Download Document

Spectroscopy
Hot self-luminous objects light
the Sun or a light bulb emit a
continuous spectrum of wavelengths.
In contract, light emitted in
low=pressure gas discharge contains
only discrete individual wavelengths,
a discrete spectrum.
These spectral
lines can be
recorded and
measured using
a grating
spectroscope.
1
The Hydrogen Spectrum
The spectrum of hydrogen, as viewed in the visible
region, shows the Balmer series of spectral lines.
Balmer did not discover them, but he did notice a
regularity in their spacing, and he found by trial and error
that they could be represented by the relation:
red
blue
Johann Jakob Balmer
(1825 – 1898)
violet
2
Classical Physics at the Limit
Rutherford’s nuclear atom model
matched experimental evidence about
the structure of atoms, but it had a
fundamental shortcoming: it was
inconsistent with Newton’s mechanics
and Maxwell’s electromagnetic theory.
In particular, on electron orbiting a
nucleus would represent an oscillating
charge that should radiate a broad
spectrum of electromagnetic radiation,
not lines. Further, the loss of energy by
such radiation should cause the orbiting
electron to spiral into the nucleus.
As the 20th century dawned, physicists could not explain the structure of
atoms, the stability of matter, discrete spectral lines, emission and
absorption spectra differences, or the origins of X-rays and radioactivity.
3
Niels Bohr’s Atomic Model
In 1911, after receiving his PhD, Niels Bohr went to Cambridge
to work in Rutherford’s laboratory. This was about a year after
Rutherford had completed his nuclear model of the atom.
Bohr wanted to “fix” the model so that the orbiting electrons
would not radiate away their energy. Starting from Einstein’s idea
of light quanta, in 1913 he proposed a radically new nuclear model of
the atom that made the following assumptions:
David Bohr
1. Atoms consist of negative electrons orbiting a small positive nucleus; Niels Henrik
(1885-1962)
2. Atoms can exist only in certain stationary states with a particular set 1922 Nobel Prize
of electron orbits and characterized by the quantum number n = 1, 2, 3, …
3. Each state has a discrete, well-defined energy En, with E1<E2<E3<…
4. The lowest or ground state E1 of an atom is stable and can persist indefinitely.
Other stationary states E2, E3, … are called excited states.
5. An atom can “jump” from one stationary state to another by emitting a photon of
frequency f = (Ef−Ei)/h, where Ei,f are the energies of the initial and final states.
6. An atom can move from a lower to a higher energy state by absorbing energy in an
inelastic collision with an electron or another atom, or by absorbing a photon.
7. Atoms will seek the lowest energy state by a series of quantum jumps between
states until the ground state is reached.
4
The Bohr Model
The implications of the Bohr model are:
1. Matter is stable, because there are no states
lower in energy than the ground state;
2. Atoms emit and absorb a discrete spectrum of
light, only photons that match the interval
between stationary states can be emitted or
absorbed;
3. Emission spectra can be produced by collisions;
4. Absorption wavelengths are a subset of the
emission wavelengths;
5. Each element in the periodic table has a
different number of electrons in orbit, and
therefore each has a unique signature of
spectral lines.
5
Example: Wavelength of
an Emitted Photons
An atom has stationary states with energies Ej = 4.0 eV and Ek = 6.0 eV.
What is the wavelength of a photon emitted in a quantum jump from state
k to state j?
hc = (4.14 ×10−15 eV s)(3.00 ×108 m/s)(109 nm/m) = 1, 242 eV nm
∆Eatom = Ek − E j = (4.0 eV) − (6.0 eV) = −2.0 eV;
Ephoton = 2.0 eV;
c
hc
(1, 242 eV nm)
λ= =
=
= 621 nm
f Ephoton
(2.0 eV)
6
Energy Level Diagrams
excited states
ground state
It is convenient to represent the energy states of an atom using an energy
level diagram.
Each energy level is represented by a horizontal line at at appropriate
height scaled by relative energy and labeled with the state energy and quantum
numbers. De-excitation photon emissions are indicated by downward arrows.
Absorption excitations are indicated by upward arrows.
7
Clicker Question
A photon with a wavelength
of 621 nm has an energy of 2.0
eV.
For an atom with the energy
levels shown, would you expect
to see a line with this
wavelength in the absorption
spectrum? In the emission
spectrum?
(a) yes, yes;
(b) yes, no;
(c) no, yes;
(d) no, no;
(e) cannot tell.
8
The Balmer Series
The Balmer series is now understood to be one of
several series in hydrogen. They are produced by
electron jumps from orbit n to orbit m, with m = 2 for
the Balmer series.
1 
 1
λ = (91.18 nm)  2 − 2 
m n 
−1
9
Clicker Question
What is the quantum number n of the
electron orbit in this hydrogen atom?
(a) 2;
(b) 3;
(c) 4;
(d) 6;
(e) 12.
10
The Bohr Hydrogen Atom
This is a version of the mathematical
formulation of the Bohr Model of the atom:
aelec
Felec
1 e2
v2
=
=
=
2
m
r
4πε 0 mr
2π r = nλ = n
h
h
=n
p
mv
e 2  nh 
∴
=

