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General Chemistry
M. R. Naimi-Jamal
Faculty of Chemistry
Iran University of Science & Technology
‫فصل ششم‪:‬‬
‫ساختار الکترونی اتم ها‬
Contents
6-1
6-2
6-3
6-4
6-5
6-6
6-7
Electromagnetic Radiation
Atomic Spectra
Quantum Theory
The Bohr Atom
Two Ideas Leading to a New Quantum Mechanics
Wave Mechanics
Quantum Numbers and Electron Orbitals
Contents
6-8
6-9
Quantum Numbers
Interpreting and Representing Orbitals of the
Hydrogen Atom
6-9 Electron Spin
6-10 Multi-electron Atoms
6-11 Electron Configurations
6-12 Electron Configurations and the Periodic Table
The Wave Nature of Light
• Electromagnetic waves include radio-, ultraviolet-,
infra-, and X waves etc.
• Electromagnetic waves require no medium.
• Electromagnetic radiation is characterized by its
wavelength, frequency, and amplitude.
Electromagnetic Radiation
• Electric and magnetic fields
propagate as waves through
empty space or through a
medium.
• A wave transmits energy.
An Electromagnetic Wave
… the amplitude of the “wiggle”
simply indicates field strength.
Low 
High 
Wavelength and Frequency
• Wavelength () is the distance between any
two identical points in consecutive cycles.
• Frequency (v) of a wave is the number of
cycles of the wave that pass through a point in a
unit of time. Unit = waves/s or s–1 (hertz).
Wavelength and Frequency
The relationship between wavelength and
frequency:
c = v
or
c
v=—

where c is the speed of light (3.00 × 108 m/s)
Example:
Calculate the frequency of an X-ray that
has a wavelength of 8.21 nm.
The Electromagnetic Spectrum
UV, X rays are shorter
wavelength, higher
frequency radiation.
Communications involve
longer wavelength, lower
frequency radiation.
Visible light is only
a tiny portion of
the spectrum.
Example: A Conceptual Example
Which light has the higher frequency: the bright
red brake light of an automobile or the faint green
light of a distant traffic signal?
A Continuous Spectrum
White light from a
lamp contains all
wavelengths of
visible light.
When that light is passed
through a prism, the different
wavelengths are separated.
We see a spectrum of all rainbow
colors from red to violet – a
continuous spectrum.
Atomic Spectra
A Line Spectrum
Light from an
electrical discharge
through a gaseous
element (e.g., neon
light, hydrogen lamp)
does not contain all
wavelengths.
The spectrum is
discontinuous; there
are big gaps.
We see a pattern of lines,
multiple images of the slit.
This pattern is called a
line spectrum. (duh!)
Line Spectra of Some Elements
The line emission
spectrum of an
element is a
“fingerprint” for that
element, and can be
used to identify the
element!
How might you tell if
an ore sample
contained mercury?
Cadmium?
Line spectra were a
problem; they
couldn’t be explained
using classical physics
…
The Photoelectric Effect
Light striking a
photoemissive cathode causes
ejection of electrons.
Ejected electrons reach the
anode, and the result is …
… current flow through an
external circuit.
But not “any old” light will cause ejection of electrons …
The Photoelectric Effect (cont’d)
Each photoemissive
material has a
characteristic threshold
frequency of light.
When light that is above
the threshold frequency
strikes the photoemissive
material, electrons are
ejected and current flows.
Light of low frequency
does not cause current
flow … at all.
As with line spectra, the
photoelectric effect cannot be
explained by classical physics.
Quantum Theory
Max Planck, 1900:
Light energy, like matter, is discontinuous.
E = h
h = 6.6262 x 10 -34 J.s
The Photoelectric Effect
• Albert Einstein won the 1921 Nobel Prize in Physics for
explaining the photoelectric effect.
• He applied Planck’s quantum theory: electromagnetic
energy occurs in little “packets” he called photons.
Energy of a photon (E) = hv
• The photoelectric effect arises when photons of light
striking a surface transfer their energy to surface electrons.
• The energized electrons can overcome their attraction for
the nucleus and escape from the surface …
• … but an electron can escape only if the photon provides
enough energy.
The Photoelectric Effect Explained
The electrons in a
photoemissive material
need a certain minimum
energy to be ejected.
