Download Class Summary QUANTUM THEORY

QUANTUM THEORY
General Chemistry I (2012) Lecture by B. H. Hong
Class Summary
• Studies of black-body radiation led to Plank’s hypothesis of the
quantization of electromagnetic radiation.
• The photoelectric effect provides evidence of the particle nature
of electromagnetic radiation.
• Electrons (an matter in general) have both wavelike and
particlelike properties.
• The location and momentum of a particle are complementary;
that is, the location and the momentum cannot both be known
simultaneously with arbitrary precision.
• When the wave equation is solved subject to the appropriate
boundary conditions, it is found that the particle can possess
only certain discrete energies.
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THE HYDROGEN ATOM
General Chemistry I (2012) Lecture by B. H. Hong
1.7 Wavefunctions and Energy Levels
1) An electron is described by a wavefunction governed
by the Schrödinger equation.
2) The nuclear model for an atom
⇒ To explain the ladder of energy levels suggested by
spectroscopy
THE HYDROGEN ATOM
General Chemistry I (2012) Lecture by B. H. Hong
1.8 The Principal Quantum Number
Constraints (boundary conditions) in solving the Schrödinger equation
⇒ Quantization of energies, discrete energy levels corresponding to a
set of quantum numbers
To solve the Schrödinger equation for ψ and energy levels of
an electron in a hydrogen atom, the potential energy term V
in the hamiltonian H is needed.
The solution by Schrödinger (1927)
THE HYDROGEN ATOM
General Chemistry I (2012) Lecture by B. H. Hong
1.8 The Principal Quantum Number
For other one-electron atoms such as He+, ···,
C5+ with atomic number Z,
Z2 dependence;
1)
2)
nucleus of charge Ze
Z times closer distance to the nucleus
Energy levels of a hydrogen atom
THE HYDROGEN ATOM
General Chemistry I (2012) Lecture by B. H. Hong
1.9 Atomic Orbitals
Atomic orbitals; wavefunctions of electrons in atoms
(orbital; wave-like orbit)
THE HYDROGEN ATOM
General Chemistry I (2012) Lecture by B. H. Hong
1.9 Atomic Orbitals
Tab 1.2
(needs
correctio
ns)
THE HYDROGEN ATOM
General Chemistry I (2012) Lecture by B. H. Hong
1.9 Atomic Orbitals
THE HYDROGEN ATOM
General Chemistry I (2012) Lecture by B. H. Hong
1.9 Atomic Orbitals
Fig 1.36
Detailed solutions of the Schrödinger
equation for a hydrogen atom require three
quantum numbers.
Three dimensional boundary
conditions
Principal quantum number (n);
-> energy & size
THE HYDROGEN ATOM
General Chemistry I (2012) Lecture by B. H. Hong
1.9 Atomic Orbitals
Shell; atomic orbitals of the same principal
quantum number n
As n increases, the energy (–1/n2) and the nucleus-electron
distance (n2) increase.
All the orbitals of a given shell of hydrogen-like atoms have the
same energy; degenerate
Orbital angular momentum quantum
number (l); shape
Angular momentum;
THE HYDROGEN ATOM
General Chemistry I (2012) Lecture by B. H. Hong
1.9 Atomic Orbitals
l = 0, 1, 2, ···, n – 1; n different values of l, n subshells
Subshell; group of orbitals with the same value of l
s (sharp); l = 0,
d (diffuse); l = 2,
p (principal); l = 1,
f (fundamental); l = 3,
g, h, ···
Orbital angular momentum of an electron in a subshell
=
; measure of the rate at which the electron
circulates around the nucleus
THE HYDROGEN ATOM
General Chemistry I (2012) Lecture by B. H. Hong
1.9 Atomic Orbitals
Magnetic quantum number (ml); orientation
ml = l, l – 1, ···, –l ;
2l + 1 different values of ml
3 p-orbitals; ml = –1, 0, 1 with l = 1
5 d-orbitals; ml = –2, –1, 0, +1, +2 with l = 2
The magnetic quantum number
determines the energy shift of
an atomic orbital due to an
external magnetic field
(Zeeman effect).