4πε 0 mr  rm 
1
2
e2
⇒ v =
4πε 0 mr
2
⇒ vn =
1
nh
nh
=
2π rm rm
(h ≡ h / 2π )
4πε 0 h 2
⇒ rn = n
me 2
2
4πε 0 h 2
Bohr radius: aB ≡
= 5.29 ×10−11 m = 0.0529 nm;
2
me
rn = n 2 aB ; r1 = 0.0529 nm; r2 = 0.212 nm; r3 = 0.476 nm; L
11
Hydrogen Atom Energy Levels
E = K +U =
1 2
1 qe q p
mv +
2
4πε 0 r
1 2
1 e2
= mv −
2
4πε 0 r
e2
v =
4πε 0 mr
1
2
2
rn = n aB
e2
⇒ E=−
4πε 0 2r
1
1 1 e2
⇒ En = − 2
n 4πε 0 2aB
e2
EB ≡
= 13.60 eV;
4πε 0 2aB
1
En = −
1
13.60 eV
E
=
−
; n = 1, 2, 3, K
B
2
2
n
n
12
Example: Stationary States of
the Hydrogen Atom
Can an electron in a hydrogen atom have a speed of 3.60x105 m/s? If
so what are its energy and the radius of its orbit?
v1
nh 1 h
vn =
=
= ; v1 = 2.19 × 106 m/s
rn m n maB n
v1 (2.19 ×106 m/s)
X
n1 = =
=
6.083
(not
an
integer,
no
such
state)
vn (3.60 ×105 m/s)
What about a speed of 3.65x105 m/s?
v1 (2.19 × 106 m/s)
n2 = =
= 6.000 (an integer, the state exists) OK
5
vn (3.65 ×10 m/s)
EB
E6 = − 2 = −0.38 eV;
6
r6 = 62 aB = 1.90 nm
13
Binding Energy and
Ionization Energy
The binding energy of an electron in
stationary state n is defined as the energy
that would be required to remove the
electron an infinite distance from the
nucleus. Therefore, the binding energy of
the n=1 state of hydrogen is EB = 13.60 eV.
It would be necessary to supply 13.60 eV
of energy to free the electron from the
proton, and one would say that the electron
in the ground state of hydrogen is “bound by
13.60 eV”.
The ionization energy is the energy required to remove the least
bound electron from an atom. For hydrogen, this energy is 13.60 eV. For
other atoms it will typically be less.
14
Quantization of
Angular Momentum
Actually, in constructing his atomic model, Niels Bohr did not assume
that an integer number of de Broglie wavelengths fitted into the
circumference of the orbit. (Bohr did not know about de Broglie waves
when he formulated his theory.) Instead, he assumed that the angular
momentum L of the orbit was quantized in units of ħ (=h/2π
π).
L ≡ mvr = nh;
λ=
h
h
2π hr 1
=
=
= 2π r ;
p mv
nh
n
∴ nλ = 2π r.
Thus, Bohr’s assumption that angular momentum is quantized in units
of ħ is equivalent to assuming that an integer number of de Broglie wave
lengths of the electron fits into the circumference of a Bohr orbit.
Where does L=nħ come from? It comes from the symmetry that if
you rotate an object by 360O, it should return to the same state. This
symmetry produces an angular momentum condition on the electron’s
wave function.
15
The Hydrogen Spectrum
The figure shows the
energy-level diagram for
hydrogen. The top “rung” is
the ionization limit, which
corresponds to n→∞ and to
completely removing the
electron from the atom. The
higher energy levels of
hydrogen are crowded
together just below the
ionization limit.
The arrows show a photon
absorption 1→4 transition
and a photon emission 4→2
transition.
16
Transitions
∆Eatom En − Em
f =
=
h
h
1   1  1 e 2
= − 2 
h   n  4πε 0 2aB
   1  1 e 2   
 − − 2 
 