Short wavelength (high
frequency, high energy)
photons have enough
energy per photon to
eject an electron.
A long wavelength—low
frequency—photon
doesn’t have enough
energy to eject an electron.
Analogy to the Photoelectric Effect
• Imagine a car stuck in a ditch; it takes a certain amount of
“push” to “eject” the car from the ditch.
• Suppose you push ten times, with a small amount of force
each time. Will that get the car out of the ditch?
• Likewise, ten photons, or a thousand, each with too-little
energy, will not eject an electron.
• Suppose you push with more than the required energy; the car
will leave, with that excess energy as kinetic energy.
• What happens when a photon of greater than the required
energy strikes a photoemissive material? An electron is
ejected—but with _____ _____ as ______ _____.
Example:
Calculate the energy, in joules, of a photon of violet light
that has a frequency of 6.15 × 1014 s–1.
Example:
A laser produces red light of wavelength 632.8 nm.
Calculate the energy, in kilojoules, of 1 mol of photons
of this red light.
The Bohr Atom
-B
E= 2
n
B = 2.179 x 10-18 J
‫مدل اتمی بور‬
‫فرضیه های بور‬
‫‪ -1‬الکترون مجاز است فقط بر روی سطوح ساکن و معینی به‬
‫دور هسته حرکت کند‪.‬‬
‫‪ -2‬اگر الکترون انرژی دریافت کند‪ ،‬به تراز باالتر رفته و‬
‫سپس با از دست دادن انرژی به ترازهای پایین تر سقوط‬
‫می کند و اختالف انرژی دو تراز را به صورت پرتو‬
‫منتشر می کند‪(.‬استفاده از نظریه پالنک)‬
‫‪E2  E1  h‬‬
‫‪ -3‬مدار حرکت الکترون به دور هسته دایره هایی به شعاع ‪r‬‬
‫است‪.‬‬
‫‪ -4‬ترازهای ساکن ترازهایی است که در آنها اندازه حرکت‬
‫زاویه ای الکترون مضرب صحیحی از ‪ h/2π‬باشد‪.‬‬
‫‪2‬‬
‫‪mvr  nh‬‬
v2
q1q2
m k 2
r
r
e2
v
Ze.e
m  2
r
r
2nucleus
Ze
r
(1)
2
mv
(1), ( 2) 
r
Ze 2
m( nh ) 2
2mr
r
n 2h 2
4 2 m Ze2
‫محاسبات مدل اتمی بور‬
‫الف) محاسبه شعاع‬
nh
mvr 
2
nh
v
2mr
(2)
for H at om: Z  1, n  1
r  0.53A Bohr Radius or a 0
‫محاسبات مدل اتمی بور‬
‫ب) محاسبه انرژی‬
Etot  Ek  E p
q1q2
2
1
Etot  2 m v  k
r
2
Ze
Ze
1
Etot  2

r
r
r
2
(1)
Ze
1
 2
r
Ze
mv 
r
2
E
2
2
2 2 m Z 2e 4
n 2h 2
for H atom: Z  1, n  1
2 2
n h
4 m Ze
2
2
E  313.5 kcal mol
 1.6  1019 eV
Bohr’s Equation …
• … allows us to find the energy change (Elevel) that
accompanies the transition of an electron from one energy
level to another.
Initial energy level:
Final energy level:
–B
–B
Ei = ——
Ef = ——
2
ni
nf2
• To find the energy difference, just subtract:
–B
–B
1
1
Elevel = —— – —— = B — – —
nf2
ni2
ni2
nf2
• Together, all the photons having this energy (Elevel)
produce one spectral line.
Example:
Calculate the energy of an electron in the second
energy level of a hydrogen atom.
Energy-Level Diagram
-B
-B
– 2
ΔE = Ef – Ei =
2
nf
ni
1
1
–
= B( 2
) = h = hc/λ
2
ni
nf
Example:
Calculate the energy change, in joules, that occurs
when an electron falls from the ni = 5 to the nf = 3
energy level in a hydrogen atom.
Example:
Calculate the frequency of the radiation released by the
transition of an electron in a hydrogen atom from the n
= 5 level to the n = 3 level, as in above example.
