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```實驗7 (課本實驗24)

References
 http://csep10.phys.utk.edu/astr162/lect/light/bohr.html
 清華大學物理系普物實驗課本實驗24講義

(atomic spectrum)

DE = En1 – En2 = hf = hc/l
h = 6.63 x 10-34 J-s
c = 3.00 x 108 m/s
Balmer series lines (巴爾麥系譜

Visible spectrum (可見光譜)
Ha: 32, l = 656.3 nm
Hb: 42, l = 486.1 nm
Hg: 52, l = 434.0 nm

1. 利用氫氣管(H2)高壓放電,

(spontaneous emission)
(保護儀器,注意安全)
2. 利用光譜儀(spectrometer)

dsinq = ml (m = 0, 1, 2, …)
m = 1: 三條譜線 (la, lb, lg)
m = 2: ~一條(lb)

3. 與理論值比較
1/l = (mee4/8e0h3c)(1/22 - 1/n2) (n2)
= (1/RH)(1/4 - 1/n2)
RH = 8.31 J/mol-K (Rydberg constant)
4. 利用實驗值波長代入RH, 利用
me = 9.11 x 10-31 kg
e = 1.60 x 10-19 C
e0 = 8.85 x 10-12 F/m
c = 3.00 x 108 m/s

5. 觀察其他氣體放電之可見光原子及分子光譜
DVD: 原子 (The Atom)
(The Mechanical Universe…and Beyond/MU49)
[Annenberg/CPB/www.learner.org]
Bohr’s theory of hydrogen atom (波爾氫原子理論)
Potential energy of electron bound to a proton
U = -e2/4pe0r
Total energy for circular orbit with centrifugal force mv2/r = e2/4pe0r2
E = K + U = mv2/2 + U = - e2/8pe0r
Frequency condition from spectral line (光譜頻率條件)
DE = Ei – Ef = hfif
Quantized angular momentum L of the orbiting electron (電子軌道角

L = mvr = n(h/2p) n = 1, 2, 3, …
r = n2h2e0/pme2 = n2rB (rB = 0.0529 nm = 52.9 pm Bohr’s radius)
Allowed energy states: En = -(mee4/8e02h2)/n2 = -13.6 eV/n2
Importance of the Hydrogen Atom
 The H-atom is the only atomic system that can be solved




exactly.
Much of what was learned about the H-atom, with its single
electron, can be extended to such single-electron ions as
He+ and Li2+.
The H-atom proved to be an ideal system for performing
precision tests of theory against experiment.
 Also for improving our understanding of atomic structure.
The quantum numbers that are used to characterize the
allowed states of hydrogen can also be used to investigate
more complex atoms. This allows us to understand the
periodic table.
The basic ideas about atomic structure must be well
understood before we attempt to deal with the complexities of
molecular structures and the electronic structure of solids.
J. J. Thomson Atomic Model
– Early Model (Newton’s Time) of the Atom
 The atom was a tiny, hard indestructible sphere.
 It was a particle model that ignored any internal structure.
 The model was a good basis for the kinetic theory of
gases.
 J. J. Thomson established the
charge to mass ratio for
electrons.
 His model of the atom
 A volume of positive charge.
 Electrons embedded
throughout the volume.
Rutherford’s Thin Foil Experiment
 Experiments done in 1911.
 A beam of positively charged
alpha particles hit and are
scattered from a thin foil target.
 Large deflections could not be
explained by Thomson’s model.
 Rutherford
 Planetary model based on results of
thin foil experiments
 Positive charge is concentrated in the
center of the atom, called the nucleus.
 Electrons orbit the nucleus like
planets orbit the sun
Difficulties with the Rutherford Model
 Atoms emit certain discrete characteristic frequencies of
 The Rutherford model is unable to explain this phenomena.
Rutherford’s electrons are undergoing a centripetal
acceleration.
waves of the same frequency.
as this radiation is given off.
 The electron should eventually spiral
into the nucleus.
 But the fact doesn’t.

The Bohr Theory of Hydrogen
-A Planetary Model of the Atom
 In 1913 Bohr provided an explanation
of atomic spectra that includes some
features of the currently accepted
theory.
 His model includes both classical and non-classical ideas.



