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Chapter 12
Quantum Mechanics
and Atomic Theory
Classical Physics
Classical mechanicsMaxwell’s theory of electricity,
magnetism and electro-magnetic radiation
ThermodynamicsKinetic theory
Eletromagnetic Radiation
Class radiation
Blackbody Radiation
Ultraviolet Catastrophe( 紫外線崩潰 )
The classical theory of matter, which
assumes that matter can absorb or emit
any quantity of energy, predicts a
radiation profile that has no maximum
and goes to infinite intensity at very
short wavelengths.
古典理論解釋失敗
1900 年, J.W.Rayleigh , J.H.Jeans 根據古典電動力學和統計物理理論,得出一黑體輻射公式,即 Rayleigh-Jeans law 。此公式只在低頻部分與實驗曲線比較符合,高頻部分是發散的,與實驗明顯不符,即ultraviolet catastrophe
Tkc Bv 3
38
長波長 短波長
By combining the formulae of Wien and Rayleigh, Planck announced in October 1900 a formula now known as Planck's radiation formula.
Max Planck
1858~1947 Planck initiated the
study of quantum mechanics when he announced in 1900 his theoretical research into the radiation and absorption of heat/light by a black body.
Max Planck’s Theory
The Nobel Prize in Physics 1918
1
2)(
3
3
kT
hv
e
v
c
hvR
h: Planck’s constantk: Boltzmann’s constantC: speed of lightv: frequency of light
Planck’s Impacts
In classical physics, energy is a continuous variable.
Planck defined the amount of energy, a quantum of energy, ∆E=nh
In quantum physics, the energy of a system is quantized.
Albert Einstein 1879~1955 Einstein contributed more
than any other scientist to the modern vision of physical reality. His special and general theories of relativity are still regarded as the most satisfactory model of the large-scale universe that we have.
Photoelectric Effect
Albert Einstein, The Nobel Prize in Physics 1921
KEelectron=1/2mv2=hv-hv0
hv: energy of incident photon
hv0: energy required to remove electron from metal’s surface
The wave nature of electrons
Louis de Broglie,
The Nobel Prize in
Physics 1929
光具有物質與波雙重性質λ=h/mv
Light waves
Werner Heisenberg
1901~1976 Werner
Heisenberg did important work in Quantum Mechanics as well as nuclear physics.
Uncertainty Principle
Werner Karl Heisenberg
The Nobel Prize in Physics 1932
任何一個粒子無法將位置與動量同時很精確的量測出來
)2
( 2
hpx
The Atomic Spectrum of Hydrogen
The Bohr Model
Niel Bohr The Nobel Prize in Physics 1922
218
22.178 10 ( )
:
:
ZE J
nn integer
Z atomic number
The electron in a hydrogen atom moves around the nucleus only in certain allowed circular orbits.
量子發展如同瞎子摸象Max Planck -能量不連續Albert Einstein -能階概念Louis de Broglie -粒子波動Werner Heisenberg -粒子運動測不準
Niel Bohr -光波不連續
Quantum Mechanics
By the mid-1920s, Werner Karl Heisenberg, Louis de Broglie and Erwin SchrÖdinger developed the wave mechanics or more commonly, quantum mechanics.
Tunnel Effect
Quantum effect in biological systems
Rudolph A. Marcus was awarded the 1992 Nobel Prize in Chemistry.
Marcus theory Electron transfer reactions
SchrÖdinger Equation
nvalue)nergy(eigeE: total e
ran operato:HamiltoniH
n)genfunctioorbital(eiction for ψ:wave fun
EψψH
equationeigenvalue
ˆ
ˆ
Particle in a box
m
kE
mEk
kxAmE
kxAkdx
kxAd
dx
xψd
kxAxψSuppose
hxψ
mE
dx
xψd
ψEψdx
d
mdx
d
mH
Vl energy)V(potentiaenergy)T(kinetic E
EψψH
2
2
)sin(2
)sin()sin()(
sin)(
)2
( )(2)(
)()(22
ˆ
)0(
ˆ
22
2
2
2
2
2
2
2
2
22
2
2
22
2
22
Boundary Conditions
The particle cannot be outside the box-it is bound inside the box.
In a given state the total probability of finding the particle in the box must be 1.
The wave function must be continuous.