m
πε
a
4
2
0
B  
 


e2  1
1 
=
−


4πε 0 2haB  m 2 n 2 
4πε 0 h 2
aB ≡
me 2
1
c 8πε 0 hcaB  1
1 
−
λ= =
 2
2 
f
e2
m
n


λ0 ≡
m=2
m=1
−1
8πε 0 hcaB
−8
=
9.112
×
10
m = 91.12 nm
2
e
λn →m
1 
 1
= λ0  2 − 2 
m n 
−1
17
Example: The Lyman α Forest
When astronomers look at distant quasars, they find that the light has
been strongly absorbed at the local wavelength of the “Lyman α” 1→2
transition of the Lyman series of hydrogen. This absorption tells us that
interstellar space is filled with vast clouds of hydrogen left over from the
Big Bang.
What is the wavelength of the Lyman α 1→2 absorption line in
hydrogen?
−1
λ0 = 91.12 nm
4
1 1 
−
=
(91.12 nm) = 121.5 nm
2
2 
3
1 2 
λ1→2 = λ0 
This wavelength is in the ultraviolet.
However, the cosmic recession velocity
of quasars Doppler shifts the Lyman α
line into the visible, and because the
absorption occurs at many distances, a
“forest” of absorption line is observed.
18
Hydrogen-like Atoms
The Bohr model also
works well for “hydrogenlike” atoms, i.e., atoms
with Z protons in the
nucleus and only one
orbital electron. For such
atoms, the orbits are
shifted by the increased
Coulomb force:
Velec
Ze2
=
;
4πε 0 r
EnZ
Z2
Z2
= − 2 EB = − 2 13.60 eV;
n
n
1
n 2 aB
rnZ =
;
Z
vnZ = Z
λ0Z =
v1  Z 
=   2.19 ×106 m/s;
n n
λ0
Z2
=
91.18 nm
Z2
19
Success and Failure
Bohr’s analysis of the hydrogen atom was a resounding success. By
introducing stationary states, together with Einstein’s ideas about light
quanta, Bohr was able to explain the stability of atoms, provide the first
solid understanding of discrete spectra, and justify the Balmer formula.
However, there were problems. The model failed when two or more
electrons were in orbit. The Bohr model could not predict even the
spectrum of helium, an atom with two electrons orbiting a charge-2
mass-4 nucleus. Something in Bohr’s assumptions worked correctly for a
single electron but failed when two or more electrons were involved.
It is important to distinguish between the Bohr Model, which assumes
that stationary states exist, and the Bohr hydrogen atom, which gives
concrete expressions for such stationary states. The problem for multielectron atoms was not Bohr’s model, but his method for finding the
stationary states.
The missing technique, which was developed in the mid-1920s, was
quantum mechanics.
20