‫تمرین‬
‫طول موج فوتونی را حساب کنید که از اتم هیدروژن نشر شده‪،‬‬
‫الکترون آن به تراز ‪ n=1‬سقوط کرده و در سری لیمان مشاهده‬
‫شده است‪.‬‬
‫‪ 3.4eV  (13.6eV)  10.2eV‬‬
‫‪hc‬‬
‫‪‬‬
‫‪Ephoton ‬‬
‫‪hc‬‬
‫‪1240‬‬
‫‪‬‬
‫‪‬‬
‫‪ 124nm‬‬
‫‪10.2eV 10.2‬‬
‫‪n=3‬‬
‫‪n=2‬‬
‫‪n=1‬‬
‫‪h  E2  E1‬‬
‫‪E2= -3.4 eV‬‬
‫‪E1= -13.6 eV‬‬
The Bohr Model of Hydrogen
When excited, the
electron is in a higher
energy level.
Excitation: The atom
absorbs energy that is
exactly equal to the
difference between two
energy levels.
Each circle represents an
allowed energy level for the
electron. The electron may be
thought of as orbiting at a fixed
distance from the nucleus.
Emission: The atom
gives off energy—as
a photon.
Upon emission, the
electron drops to a
lower energy level.
Line Spectra Arise Because …
Transition from
n = 3 to n = 2.
Transition from
n = 4 to n = 2.
• … each electronic
energy level in an
atom is quantized.
• Since the levels are
quantized, changes
between levels must
also be quantized.
• A specific change
thus represents one
specific energy, one
specific frequency,
and therefore one
specific wavelength.
Example:
A Conceptual Example
Without doing detailed calculations,
determine which of the four electron
transitions shown produces the
shortest-wavelength line in the
hydrogen emission spectrum.
‫ایرادهای مدل اتمی بور‬
‫‪-RH‬‬
‫‪E= 2‬‬
‫‪n‬‬
‫‪RH = 2.179 x 10-18 J‬‬
‫‪ .1‬اگر الکترون را با ماهیت دوگانه آن یعنی ماهیت موجی – ذره ای‬
‫در نظر بگیریم‪ ،‬موج اگر بر روی سطوح ساکن دایره ای بچرخد‬
‫پس از مدتی انرژی از دست داده و بر روی هسته سقوط می کند‪.‬‬
‫‪ .2‬مدل بور توضیحی برای پیوندهای شیمیایی ندارد‪.‬‬
‫‪ .3‬مدل بور فقط برای اتم هیدروژن ‪ Z=1‬مصداق دارد و طیف‬
‫عنصرهای دیگر را تفسیر نمی کند‪.‬‬
Two Ideas Leading to a New Quantum
Mechanics
• Wave-Particle Duality.
– Einstein suggested particle-like properties of
light could explain the photoelectric effect.
– But diffraction patterns suggest photons are
wave-like.
• deBroglie, 1924
– Small particles of matter may at times display
wavelike properties.
‫دوگانگی موج ‪ -‬ذره‬
‫لویی دوبروی ‪1892-1987‬‬
‫‪h‬‬
‫‪‬‬
‫‪mv‬‬
‫‪h‬‬
‫‪‬‬
‫‪mv ‬‬
‫رابطه دوبروی برای فوتونها ثابت شد ولی دوبروی نشان داد که‬
‫این رابطه برای هر ذره ای که مومنتوم داشته باشد‪ ،‬صادق است‪.‬‬
‫مثل هسته‪ ،‬الکترون‪ ،‬اتم و حتی یک توپ فوتبال!!!‬
‫‪„Should be able to see‬‬
‫‪interference and diffraction‬‬
‫”!!‪for material particles‬‬
Example 7.9
Calculate the wavelength, in meters and nanometers,
of an electron moving at a speed of 2.74 × 106 m/s.
The mass of an electron is 9.11 × 10–31 kg, and 1 J = 1
kg m2 s–2.
Uh oh …
• de Broglie just messed up the Bohr model of the
atom.
• Bad: An electron can’t orbit at a “fixed distance” if
the electron is a wave.
– An ocean wave doesn’t have an exact location—neither
can an electron wave.
• Worse: We can’t even talk about “where the
electron is” if the electron is a wave.
• Worst: The wavelength of a moving electron is
roughly the size of an atom! How do we describe
an electron that’s too big to be “in” the atom??
The Uncertainty
Principle
Werner Heisenberg: We can’t
know exactly where a moving
particle is AND exactly how
fast it is moving at the same
time.