1)(sin)(
1. bemust stategiven ain y probabilit
totalThe box. ain particle a finding ofy probabilit
relative themeansfunction wave theof square The
)sin()(
)sin()sin()(0)()0(
)sin()(
and constant thedefine
0
22
0
2
dxxL
nAdxxψ
xL
nAxψ
L
nknkL
nAkLALψLψψ
kxAxψ
Ak
LL
)sin(
2)(
)2
( 82
)(
2
1
2
1)
2sin(
4
1
2
1)(sin
2sin4
1
2
1sin
1
)(sin
2
2222
20
00
2
2
20
2
xL
n
Lxψ
h
mL
hn
mL
n
E
LA
ALx
L
n
L
nxxdx
L
n
cxcxcxdx
Adxx
L
n
n
LLL
L
Three energy levels
Three Dimension Box
z)L
πn(y)
L
πn(x)
L
πn(
LLLz)yψ(x
) L
n
L
n
L
n (
m
hE
z
z
y
y
x
x
zyx
z
z
y
y
x
x
sinsinsin8
,,
8 2
2
2
2
2
22
Degenercy
221 212 122
211 121 112
111
E Lx=Ly=Lz
The spherical polar coordinate system
The Wave Function for the Hydrogen Atom
)(10178.2)8
(
)(ˆ
sin
1cot
112
2ˆ
2ˆˆˆ
ˆ
2
218
20
4
2
2
2
2
2222
2
22
22
22
2
n
ZJ
h
me
n
ZE
r
ZerVV
rrrrrrT
r
ZeVTH
EΨΨH
)R(r)Θ(r)Θ(Ψ
n
The Physical Meaning of a Wave Function
The square of the function evaluated at a particular point in space indicates the probability of finding an electron near that point.
Ψ2:probability distribution
dv: small volume element2
1 1 1 12
2 2 2 2
[ ( , , )]
[ ( , , )]
r dv N
r dv N
The probability
distribution for the
hydrogen 1s orbital
Calculate the probability
at points along a line
drawn outward in any
direction from nucleus.
number of nodes=n-1
nodes
The radial probability
distribution for the
hydrogen 1s orbital
in spherical shell
Bohr radius: 0.529Å
Denoted by a00.529Å
)4()
34
()( 22
3
22 rRdr
rdR
dr
dVR
Relative Orbital Size
The normally accepted arbitrary definition of the size of the hydrogen 1s orbital is the radius of the sphere that encloses 90% of the total electron probability.
For H(1s), r(1s)=2.6a0=1.4Å
Quantum Numbers
n: the principal quantum number
l: the angular momentum quantum
number
Ml: the magnetic quantum number
Quantum Numbers principal quantum number (n)
Have integral values (1,2,3…) It is related to the size and energy of orbital. As n increases, the orbital becomes larger
and the electron spends more time farther from the nucleus
An increase in n also means higher energy
Quantum Numbers angular momentum quantum number (l) Have integral values from 0 to n-1. Determines the shapes of the atomic orbital.
Value Letter Used
0 s
1 p
2 d
3 f
4 g
Quantum Numbers magnetic quantum number (ml)
Have integral values between l and –l, including zero.
Relates to the orientation in space of angular momentum associated with the orbital.
2Px
n value
l Value
Orientation in space
Electron Spin and Pauli Principle
electron spin quantum number (ms): +1/2 and -1/2
Pauli Principle: In a given atom, no two electrons can have the same set of four quantum numbers (n, l, ml, ms)
Polyelectronic Atoms
the kinetic energy of the electrons as they move around the nucleus
the potential energy of attraction between the nucleus and the electrons
the potential energy of repulsion between the two electrons
First ionization energy: 2372 kJ/molSecond ionization energy: 5248 kJ/mol
Hydrogen Like Approximation Zeff=Zactual-(effect of electron repulsions) 1<Zeff<2 Substituting Zeff in place of Z=1 in the hydrogen wave mecha
nical equations
2+ e-
e-
Zeff+
e-
Actual He atom Hypothetical He atom
Valence electrons and Core electrons
Valence electrons: the electrons in the outermost principal quantum level of an atom.
Core electrons: inner electrons
Penetration effect
Core electrons
Na:1s22s22p63s1
K: 1s22s22p63s23p64s1
Aufbau principle
The physical and chemical properties of elements is determined by the atomic structure.
The atomic structure is, in turn, determined by the electrons and which shells, subshells and orbitals they reside in.
The rules of placing electrons within shells is known as the Aufbau principle.
As protons are added one by one to the nucleus to build up the elements, electrons are similarly added to these atomic orbitals.
Ionization EnergyX(g) →X+(g)+e-
The ionization energy for a particular electron in an atom is a source of information about the energy of the orbital it occupies in the atom.
Resulting ion will not reorganize in response to the removal of an electron.
The ionization energies do provide information that is quite useful in testing the orbital model of the atom.
Ionization of Al
Al(g) →Al+(g)+e- I1=580 kJ/mol
Al+(g) →Al+2(g)+e- I2=1815 kJ/mol
Al+2(g) →Al+3 +e- I3=2740 kJ/mol
Al+3(g) →Al+4 +e- I4=11600 kJ/mol
(1) 1s22s22p63s23p1 →1s22s22p63s2
(2) 1s22s22p63s2 → 1s22s22p63s1
(3) 1s22s22p63s1→1s22s22p6
(4) 1s22s22p6→ 1s22s22p5
The values of first ionization energy for the elements in the first five periods.
○
○
As n increases, the size of the orbital increases, and the electron is easier to remove.
The electron affinity for atoms among the first 20 elements that form stable, isolated X- ions.