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氫原子光譜之分析
氫原子的 Bohr 理論模型
利用分光儀分析光譜
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Importance of the Hydrogen Atom
• A structural model can also be used to describe a very small-scale system, the atom
(利用結構模型描述原子系統 )• The hydrogen atom is the only atomic system
that can be solved exactly ( 氫原子為原子系統中唯一有正確解的原子 )• Much of what was learned about the hydrogen
atom, with its single electron, can be extended to such single-electron ions as He+ and Li2+
( 氫原子模型的理論計算可推廣至擁有單一電子的離子 )
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Light From an Atom
• The electromagnetic waves emitted from the atom can be used to investigate its structure and properties– Our eyes are sensitive to visible light– We can use the simplification model of a wave
to describe these emissions
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物質 (原子 )的電子能階
激發態不穩定
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Emission Spectra Examples
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The Bohr Theory of Hydrogen
• 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
E h f
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Niels Bohr
• 1885 – 1962• An active participant in
the early development of quantum mechanics
• Headed the Institute for Advanced Studies in Copenhagen
• Awarded the 1922 Nobel Prize in physics– For structure of atoms
and the radiation emanating from them
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Bohr’s Assumptions for Hydrogen• The electron moves in circular orbits around the
proton under the electric force of attraction. • ( 電子對原子核做圓周運動 )
• Only certain electron orbits are stable and these are the only orbits in which the electron is found.
• ( 電子在特定的軌道才能穩定的運動 )
• Radiation is emitted by the atom when the electron makes a transition from a more energetic initial state to a lower-energy orbit.
• ( 電子在特定軌道之間躍遷時,吸收或發出光子 )
• The size of the allowed electron orbits is determined by a condition imposed on the electron’s orbital angular momentum.
Quantization of angular momentum
v
( )
eL m r
n
��������������
特定的軌道大小由電子的角動量決定
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Bohr’s Assumptions for Hydrogen• ( 電子對原子核做圓周運動 )
• ( 電子在特定的軌道才能穩定的運動 )
• ( 電子在特定軌道之間躍遷時,吸收或發出光子 )
Quantization of angular momentum
v
( )
eL m r
n
��������������
特定的軌道大小由電子的角動量決定
22
0
1 1v
2 4
eE K U m
r
2 2
20
v 1
4
emr r
i fE E h f
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Wave Properties of Particles
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Louis de Broglie• 1892 – 1987• Originally studied
history• Was awarded the
Nobel Prize in 1929 for his prediction of the wave nature of electrons
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Wave Properties of Particles• Louis de Broglie postulated that because
photons have both wave and particle characteristics, perhaps all forms of matter have both properties
• The de Broglie wavelength of a particle is
h h
p mv
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de Broglie wavelength of a particle
(1) Baseball m=1kg ; v=10 m/s
(2) Electron
34356.63 10
6.63 1010 /
h J sm
p kg m s
2 191. v 100 100 1.6 10
2K E m eV J
34
31 19
10
6.63 10
2 ( . ) 2 9.11 10 100 1.6 10
1.2 10
h h J s
p m K E kg J
m
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wave nature of electrons
L
L = 3λ
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2
(1)2
de Broglie wavelength
v (2)v
v
v
2
where n=1,2,3,2
r n
nr
h h hm
p m
L r p
r m
m r
h n
nhn
������������������������������������������
Quantization of angular momentum
v
( )
eL pr m r
n
��������������
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Mathematics of Bohr’s Assumptions and Results
• Electron’s orbital angular momentum
where n = 1, 2, 3,… -------- (2)• The total energy of the atom is
• The centripetal force
• The total energy
vm r n
22
0
1 1v
2 4
eE K U m
r
2 2
20
------(11
4)
v emr r
2
0
1 1
2 4
eE
r
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Bohr Radius
v vn
m r nmr
2
2
2 2 2
20 0
1 1
4 4
v
n
me em mr rr
r
r
220
2 whe 1,2,3,4...re
.
.
nn
hr
m e
n
The radii of the Bohr orbits are quantized
When n = 1, the orbit has the smallest radius, called
the Bohr radius,
n is called a quantum number
20
0 2 0.0529 nm h
am e
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2
020
2
2
n
hr a
m e
nn
0
2
4
2
0
2 2
1 1
2 4
8
wher 1,2,3,4.....( quantum number )e
nn
n
eE
r
meE
h n
n
Radii and Energy of Orbits
2
13.606nE eV
n
n=1
n=2
n=3
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Wavelength of Emitted Photons
• The wavelengths are found by
• RH = Rydberg constant
0
4
2 2 22
1 1
8i ff i
h c meE E h f
h n n
0
2 23 2
4
22
1 1 1 1 1
8 f i f i
me
h n n n nc
HR
RH = 1.097373 2 x 107 m-1
ni
nf
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Energy Level Diagram
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Balmer Series
– H is red, = 656.3 nm
– H is green, = 486.1 nm
– H is blue, = 434.1 nm
– H is violet, = 410.2 nm
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0
4
2 38
me
h c HR
0
2 23 2
4
22
1 1 1 1 1
8 f i f i
me
h n n n nc
HR
計算 RH 與 h
• 利用 ,先算 RH 。
• 再利用 0
43
28
meh
c
HR
0
43
2:
8
meh
c
-99 -10210 10提示 先整理 或算出HR
計算 h
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Extension to Other Atoms
• Bohr extended his model for hydrogen to other elements in which all but one electron (He+ and Li2+) had been removed
0
1 ; Reduced
4 massn e
n e
M m
M m
21
v2
2eE K U
Z
r
2 2
20
v 1
4
Ze
r r
0
2 22
8E
nh
2 4
n
Z e
charge of nucleusZe =
v r n
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分光儀
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光柵分光儀Diffraction Grating Spectrometer
sin ; 0, 1...d m m
sind
m
瞄準器 望遠鏡
透鏡
物鏡目鏡
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Emission Spectra Examples
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• End of LectureEnd of Lecture
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雙狹縫干涉
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Young’s Double Slit Experiment, Schematic
• Thomas Young first demonstrated interference in light waves from two sources in 1801
• The narrow slits, S1 and S2 act as sources of waves
• The waves emerging from the slits originate from the same wave front and therefore are always in phase
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雙狹縫干涉sin ; 0, 1, 2...d n n 建設性干涉 破壞性干涉
1sin ( ) ; 0, 1, 2...