He applied Planck’s ideas of quantized energy levels to
orbiting electrons.
In this model, the electrons are generally confined to
stable, nonradiating orbits called stationary states.
Used Einstein’s concept of the photon to arrive at an
expression for the frequency of radiation emitted when
the atom makes a transition.
 The Bohr Model is probably familiar as the "planetary model"
of the atom.
 for example, is used as a symbol for atomic energy (a bit of
a misnomer, since the energy in "atomic energy" is actually
the energy of the nucleus, rather than the entire atom).
 In the Bohr Model the neutrons and protons (symbolized by
red and blue balls in the adjacent image) occupy a dense
central region called the nucleus, and the electrons orbit the
nucleus much like planets orbiting the Sun (but the orbits are
not confined to a plane as is approximately true in the Solar
System).
 The adjacent image is not to scale since in the realistic case
than the radius of the entire atom, and as far as we can tell
electrons are point particles without a physical extent.
 This similarity between a planetary model and the Bohr
Model of the atom ultimately arises
because the attractive gravitational force in a solar
system and
 the attractive Coulomb (electrical) force between the
positively charged nucleus and the negatively charged
electrons in an atom are mathematically of the same
form.
 The form is the same, but the intrinsic strength of the
Coulomb interaction is much larger than that of the
gravitational interaction;
 in addition, there are positive and negative electrical
charges so the Coulomb interaction can be either
attractive or repulsive, but gravitation is always
attractive in our present Universe.
The Orbits Are Quantized
-Quantized energy levels in hydrogen
 The basic feature of quantum mechanics that is




incorporated in the Bohr Model.
That is completely different from the analogous planetary
model is that the energy of the particles in the Bohr atom
is restricted to certain discrete values.
One says that the energy is quantized.
This means that only certain orbit
orbits in between simply
don't exist.
Quantized energy levels
in hydrogen
Quantized Energy Levels in the
hydrogen atom
 These energy levels are labeled by an integer n that is
called a quantum number.
 The lowest energy state is generally termed the ground
state.
 The states with successively more energy than the ground
state are called the first excited state, the second excited
state, and so on.
 Beyond an energy called the ionization potential the single
electron of the hydrogen atom is no longer bound to the
atom.
 Then the energy levels form a continuum.
 In the case of hydrogen, this continuum starts at 13.6 eV
above the ground state ("eV" stands for "electron-Volt", a
common unit of energy in atomic physics).
Atomic Excitation and De-excitation
 Atoms can make transitions between the orbits allowed
by quantum mechanics by absorbing or emitting exactly
the energy difference between the orbits.
Excitation by absorption of
light and de-excitation by
emission of light
 Atoms can make transitions between the orbits allowed
by quantum mechanics by absorbing or emitting exactly
the energy difference between the orbits.
 In each case the wavelength of the emitted or absorbed
light is exactly such that the photon carries the energy
difference between the two orbits.
 This energy may be calculated by dividing the product
of the Planck constant and the speed of light hc by the
wavelength of the light).
 Thus, an atom can absorb or emit only certain discrete
wavelengths (or equivalently, frequencies or energies).
 Here is a Shockwave movie of atomic absorption and
emission in
 Here is a Java applet illustrating atomic absorption and
emission.
Separation of light by a prism
according to wavelength
 Based on the Bohr atom, isolated atoms can absorb and
emit packets of electromagnetic radiation having discrete
energies dictated by the detailed atomic structure of the
atoms.
 When the corresponding light is passed through a prism
or spectrograph it is separated spatially according to
wavelength l.
Continuum, Emission &
Absorption Spectra
 The corresponding spectrum may exhibit a continuum,
or may have superposed on the continuum bright lines
(an emission spectrum) or dark lines (an absorption
spectrum), as illustrated in the following figure.
Continuous Spectrum
Emission Spectra
Absorption spectra
Origin of Continuum, Emission &
Absorption Spectra
 The emission spectra are produced by thin gases in which
the atoms do not experience many collisions (because of the
low density).
 The emission lines correspond to photons of discrete energies
that are emitted when excited atomic states in the gas make
transitions back to lower-lying levels.
 A continuum spectrum results when the gas pressures are
higher. Generally, solids, liquids, or dense gases emit light at
all wavelengths when heated.
 An absorption spectrum occurs when light passes through a
cold, dilute gas and atoms in the gas absorb at characteristic
frequencies; since the re-emitted light is unlikely to be emitted
in the same direction as the absorbed photon, this gives rise
to dark lines (absence of light) in the spectrum.
Sources of continuous, emission, and
absorption spectra

