Transcript Topic 7_1_Ext C__The Bohr theory of the hydrogen atom

Topic 7.1 Extended
C –The Bohr theory of the hydrogen atom
When a gas in a tube is subjected to a voltage, the
gas ionizes, and emits light.
We can analyze that light by looking at it through a
spectroscope.
A spectroscope acts similar to a prism, in that it
separates the incident light into its constituent
wavelengths.
For example, barium gas in a gas discharge tube will
produce an emission spectrum that looks like this:
4000
4500
5000
5500
6000
6500
7000
The emission spectrum is really an elemental
fingerprint - it uniquely identifies the element
producing it.
FYI: The wavelengths are given in angstroms Å. 1 Å = 10-10 m.
7500
Extended
FYI: Note the possibleTopic
fingerprint7.1
for calcium
as an absorbing constituent
of the
atmosphere.
C Sun's
–Theouter
Bohr
theory of the hydrogen atom
Calcium gas produces this spectrum:
4000
4500
5000
5500
6000
6500
7000
7500
Not only do glowing gases emit spectral lines, but cool
gases absorb light in the same wavelengths and produce
what is called an absorption spectrum.
For example, the Sun produces a continuous spectrum
that looks like this...
4000
4500
5000
5500
6000
6500
7000
7500
...with characteristic absorption spectral lines,
revealing what non-glowing elements are in the Sun's
atmosphere.
FYI: Balmer studied the middle series, which has parts in the visible
spectrum. Obviously, the series is in his name.
Topic 7.1 Extended
C –The Bohr theory of the hydrogen atom
In the late 1800s a Swedish physicist by the name of
J.J. Balmer observed the spectrum of hydrogen - the
simplest of all the elements:
Emission Spectra of Hydrogen
0
200
Lyman
Series
UV
400
600
Balmer
Series
800
1000
1200
1400
1600
1800
2000
Paschen
Series
IR
The general spectral signature is divided up into
natural groups of spectral lines called series, falling
roughly in the UV (ultraviolet), Visible, and IR
(infrared) ranges of wavelengths.
FYI: Each series is characterized by spectral line "bunching" at the smaller
wavelengths, and "spreading" at the larger wavelengths.
Topic 7.1 Extended
C –The Bohr theory of the hydrogen atom
In fact, Balmer found an
Balmer Series
empirical formula that
predicted the allowed
spectral wavelengths for the
Balmer series of the hydrogen
atom:
1 = R 1 - 1
n2
22

for n = 3,4,5,...
Balmer
Series
where R = 1.09710-2 nm-1 is called the Rydberg
constant.
FYI: The visible spectrum for hydrogen was found to fit this formula, but it
was NOT understood why.
FYI: Similar formulas were found to fit the other two series.
FYI: Note that we are using the relation U = -kqQ / r for two point charges.
Recall that this energyTopic
is negative
sinceExtended
q and Q are oppositely charged.
7.1
FYI:
to this point
in Bohr's
derivation,
mechanics has atom
been
C Up–The
Bohr
theory
of classical
the hydrogen
used.
An explanation was finally given
in 1913 by the Danish physicist
Niels Bohr.
r
Bohr postulated that the single
electron was held in a circular
+
orbit about the single proton in
the hydrogen nucleus by the
Coulomb force:
Fc = FE
ke2
ke2
mv2
1
2
 2mv =
=
2r
r2
r
The total mechanical E energy of the hydrogen atom
is given by E = K + U so that
ke2
1
2
E =
mv 2
r
2
ke2
2 ke2
ke
Energy in
E = E =
H Atom
2r
2r
2r
FYI: We subscript the r to indicate that the electron can only orbit the
Topic
7.1
Extended
nucleus at certain quantized
radii,
determined
by the principal quantum
number.
C –The Bohr theory of the hydrogen atom
Recall that the angular momentum
l of a point mass moving in a
circle of radius r is given by
l = mvr
Bohr then postulated the radical
idea that the angular momentum of
the electron was quantized, just
like light.
He stated that the angular momentum
of the electron can only carry the
discrete values given by
r
+
h
Principal Quantum
for n = 1,2,3,...
Number - H Atom
2
2
ke
1
and the previous equation we can
From
mv2 =
2
2r
eliminate v, solving for r:
mvr = n
h2
rn =
n2 for n = 1,2,3,...
