Download Bohr vs. Correct Model of Atom

Physics 102: Lecture 24
Bohr vs. Correct Model of Atom
Today’s Lecture will cover Ch 27.6-7,
28.6
Physics 102: Lecture 24, Slide 1
Science fiction
The Bohr model is complete nonsense.
Electrons do not circle the nucleus in little planetlike orbits.
The assumptions injected into the Bohr model
have no basis in physical reality.
BUT the model does get some of the numbers
right for SIMPLE atoms…
Physics 102: Lecture 24, Slide 2
What does work, 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
Physics 102: Lecture 24, Slide 3
Preflight 24.1
h 2 1 n2
n2
rn  ( )
 (0.0529nm)
2
2 mke Z
Z
Bohr radius
If the electron in the hydrogen atom was 207 times
heavier (a muon), the Bohr radius would be
1) 207 Times Larger
2) Same Size
3) 207 Times Smaller
(Z =1 for hydrogen)
Physics 102: Lecture 24, Slide 4
Preflight 24.1
h 2 1 n2
n2
rn  ( )
 (0.0529nm)
2
2 mke Z
Z
Bohr radius
If the electron in the hydrogen atom was 207 times
heavier (a muon), the Bohr radius would be
1) 207 Times Larger
h 2 1
Bohr Radius  ( )
2
2

mke
2) Same Size
3) 207 Times Smaller
This “m” is electron
mass, not proton mass!
Physics 102: Lecture 24, Slide 5
Preflight 24.2
A single electron is orbiting around a nucleus
with charge +3. What is its ground state (n=1)
energy? (Recall for charge +1, E= -13.6 eV)
1)
2)
3)
E = 9 (-13.6 eV)
E = 3 (-13.6 eV)
E = 1 (-13.6 eV)
Physics 102: Lecture 24, Slide 6
Preflight 24.2
A single electron is orbiting around a nucleus
with charge +3. What is its ground state (n=1)
energy? (Recall for charge +1, E= -13.6 eV)
1)
2)
3)
E = 9 (-13.6 eV)
E = 3 (-13.6 eV)
E = 1 (-13.6 eV)
32/1 = 9
Z2
E n  13.6eV 2
n
Note: This is LOWER energy since negative!
Physics 102: Lecture 24, Slide 7
Transitions + Energy Conservation
• Each orbit has a specific energy:
• 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
Physics 102: Lecture 24, Slide 8
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
JAVA
Physics 102: Lecture 24, Slide 9
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!
Physics 102: Lecture 24, Slide 10
Preflight 24.3
Electron A falls from energy level n=2 to energy level
n=1 (ground state), causing a photon to be emitted.
Electron B falls from energy level n=3 to energy level
n=1 (ground state), causing a photon to be emitted.
n=3
n=2
Which photon has more energy?
•
Photon A
• Photon B
A
B
n=1
Physics 102: Lecture 24, Slide 11
Preflight 24.3
Electron A falls from energy level n=2 to energy level
n=1 (ground state), causing a photon to be emitted.
Electron B falls from energy level n=3 to energy level
n=1 (ground state), causing a photon to be emitted.
n=3
n=2
Which photon has more energy?
•
Photon A
• Photon B
A
B
n=1
Physics 102: Lecture 24, Slide 12
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 102: Lecture 24, Slide 13
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
Physics 102: Lecture 24, Slide 14
n=1
ACT: Spectral Line Wavelengths
Compare the wavelength of a photon produced
from a transition from n=3 to n=2 with that of a
photon produced from a transition n=2 to n=1.
1
l32 < l21
2
l32 = l21
3
l32 > l21
n=3
n=2
n=1
Physics 102: Lecture 24, Slide 15
ACT: Spectral Line Wavelengths
Compare the wavelength of a photon produced
from a transition from n=3 to n=2 with that of a
photon produced from a transition n=2 to n=1.
1
l32 < l21
2
l32 = l21
3
l32 > l21
E32 < E21
Physics 102: Lecture 24, Slide 16
so
n=3
n=2
l32 > l21
n=1
Preflight 24.4
The electrons in a large group of hydrogen atoms
are excited to the n=3 level. How many spectral
lines will be produced?
(1)
(2)
(3)
(4)
(5)
(6)
n=3
n=2
n=1
Physics 102: Lecture 24, Slide 17
Preflight 24.4
The electrons in a large group of hydrogen atoms
are excited to the n=3 level. How many spectral
lines will be produced?
(1)
(2)
(3)
(4)
(5)
(6)
n=3
n=2
n=1
Physics 102: Lecture 24, Slide 18
Preflights 24.6, 24.8
So what keeps the electron from “sticking” to the nucleus?
Centripetal Acceleration
Pauli Exclusion Principle
Heisenberg Uncertainty Principle
To be consistent with the Heisenberg Uncertainty Principle, which
of these properties can not be quantized (have the exact value
known)? (more than one answer can be correct)
Electron Orbital Radius
Electron Energy
Electron Velocity
Electron Angular Momentum
Physics 102: Lecture 24, Slide 19
Preflights 24.6, 24.8
So what keeps the electron from “sticking” to the nucleus?
Centripetal Acceleration
Pauli Exclusion Principle
Heisenberg Uncertainty Principle
To be consistent with the Heisenberg Uncertainty Principle, which
of these properties can not be quantized (have the exact value
known)? (more than one answer can be correct)
Electron Orbital Radius
Would know location
Electron Energy
Electron Velocity
Electron Angular Momentum
Physics 102: Lecture 24, Slide 20
Would know momentum
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)
Coming Soon!
ms = spin (-½ , +½)
Physics 102: Lecture 24, Slide 21
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).
Physics 102: Lecture 24, Slide 22
JAVA Links
• Bohr Atom
• Debroglie Atom
• Schroedinger Atom
Physics 102: Lecture 24, Slide 23
Bohr’s Model
JAVA
• 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
DeBroglie
Physics 102: Lecture 24, Slide 24
Bohr’s Derivation
Physics 102: Lecture 24, Slide 25
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
Physics 102: Lecture 24, Slide 26
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
Physics 102: Lecture 24, Slide 27
Note:
rn has Z
En has Z2
So Why is Bohr Wrong?
• Bohr gets energy levels correct, and also
approximate size.
• Quantized electron velocity and radius
violates Heisenberg uncertainty principle.
• It is “LUCKY” that Bohr model got
anything correct!
Physics 102: Lecture 24, Slide 28
Let’s try it!
The smaller space you try to confine a particle,
the more energy it takes.
– Small space => small D x
h
DxDp 
– Small Dx => large Dp
2
qq
– Large Dp => large KE ...
U k
r
• Electric Potential Energy
p2
KE 
• Electron’s Kinetic Energy
2m
Note: p  Dp and Dx  r
Substitute into KE formula:
1  h 
KE 


2m  2r 
Physics 102: Lecture 24, Slide 29
2
R (nm) U (eV) KE
.0053
.053
.53
U+KE
-27.2 +13.6 -13.6
See you later!
• Read Textbook Section 28.7
Physics 102: Lecture 24, Slide 30