The photon that will
enter the microscope,
so that we might
“see” the electron …
… has enough momentum
to deflect the electron.
The act of measurement
has interfered with the
electron’s motion.
‫‪The Uncertainty Principle‬‬
‫الکترون ساکن‬
‫فوتون ورودی با اندازه‬
‫حرکت ‪ P‬و طول موج ‪l‬‬
‫الکترون عقب رانده شده که‬
‫مقداری از اندازه حرکت و‬
‫انرژی فوتون را حمل می کند‬
‫بیان ریاضی اصل عدم قطعیت‬
Δx ~ λ
Δmv ~ p
Δmv.Δx ~ p. λ
Δmv.Δx ≥ h/4
‫تمرین‬
‫اگر بخواهیم بیشترین خطای ما در تعیین مکان الکترون تنها‬
‫‪ 0.05 Å‬باشد‪ ،‬در تعیین سرعت آن دچار چقدر خطا خواهیم‬
‫شد؟‬
The Uncertainty Principle
• A wave function doesn’t tell us where the electron is.
The uncertainty principle tells us that we can’t know
where the electron is.
• However, the square of a wave function gives the
probability of finding an electron at a given location
in an atom.
• Analogy: We can’t tell where a single leaf from a
tree will fall. But (by viewing all the leaves under
the tree) we can describe where a leaf is most likely
to fall.
Wave Functions
• Erwin Schrödinger: We can describe the
electron mathematically, using quantum
mechanics (wave mechanics).
• Schrödinger developed a wave equation to
describe the hydrogen atom.
• An acceptable solution to Schrödinger’s
wave equation is called a wave function.
• A wave function represents an energy state
of the atom.
9-6 Wave Mechanics
• Standing waves.
– Nodes do not undergo displacement.
2L
λ=
, n = 1, 2, 3…
n
Wave Functions
• ψ, psi, the wave function.
– Should correspond to a
standing wave within the
boundary of the system being
described.
• Particle in a box.
ψ 
2
 n x 
sin 

L
 L 
Probability of Finding an Electron
Wave Functions for Hydrogen
• Schrödinger, 1927
Eψ = H ψ
– H (x,y,z) or H (r,θ,φ)
ψ(r,θ,φ) = R(r) Y(θ,φ)
R(r) is the radial wave function.
Y(θ,φ) is the angular wave function.
9-8 Interpreting and Representing the
Orbitals of the Hydrogen Atom.
Quantum Numbers and Atomic Orbitals
• The wave functions for the hydrogen atom contain
three parameters called quantum numbers that
must have specific integral values.
• A wave function with a given set of these three
quantum numbers is called an atomic orbital.
• These orbitals allow us to visualize the region in
which the electron “spends its time.”
• When values are assigned to the three quantum
numbers, a specific atomic orbital has been
defined.
Quantum Numbers: n
The principal quantum number (n):
• Is independent of the other two quantum numbers.
• Can only be a positive integer (n = 1, 2, 3, 4, …)
• The size of an orbital and its electron energy
depend on the value of n.
• Orbitals with the same value of n are said to be in
the same principal shell.
Quantum Numbers: l
The orbital angular momentum quantum number (l):
• Determines the shape of the orbital.
• Can have positive integral values from 0, 1, 2, … (n – 1)
• Orbitals having the same values of n and of l are said to be
in the same subshell.
Value of l
0
1
2
3
Subshell
s
p
d
f
• Each orbital designation represents a different region of
space and a different shape.
Quantum Numbers: ml
The magnetic quantum number (ml):
• Determines the orientation in space of the
orbitals of any given type in a subshell.
• Can be any integer from –l to +l
• The number of possible values for ml is
(2l + 1), and this determines the number of
orbitals in a subshell.
Example:
Considering the limitations on values for the various
quantum numbers, state whether an electron can be
described by each of the following sets. If a set is not
possible, state why not.
(a) n = 2, l = 1, ml = –1
(c) n = 7, l = 3, ml = +3
(b) n = 1, l = 1, ml = +1
(d) n = 3, l = 1, ml = –3
Example:
Consider the relationship among quantum numbers and
orbitals, subshells, and principal shells to answer the
following. (a) How many orbitals are there in the 4d
subshell? (b) What is the first principal shell in which f
orbitals can be found? (c) Can an atom have a 2d
subshell? (d) Can a hydrogen atom have a 3p subshell?