2d n n
dd
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1 1 0 1
2 2 0 2
1 2
2
0
( , ) sin( )
( , ) sin( )
1
2
E l t E kl t
E l t E kl t
E E E
I c E
合
合 振幅
0
20 0
( , ) sin( )
2 ; 2 ;
1
2
E x t E kx t
k f c fk
I c E
0 1 0 2sin( ) sin( )E E kl wt E kl wt 合
0 2 10 1 1sin( ) sin{ ( )}E E kl wt E kl kt klw l 合
2 1( ) 2
sin 2
sin
2
; 0, 1, 2...
k l l lk
n
n
d n
d n
建設性干涉
0 1
2
max 0 0
2 sin( )
12
2
E E kl wt
I c E
合
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2 1( ) (2 1)
sin (2
; 0, 1,
2 1)
2..1
sin ( )2
.
k l l l n
d
d n
n
n
k
破壞性干涉
0 2 10 1 1sin( ) sin{ ( )}E E kl wt E kl kt klw l 合
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雙狹縫干涉sin ; 0, 1, 2...d n n 建設性干涉 破壞性干涉
1sin ( ) ; 0, 1, 2...
2d n n
dd
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Diffraction Grating
• The condition for maxima is
• The integer n is the order number of the diffraction pattern
• If the incident radiation contains several wavelengths, each wavelength deviates through a specific angle
Bright
sin ; 0, 1, 2...d n n
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• 講義結束
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分光儀
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Diffraction Grating Spectrometer
• The collimated beam is incident on the grating
• The diffracted light leaves the gratings and the telescope is used to view the image
• The wavelength can be determined by measuring the precise angles at which the images of the slit appear for the various orders
sin ; 0, 1, 2...d m m
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Emission Spectra Examples
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雙狹縫干涉
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Young’s Double Slit Experiment, Schematic
• Thomas Young first demonstrated interference in light waves from two sources in 1801
• The narrow slits, S1 and S2 act as sources of waves
• The waves emerging from the slits originate from the same wave front and therefore are always in phase
![Page 41: 氫原子光譜之分析 氫原子的 Bohr 理論模型 利用分光儀分析光譜. Importance of the Hydrogen Atom A structural model can also be used to describe a very small-scale](https://reader033.vdocuments.pub/reader033/viewer/2022061313/5697bfc71a28abf838ca7c75/html5/thumbnails/41.jpg)
1 1 0 1
2 2 0 2
1 2
2
0
( , ) sin( )
( , ) sin( )
1
2
E l t E kl t
E l t E kl t
E E E
I c E
合
合 振幅
0
20 0
( , ) sin( )
2 ; 2 ;
1
2
E x t E kx t
k f c fk
I c E
0 1 0 2sin( ) sin( )E E kl wt E kl wt 合
0 2 10 1 1sin( ) sin{ ( )}E E kl wt E kl kt klw l 合
2 1( ) 2
sin 2
sin
2
; 0, 1, 2...
k l l lk
n
n
d n
d n
建設性干涉
0 1
2
max 0 0
2 sin( )
12
2
E E kl wt
I c E
合
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2 1( ) (2 1)
sin (2
; 0, 1,
2 1)
2..1
sin ( )2
.
k l l l n
d
d n
n
n
k
破壞性干涉
0 2 10 1 1sin( ) sin{ ( )}E E kl wt E kl kt klw l 合
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雙狹縫干涉sin ; 0, 1, 2...d n n 建設性干涉 破壞性干涉
1sin ( ) ; 0, 1, 2...
2d n n
dd
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Diffraction Grating
• The condition for maxima is
• The integer n is the order number of the diffraction pattern
• If the incident radiation contains several wavelengths, each wavelength deviates through a specific angle
Bright
sin ; 0, 1, 2...d n n
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Diffraction Grating Spectrometer
• The collimated beam is incident on the grating
• The diffracted light leaves the gratings and the telescope is used to view the image
• The wavelength can be determined by measuring the precise angles at which the images of the slit appear for the various orders
sin ; 0, 1, 2...d m m
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Emission Spectra Examples
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Emission Spectra Examples
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• 講義結束