Download (n=1).

Physics 1161: Pre-Lecture 30
Models of the Atom
Sections 31-1 – 31-4
The Bohr Model
What works, approximately
Hydrogen-like energy levels (relative to a free electron
that wanders off):
mk 2e4 Z 2
13.6  Z 2
En  

eV  where  h / 2 
2
2
2
2
n
n
Typical hydrogen-like radius (1 electron, Z protons):
2
2
h
1
n


2
rn  

0.0529
nm
n



2
 2  mke Z
Transitions + Energy Conservation
• Each orbit has a specific energy:
En= -13.6 Z2/n2
• Photon emitted when electron jumps from
high energy to low energy orbit. Photon
absorbed when electron jumps from low
energy to high energy:
| E1 – E2 | = h f = h c / l
Line Spectra
In addition to the continuous blackbody
spectrum, elements emit a discrete set of
wavelengths which show up as lines in a
diffraction grating.
This is how neon signs work!
Which lamp is Hydrogen?
Better yet…
Wavelengths can be predicted!
Spectral Line Wavelengths
Calculate the wavelength of photon emitted when
an electron in the hydrogen atom drops from the
n=2 state to the ground state (n=1).
n=3
n=2
n=1
Physics 1161: Lecture 24, Slide 5
Spectral Line Wavelengths
Calculate the wavelength of photon emitted when
an electron in the hydrogen atom drops from the
n=2 state to the ground state (n=1).
hf  E2  E1  3.4eV  (13.6eV)  10.2eV
E2= -3.4 eV
n=3
n=2
Ephoton 
hc
l
hc
1240
l

 124nm
10.2eV 10.2
E1= -13.6 eV
n=1
Quantum Mechanics
• Predicts available energy states agreeing with
Bohr.
• Don’t have definite electron position, only a
probability function.
• Orbitals can have 0 angular momentum!
• Each electron state labeled by 4 numbers:
n = principal quantum number (1, 2, 3, …)
l = angular momentum (0, 1, 2, … n-1)
ml = component of l (-l < ml < l)
ms = spin (-½ , +½)
Summary
• Bohr’s Model gives accurate values for
electron energy levels...
• But Quantum Mechanics is needed to describe
electrons in atom.
• Electrons jump between states by emitting or
absorbing photons of the appropriate energy.
• Each state has specific energy and is labeled
by 4 quantum numbers (next time).
JAVA Links
• Bohr Atom
• Debroglie Atom
• Schroedinger Atom
Bohr’s Model
• Mini Universe
• Coulomb attraction produces centripetal
acceleration.
– This gives energy for each allowed radius.
• Spectra tells you which radii orbits are
allowed.
– Fits show this is equivalent to constraining
angular momentum L = mvr = n h
Bohr’s Derivation 1
mv 2 kZe 2
Circular motion
 2
r
r
Total energy
2
1
kZe
mv 2 
2
2r
1 2 kZe 2
kZe 2
E  mv 

2
r
2r
h
Quantization of
(mvr )n  mv n rn  n
angular momentum:
2
h
vn  n
2mrn
Bohr’s Derivation 2
Use
h
vn  n
in
2mrn
2
kZe
mv n2 
rn
2
h
1
n
rn  n 2 ( )2
 (0.0529nm )
2
2 mkZe
Z
“Bohr radius”
kZe 2
Substitute for rn in E n  
2rn
Z2
E n  13.6eV 2
n
Note:
rn has Z
En has Z2