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Physics 1C · 2011. 12. 2. · The only series that lies in the visible range (390 – 750nm) is...
Transcript of Physics 1C · 2011. 12. 2. · The only series that lies in the visible range (390 – 750nm) is...
Physics 1C Lecture 29A
"Splitting the atom is like trying to shoot a gnat in the Albert Hall at night and using ten million rounds of
ammunition on the off chance of getting it. That should convince you that the atom will always be a sink of energy
and never a reservoir of energy." --Ernest Rutherford
Outline
Last time – particle in a box & quantized energy
wavefunctions and probability density
tunneling & electron transfer
Balmer series & emission spectra
Hydrogen atom
Atomic spectra - emission and absorption
LASER
Hydrogen Atom In 1913, Neils Bohr explained atomic spectra by
utilizing Rutherford’s Planetary model and
quantization.
In Bohr’s theory for the
hydrogen atom, the
electron moves in
circular orbit around the
proton.
The Coulomb force
provides the centripetal
acceleration for
continued motion.
Hydrogen Atom Only certain electron orbits are stable.
In these orbits the
atom does not emit
energy in the form of
electromagnetic
radiation.
Radiation is only
emitted by the atom
when the electron
“jumps” between
stable orbits.
Hydrogen Atom The electron will move from a more energetic
initial state to less energetic final state.
The frequency of the
photon emitted in the
“jump” is related to the
change in the atom’s
energy:
If the electron is not
“jumping” between
allowed orbitals, then
the energy of the atom
remains constant.
Hydrogen Atom Bohr then turned to conservation of energy of the
atom in order to determine the allowed electron
orbitals.
The total energy of the atom will be:
But the electron is undergoing centripetal
acceleration (Newton’s second law):
Angular Momentum Recall from classical mechanics that there was this variable known as angular momentum, L.
Angular momentum, L, was defined as: L = I ω where I was rotational inertia and ω was angular velocity.
For an electron orbiting a nucleus we have that:
Giving us:
Hydrogen Atom Bohr postulated that the electron’s orbital angular
momentum must be quantized as well:
where ħ is defined to be h/2π.
This gives us a velocity of:
Substituting into the last equation from two slides
before:
Hydrogen Atom Solving for the radii of Bohr’s orbits gives us:
The integer values of n = 1, 2, 3, … give you the
quantized Bohr orbits.
Electrons can only exist in certain allowed orbits
determined by the integer n.
When n = 1, the orbit has the smallest radius, called
the Bohr radius, ao.
ao = 0.0529nm
Hydrogen Atom: Bohr’s Theory We know that the radii of the Bohr orbits in a hydrogen
atom are quantized:
Recall the Bohr radius (ao = 0.0529nm).
So, in general we have: rn = n2ao
The total energy of the atom can be expressed as :
rn n2 2
mekee2
E tot KE PE elec 1
2mev
2 kee2
r
(assuming the
nucleus is at rest)
Hydrogen Atom Plus, from centripetal acceleration we found that:
Putting this back into energy we get:
mev2 ke
e2
r
E tot 1
2mev
2 kee2
r
1
2kee2
r
ke
e2
r
E tot 1
2kee2
r
But we can go back to the result for the radius
(rn = n2 ao) to get a numerical result.
Hydrogen Atom
This is the energy of any quantum state (orbit).
Please note the negative sign in the equation.
When n = 1, the total energy is –13.6eV.
This is the lowest energy state and it is called the
ground state.
The ionization energy is the energy needed to
completely remove the electron from the atom.
The ionization energy for hydrogen is 13.6eV.
E tot 1
2kee2
n2ao
13.6 eV
n2
Hydrogen Atom So, a general expression
for the radius of any orbit in
a hydrogen atom is:
rn = n2ao
note that the orbits are not
increasing linearly with n.
The energy of any orbit is:
Hydrogen Atom
What are the first four energy levels for the hydrogen atom?
When n = 1 => E1 = –13.6eV.
When n = 2 => E2 = – 13.6eV/22 = – 3.40eV.
When n = 3 => E3 = – 13.6eV/32 = – 1.51eV.
When n = 4 => E4 = – 13.6eV/42 = – 0.850eV.