2
2
4 ke m
Allowed Radii
- H Atom
Topic 7.1 Extended
C –The Bohr theory of the hydrogen atom
We can then take our energy equation for the hydrogen
atom and substitute our allowed values for r:
ke2
E = 2r
Energy Quantization
22k2e4m 1 for n = 1,2,3,...
En = in the Bohr
2
h2
n
Hydrogen Atom
Everything in the parentheses is a constant whose
value we know. We can then rewrite both rn and En
like this:
-13.6
The Bohr
En =
eV
n2
Hydrogen
2
rn = 0.0529n nm
Atom
FYI: We call the lowest energy level (n = 1) the GROUND STATE.
Topic
7.1(n =Extended
We call the next highest
energy level
2) the 1ST EXCITED STATE.
WeCcall–The
the nextBohr
highest energy
level of
(n = 3)
the 2ND
EXCITED STATE.
theory
the
hydrogen
atom
Et cetera.
What are the orbital radius and energy of an electron
in a hydrogen atom characterized by principal quantum
number 3?
-13.6
E
=
eV
n
rn = 0.0529n2 nm
n2
-13.6
r3 = 0.052932 nm
E3 =
eV
32
r3 = 0.4761 nm
E3 = -1.51 eV
What is the change in energy if the electron "drops"
to the energy characterized by principal quantum
number 2?
-13.6
E = (-3.4 - -1.51) eV
E2 =
eV
22
E = -1.89 eV
E2 = -3.4 eV
FYI: In general, if an electron "drops" from a higher quantum state to a
lower one, the hydrogen atom experiences a net loss of energy.
Topic 7.1 Extended
C –The Bohr theory of the hydrogen atom
What is the orbital velocity of an electron in the
second excited state (n = 3)?
From the previous slide r3 = 0.4761 nm.
h
Then
mvr = n
2
nh
v =
2mr
3(6.6310-34)
v =
2(9.1110-31)(0.476110-9)
v = 7.30105 m/s
What then is the centripetal acceleration of the
electron?
(7.30105)2
v2
= 1.12 1021 m s-2
=
ac =
-9
0.476110
r
FYI: Classical theory predicts that electromagnetic radiation is created by
accelerating charges. Since the hydrogen atom only radiates when its
electron "drops" from one excited state to a less energetic state, Bohr
postulated that "the hydrogen electron does NOT radiate energy when it is
in one of its bound states (allowed by n). It only does so when "dropping"
from a higher state to a lower state."
EXAMPLE: If a photon having an energy of E = +12.09 eV is absorbed by a
Topic
7.1
Extended
hydrogen atom in its ground
state,
the electron
will "jump" up to the second
excited
state (nBohr
= 3) since
E = (-1.51
12.09 eV.
C –The
theory
of- -13.6)
the= +hydrogen
atom
Consider a plot
of energies for
n = 1 to :
Excited
States
-13.6
En =
eV
n2
FYI: Bohr's theory only
allows electrons in the
hydrogen atom to absorb or
emit photons having energies
equal to the difference
between any two of the
Ground
allowed states shown here. State
n=
0.00 eV
n=5
n=4
-0.544 eV
-0.850 eV
n=3
-1.51 eV
n=2
-3.40 eV
n=1
-13.6 eV
EXAMPLE: If an electron at r3 suddenly "drops" to the ground state, the
hydrogen atom LOSES energy having a value of
E = (-13.6 - -1.51) = -12.09 eV.
To conserve energy, a PHOTON having an energy of
E = +12.09 eV
is released.
FYI: The Lyman Series has as its final state the GROUND STATE.
Topic
7.1
The Balmer Series has
as its final
state Extended
the FIRST EXCITED STATE.
The
as its final state
SECOND
EXCITEDatom
STATE.
C Paschen
–The Series
Bohrhastheory
of the
the
hydrogen
0.00 eV
n=
n=5
-0.544 eV
So how do the three
-0.850 eV
series of hydrogen Second n = 4
spectra relate to Excited n = 3
-1.51 eV
State
the Bohr model?
Paschen
Consider the
First
n=2
-3.40 eV
Series
Excited
following transiIR
State
tions of hydrogen
Balmer
from higher to lower
Series
states:
Visible
Each transition
Lyman
gives off a photon Ground n = 1
-13.6 eV
Series
State
of a different
UV
wavelength.
0
200
Lyman
Series
UV
400
600
Balmer
Series
800
1000
1200
1400
Paschen
Series
IR
1600
1800
2000