THE HYDROGEN ATOM
General Chemistry I (2012) Lecture by B. H. Hong
1.9 Atomic Orbitals
Hierarchy of shells, subshells, and orbitals
Tab 1.3
Fig 1.30
s-Orbitals; l = 0, no angular momentum
Only one orbital in each subshell
Spherically symmetric;
Non-zero probability of being found at the nucleus; R2(0) ≠ 0
n – 1 radial nodes
Figs 1.33
Figs 1.34
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THE HYDROGEN ATOM
General Chemistry I (2012) Lecture by B. H. Hong
1.9 Atomic Orbitals
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THE HYDROGEN ATOM
General Chemistry I (2012) Lecture by B. H. Hong
1.9 Atomic Orbitals
Fig 1.31
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THE HYDROGEN ATOM
General Chemistry I (2012) Lecture by B. H. Hong
1.9 Atomic Orbitals
Radial distribution function;
probability density for finding an electron at a given radius summed
over all directions
P(r)δr; probability of finding an electron anywhere in a thin shell of
radius r and thickness δr
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THE HYDROGEN ATOM
General Chemistry I (2012) Lecture by B. H. Hong
1.9 Atomic Orbitals
For any s-orbitals of hydrogen-like
atoms,
Thus,
Fig 1.32
P(r) of the 1s-orbital of hydrogen reaches a maximum at r = a0,
the Bohr radius.
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THE HYDROGEN ATOM
General Chemistry I (2012) Lecture by B. H. Hong
1.9 Atomic Orbitals
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THE HYDROGEN ATOM
General Chemistry I (2012) Lecture by B. H. Hong
1.9 Atomic Orbitals
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THE HYDROGEN ATOM
General Chemistry I (2012) Lecture by B. H. Hong
1.9 Atomic Orbitals
Boundary surface;
a smooth surface that encloses most (typically 90%) of the
electron cloud
All s-orbitals have spherical boundary surfaces
(with internal structures such as nodes).
Figs 1.33, 1.35
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THE HYDROGEN ATOM
General Chemistry I (2012) Lecture by B. H. Hong
1.9 Atomic Orbitals
p-Orbitals;
3 orbitals in each subshell;
⇒ The Boundary surface of
each p orbital has two lobes
and one nodal plane.
⇒ A p-electron will never be
found at the nodal plane nor
at the nucleus.
Figs 1.36
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THE HYDROGEN ATOM
General Chemistry I (2012) Lecture by B. H. Hong
1.9 Atomic Orbitals
d-Orbitals;
5 orbitals in each subshell;
Figs 1.37
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THE HYDROGEN ATOM
General Chemistry I (2012) Lecture by B. H. Hong
1.9 Atomic Orbitals
f-Orbitals;
Figs 1.38
Total number of orbitals in a shell = n2
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THE HYDROGEN ATOM
General Chemistry I (2012) Lecture by B. H. Hong
1.9 Atomic Orbitals
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THE HYDROGEN ATOM
General Chemistry I (2012) Lecture by B. H. Hong
1.10 Electron Spin
Box 1.1 Stern-Gerlach Experiment (1922, 1943)
Unexpected splitting of an Hg atomic beam
under an inhomogeneous magnetic field
⇒ Quantum mechanical electron spin
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THE HYDROGEN ATOM
General Chemistry I (2012) Lecture by B. H. Hong
1.10 Electron Spin
One of the biggest impacts on modern physics
Hyperfine structures; Goudsmit and Uhlenbeck (1925)
Nuclear magnetic resonance; Rabi (1937, 1944)
Separated oscillatory fields; Ramsey (1989)
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THE HYDROGEN ATOM
General Chemistry I (2012) Lecture by B. H. Hong
1.10 Electron Spin
Spin angular momentum quantum number (s);
Spin magnetic quantum number (ms);
For an electron; s = ½
Fig 1.40
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THE HYDROGEN ATOM
General Chemistry I (2012) Lecture by B. H. Hong
1.11 The Electronic Structure of Hydrogen
The state of an electron in a hydrogen atom is defined by a set
of the four quantum numbers:
{n, l, ml, ms}
Ground state; 1s-electron,
{n = 1, l = 0, ml = 0, ms = ½ or –½}
Excited states; n > 1, higher energy, greater distance between the
electron and the nucleus
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THE HYDROGEN ATOM
General Chemistry I (2012) Lecture by B. H. Hong
Class Summary
• The energy levels of a hydrogen atom are defined by the
principal quantum number, n=1,2,…, and form a converging
series.
• The location of an electron in n atom is described by a
wavefunction known as an atomic orbital; atomic orbitals are
designated by the quantum numbers n, l, and ml and fall into
shells and subshells.
• An electron has the property of spin; the spin is described by the
quantum number ms , which may have one of two values.
• The state of an electron in a hydrogen atom is defined by the
four quantum numbers n, l, ml , and ms ; as the value of n
increases, the size of the atom increases.
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