Download Lecture 15: Bohr Model of the Atom

Lecture 15: Bohr Model of the
Atom
•  Reading: Zumdahl 12.3, 12.4
•  Recommended Problems: 12.35, 12.37, 12.39,
12.41
•  Outline
–  Review of black body radiation and applications
–  Emission spectrum of atomic hydrogen.
–  The Bohr model.
–  Extension to higher atomic number.
Summary of Black Body Radiation
•  A black body is simulated by a hollow metal chamber.
When heated it glows red, because atoms in the metal
oscillate at frequencies around the low frequency end of
the visible spectrum. As the temperature is increased, more
atoms vibrate at higher frequencies, higher frequencies of radiation are emitted in the cavity until the metal becomes “white hot”. •  Application: –  You can measure the temperatures of hot objects (e.g. stars) from the
radiation they emit.
Summary of Black Body Radiation
•  Classical vs. Quantum Theory
of Black Body Radiation
Property
Classical Theory
Planck’s Quantum
Theory
Energy of Oscillator
Average Energy of
an Oscillator: <E>
Change in energy
(ΔE)
anything
Intensity of radiation
in the cavity at λ
Result
Ultraviolet
catastrophe
Agrees with data
No catastrophe
Einstein’s Theory of the Heat Capacity of
Vibration (CV=5R/2+CV,vib)
•  Einstein used Planck’s theory of quantized oscillations to
calculate the vibrational heat capacity. Einstein showed
that the heat capacity for vibrational motions is
temperature dependent when the motions are quantized.
So for an ideal diatomic gas CV=5R/2 at T=100K
But CV=5R/2+R=7R/2 at T=2000K…
Big Achievement of the Quantum
Approach: Atomic structure
•  Atoms were known in the late 19th century to be composed
of negatively charged electrons and positively charged
protons. But how these particles were arranged in the atom
was not understood. •  By the early 20th century, experiments showed that the
protons were located within a very small volume called the
nucleus, while the electrons moved about the nucleus in
some complicated way. This was called the nuclear atom
model.
•  Problem: Classical physics could not explain the stability
of the nuclear atom, nor its interactions with light.
•  Quantization explained all this.
Continuous vs. Discrete Spectra
When the light emitted
from excited atoms is
passed through a
prism, we see discrete
bands of color at
specific wavelengths.
White light passed
through a prism gives a
continuous
spectrum ...all visible
wavelengths are
present.
H
Li
Ba
Atomic Emission
When we heat a sample of an element, the atoms become excited. When
the atom relaxes it emits visible light. The color of the light depends on
the element.
Li
When the light emitted
from excited atoms is
passed through a
prism, we see discrete
bands of color at
specific wavelengths.
Na
K
Ca
Sr
H
Li
Ba
An atom absorbs light exciting
electron motions to higher energies
E. Eventually, the excited atom
“relaxes” when its electrons go from
high E to low E by emitting energy.
We can determine the energy
difference (ΔE) between the
electronic energy levels by
measuring the wavelength of the
emitted radiation.
ΔE = hν=hc/λ ◊ λ = hc/ ΔE
If λ = 440 nm, ΔΕ = 4.5 x 10-19
J
Emission
Mechanism of Atomic Emission
Emission spectrum of H
Light Bulb
Hydrogen Lamp
Quantized, not continuous
Emission spectrum of H (cont.)
“Quantized” spectrum
ΔE
ΔE
“Continuous” spectrum
Any ΔE is
possible
Only certain
ΔE are
allowed
Emission spectrum of H (cont.)
We can use the emission spectrum to determine the energy levels for the hydrogen atom.
Rydberg Model
•  Johann Rydberg extends
the Balmer model by
finding more emission
lines outside the visible
region of the spectrum:
n1 = 1, 2, 3, …..
n2 = n1+1, n1+2, …
Ry =3.29x1015s-1
•  This suggests that the energy levels of the H atom
are proportional to 1/n2
The Bohr Model
•  These spectroscopic observations had to be reconciled with
the model of the nuclear atom: that in the atom electrons
were separated in space from a single massive, positively
charged nucleus. •  The problem with this classical view of an electron
orbiting around a nucleus like a planet around a sun: such
an atom is unstable and does not explain the emission
spectra.
•  Unlike an orbiting planet, an orbiting electron would lose
energy quickly because it radiates. The orbit of such an
electron would decay in less than 10-12 s and due to the
Coulombic attraction, would collide with the nucleus. The
atom would collapse!
The Bohr Model: The Ideas
•  To address this perceived instability of the nuclear atom,
and to explain the atomic emission data, Niels Bohr
postulated that electrons do not radiate if they occupy
certain “allowed”orbits.This thinking leads to…
Quantized Angular momentum: Quantized Orbital radii: Quantized Energy:
n is called a quantum number The Bohr Model of
the atom
Principle Quantum number: n
An “index” of the energy levels
available to the electron.
The Bohr Model (cont.)
• Energy levels get closer together
as n increases
• at n = infinity, E = 0
The Bohr Model (cont.)
• We can use the Bohr model to predict what ΔE is
for any two energy levels
The Bohr Model (cont.)
• Example: At what wavelength will emission from
n = 4 to n = 1 for the H atom be observed?
1
4
The Bohr Model (cont.)
• Example: Estimate the wavelength of light that
will result in removal of the e- from H.
∞
1
Extension to Higher Z
• The Bohr model can be extended to any single electron system….must keep track of Z
(atomic number).
Z = atomic number
n = integer (1, 2, ….)
• Examples: He+ (Z = 2), Li+2 (Z = 3), etc.
Extension to Higher Z (cont.)
• Example: At what wavelength will emission from
n = 4 to n = 1 for the He+ atom be observed?
2
1
4
Bohr Model: The Good and the Bad
•  Good:
–  Good illustration of quantization approach
–  Correctly predicts the emission spectrum of H
–  Correctly predicts the quantized energy
–  Explains stability of nuclear atom
•  Bad:
–  No physical explanation for quantized orbitals
–  Only works for single electron atoms (H, He+)
–  Angular momentum only correct for large n
A more general theory is needed.