The emission spectra are produced by thin gases in which the atoms do not
experience many collisions (because of the low density).
 The emission lines correspond to photons of discrete energies that are emitted when
excited atomic states in the gas make transitions back to lower-lying levels.
 An absorption spectrum occurs when light passes through a cold, dilute gas and
atoms in the gas absorb at characteristic frequencies; since the re-emitted light is
unlikely to be emitted in the same direction as the absorbed photon, this gives rise to
dark lines (absence of light) in the spectrum.
Continuous Spectrum
Hot Gas
Cold Gas
Emission Spectra
Absorption spectra
Hydrogen
Emission &
Absorption
Series
(visible light)
Hydrogen emission series
(UV spectrum)
Hydrogen Emission & Absorption Series
 The spectrum of hydrogen is particularly important in






astronomy because most of the Universe is made of
hydrogen.
Emission or absorption processes in hydrogen give rise to
series, which are sequences of lines corresponding to
atomic transitions, each ending or beginning with the same
atomic state in hydrogen.
The Balmer Series involves transitions starting (for
absorption) or ending (for emission) with the first excited
state of hydrogen.
The Lyman Series involves transitions that start or end with
the ground state of hydrogen.
Because of the details of hydrogen's atomic structure,
the Balmer Series is in the visible spectrum and
the Lyman Series is in the the UV.
 Because of the details of hydrogen's atomic
structure,
 the Balmer Series is in the visible spectrum and
 the Lyman Series is in the the UV.
 The Balmer lines are designated by H with a
greek subscript Hi in order of decreasing
wavelength.
 Thus the longest wavelength Balmer transition is
designated H with a subscript alpha, Ha.
 the second longest H with a subscript beta, Hb,
 and so on, Hg, H….
Electron Transitions
 An electron transition in hydrogen between quantized
energy levels with different quantum numbers n yields
a photon by emission with quantum energy:
This is often expressed in terms of the inverse wavelength or
"wave number" as follows:
Quantized Energy States
 The electrons in free atoms can will be found in only certain
discrete energy states. These sharp energy states are associated
with the orbits or shells of electrons in an atom, e.g., a hydrogen
atom. One of the implications of these quantized energy states is
that only certain photon energies are allowed when electrons jump
down from higher levels to lower levels, producing the hydrogen
spectrum. The Bohr model successfully predicted the energies for
the hydrogen atom, but had significant failures that were corrected
by solving the Schrodinger equation for the hydrogen atom.
Hydrogen
Energy Levels
Basic Structure of the Hydrogen
Energy Levels
 It can be calculated from the Schrodinger equation.
 The energy levels agree with the earlier Bohr model, and




agree with experiment within a small fraction of an
electron volt.
If you look at the hydrogen energy levels at extremely
high resolution, you do find evidence of some other small
effects on the energy.
The 2p level is split into a pair of lines by the spin-orbit
effect.
The 2s and 2p states are found to differ a small amount
in what is called the Lamb shift.
And even the 1s ground state is split by the interaction of
electron spin and nuclear spin in what is called hyperfine
structure.
Balmer Line Series
in Visible Spectrum
 1885 - Johann Jacob Balmer
 Analyzed the hydrogen spectrum and
found that hydrogen emitted four bands
of light within the visible spectrum:
 Balmer found that the data fit to the
following equation:

l = wavelength (nm)
 RH = Rydberg's constant
= 1.09678 x 10-2 nm-1
 n1 = the lower energy level
 n2 = the higher energy level
Wavelength
(nm)
Color
656.2
red
486.1
blue
434.0
blue-violet
410.1
violet
For example, to calculate the wavelength
of light emitted when the electron in a
hydrogen atom falls from the fourth energy
level to the second energy level:
Each series is named after its discoverer:
 The Lyman series is the wavelengths in the ultra violet (UV)




spectrum of the hydrogen atom, resulting from electrons
dropping from higher energy levels into the n = 1 orbit.
The Balmer series is the wavelengths in the visible light
spectrum of the hydrogen atom, resulting from electrons falling
from higher energy levels into the n = 2 orbit.
The Paschen series is the wavelengths in the infrared
spectrum of the hydrogen atom, resulting from electrons falling
from higher energy levels into the n = 3 orbit.
The Brackett series is the wavelengths in the infrared
spectrum of the hydrogen atom, resulting from electrons falling
from higher energy levels into the n = 4 orbit.
The Pfund series is the wavelengths in the infrared spectrum
of the hydrogen atom, resulting from electrons falling from
higher energy levels into the n = 5 orbit.
Absorption Spectrum
 1814 - Joseph von Fraunhofer
 Studied the absorption spectrum of the light given off by
the sun.
 Absorption Spectrum - The spectrum of dark lines
against a light background that results from the absorption
of selected frequencies of the electromagnetic radiation by
an atom or molecule.
The Balmer Series of Hydrogen consists of four visible lines.
The Balmer Series of Hydrogen (H)
consists of four visible lines
 The helium (He) spectrum is somewhat
more complex than that of hydrogen.
The neon (Ne) spectrum is dominated
by red lines.
 The sodium (Na) spectrum consists of
one very bright yellow line.
The mercury (Hg) spectrum
```
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