Note that the energy levels get closer together as n increases (similar to how the wavelengths got closer in atomic spectra).
When the atom releases a photon it will experience a transition from an initial higher energy level (ni) to a final lower energy level (nf).
Hydrogen Atom The energies can be
compiled in an energy level
diagram.
As the atom is in a higher
energy state and moves to
a lower energy state it will
release energy (in the form
of a photon).
The wavelength of this
photon will be determined
by the starting and ending
energy levels.
Hydrogen Atom The photon will have a
wavelength λ and a
frequency f:
To find the wavelengths
for an arbitrary transition
from one orbit with nf to
another orbit with ni, we
can generalize Rydberg’s
formula:
f E i E f
h
2
i
2
f
Hn
1
n
1R
1
Hydrogen Atom The wavelength will be
represented by a different
series depending on your
final energy level (nf).
For nf = 1 it is called the
Lyman series (ni = 2,3,4,...).
For nf = 2 it is called the
Balmer series (ni = 3,4,5...).
For nf = 3 it is called the
Paschen series (ni = 4,5,...).
Clicker Question 29A-1 When a cool gas is placed between a glowing wire
filament source and a diffraction grating, the
resultant spectrum from the grating is which one of
the following?
A) line emission.
B) line absorption.
C) continuous.
D) monochromatic.
E) de Broglie.
Atomic Spectra Example
What are the first four wavelengths for the
Lyman, Balmer, and Paschen series for the
hydrogen atom?
Answer
The final energy level for either series will be
nf = 1 (Lyman), nf = 2 (Balmer), and nf = 3
(Paschen).
Atomic Spectra Answer
Turn to the generalized Rydberg equation:
For the Lyman series we have:
2
i
2
f
Hn
1
n
1R
1
1
RH
1
1
1
ni2
RH
ni2
ni2
1
ni2
1
RH
ni2 1
ni2
n 1
RH
ni2
ni2 1
1
1.097107 m-1
ni2
ni2 1
Atomic Spectra Answer
Finally for the Lyman series:
1 1
1.097107 m-1
22
22 1
121 nm
2 1
1.097107 m-1
32
32 1
103 nm
3 1
1.097107 m-1
42
42 1
97.2 nm
4 1
1.097107 m-1
52
52 1
95.0 nm
For the Balmer series we have from before:
λ1 = 656nm, λ2 = 486nm, λ3 = 434nm, λ4 = 410nm.
Atomic Spectra Answer
For the Paschen series we have:
1
RH
1
9
1
ni2
RH
ni2
9ni2
9
9ni2
n 1
1.097107 m-1
9ni2
ni2 9
1 1
1.097107 m-1
9 42 42 9
1880 nm
2 1
1.097107 m-1
9 52 52 9
1280 nm
3 1
1.097107 m-1
9 62 62 9
1090 nm
4 1
1.097107 m-1
9 72 72 9
1010 nm
Atomic Spectra
The only series that lies in the visible range (390 – 750nm) is the Balmer series.
The Lyman series lies in the ultraviolet range and the Paschen series lies in the infrared range.
We can extend the Bohr hydrogen atom to fully describe atoms that are “close” to hydrogen.
These hydrogen-like atoms are those that only contain one electron. Examples: He+, Li++, Be+++
In those cases, when you have Z as the atomic number of the element (Z is the number of protons in the atom), you replace e2 with (Ze)2 in the hydrogen equations.
Clicker Question 29A-2 Consider a hydrogen atom, a singly-ionized helium
atom, a doubly-ionized lithium atom, and a triply-
ionized beryllium atom. Which atom has the lowest
ionization energy?
A) hydrogen
B) helium
C) lithium
D) beryllium
E) the ionization energy is the same for all four
E tot 1
2kee2
r
rn n2 2
mekee2
Laser
An acronym for Light Amplification by Stimulated Emission of Radiation.
A laser is a light source that produces a focused, collimated, monochromatic beam of light.
The laser operates using the principle of stimulated emission of light.
In 1917, Albert Einstein established the theoretic foundations for the laser.
For Next Time (FNT)
Finish reading Chapter 29
Start homework on Chapter 29