The 1s Orbital
• The 1s orbital (n = 1, l = 0, ml = 0) has spherical
symmetry.
• An electron in this orbital spends most of its time near the
nucleus.
Spherical symmetry;
probability of finding
the electron is the same
in each direction.
The electron
cloud doesn’t
“end” here …
… the electron just
spends very little
time farther out.
Analogy to the 1s Orbital
Highest “electron
density” near the
center …
… but the electron
density never drops to
zero; it just decreases
with distance.
The 2s Orbital
• The 2s orbital has two concentric, spherical regions of
high electron probability.
• The region near the nucleus is separated from the outer
region by a node—a region (a spherical shell in this case)
in which the electron probability is zero.
s orbitals
p Orbitals
p Orbitals
The Five d Orbitals
Five values of ml (–2,
–1, 0, 1, 2) gives five d
orbitals in the d
subshell.
Electron Spin: A Fourth Quantum Number
The Stern-Gerlach Experiment Demonstrates
Electron Spin
The magnet
splits the beam.
Silver has 47 electrons
(odd number). On
average, 23 electrons will
have one spin, 24 will
have the opposite spin.
These silver atoms each
have 24 +½-spin electrons
and 23 –½-spin electrons.
These silver atoms each
have 23 +½-spin electrons
and 24 –½-spin electrons.
Multi-electron Atoms
• Schrödinger equation was for only one e-.
• Electron-electron repulsion in multielectron atoms.
• Hydrogen-like orbitals (by approximation).
Electron Configurations
• Aufbau process.
– Build up and minimize energy.
• Pauli exclusion principle.
– No two electrons can have all four quantum
numbers alike.
• Hund’s rule.
– Degenerate orbitals are occupied singly first.
Atomic Subshell Energies and Electron
Assignments
• We need to predict the distribution of
electrons in atoms with more than one
electron
• Use a building up principle to assign
electrons to shells of higher and higher
energy
• Assign electrons so the total energy of the
atom is as low as possible
Orbital Energies
Aufbau Process and Hunds Rule
C
Filling p Orbitals
Order of Subshell Energies and Electron
Assignments
• Subshell energies for multielectron atoms
depend on both n and l
• Subshells n = 3, 3s<3p<3d energy
– Electrons are assigned to subshells in order of
increasing “n+l” value
– For two subshells with the same “n+l” value,
electrons are assigned first to the subshell of
lower n
Orbital Filling
Electron Configurations of the Transition
Elements
• Must use d or f subshells
– If you fill a d subshell it is a transition element
– Filling 4f is lanthanides
– Filling 5f is actinides
Electron Configurations of the Main Group
Elements
• When electrons are assigned to p, d, or f orbitals,
each successive electron is assigned to a different
orbital of the subshell, and has the same spin as
the last one
– Pattern continues until the subshell is half full
– Hund’s rule – most stable arrangement of electrons is
that with the maximum number of unpaired electrons,
all with the same spin direction
Electron Configurations of Ions
• To form a cation, remove an electron
– Electron is removed from the highest value of n
– If more than one electron must be removed and
they have the same value of n, the electrons are
removed starting with the highest value of l
• To form an anion, add an electron
Chapter 6 Questions
8, 14, 17, 22, 31,
34, 39, 41, 45, 50,
61, 65, 66
Bohr’s Hydrogen Atom
• Niels Bohr followed Planck’s and Einstein’s lead by
proposing that electron energy (En) was quantized.
• The electron in an atom could have only certain allowed
values of energy (just as energy itself is quantized).
• Each specified energy value is called an energy level of the
atom:
En = –B/n2
– n is an integer, and B is a constant (2.179 × 10–18 J)
– The negative sign represents force of attraction.
• The energy is zero when the electron is located infinitely
far from the nucleus.
De Broglie’s Equation
• Louis de Broglie’s hypothesis stated that an object in
motion behaves as both particles and waves, just as light
does.
• A particle with mass m moving at a speed v will have a
wave nature consistent with a wavelength given by the
equation:
 = h/mv
• This wave nature is of importance only at the
microscopic level (tiny, tiny m).
• De Broglie’s prediction of matter waves led to the
development of the electron microscope.