’s
Z= Number of protons
http://www.jcrystal.com/steffenweber/
gallery/sh/orbitals4.htm