Daniel L. RegerScott R. GoodeDavid W. Ball
www.cengage.com/chemistry/reger
Chapter 7Electronic Structure
• Waves are periodic disturbances – they repeat at regular intervals of time and distance.
Waves
• Wavelength () is the distance between one peak and the next.• Frequency () is the number of waves
that pass a fixed point each second.
Properties of Waves
• Light or electromagnetic radiation consists of oscillating electric and magnetic fields.
Electromagnetic Radiation
• All electromagnetic waves travel at the same speed in a vacuum, 3.00×108 m/s.• The speed of a wave is the product of its
frequency and wavelength, so for light:
• So, if either the wavelength or frequency is known, the other can be calculated.
Speed of Light
83.00 10 m/sc
• An FM radio station broadcasts at a frequency of 100.3 MHz (1 Hz = 1 s-1). Calculate the wavelength of this electromagnetic radiation.
Example: Electromagnetic Radiation
• Visible light is only a very small portion of the electromagnetic spectrum.• Other names for regions are gamma rays,
x rays, ultraviolet, infrared, microwaves, radar, and radio waves.
Kinds of Electromagnetic Radiation
• In 1900, Max Planck proposed that there is a smallest unit of energy, called a quantum. The energy of a quantum is
where h is Planck’s constant, 6.626×10-34 J·s.
Quantization of Energy
E h
The Photoelectric Effect• The photoelectric effect: the process in
which electrons are ejected from a metal when it is exposed to light.• No electrons are ejected by light with a
frequency lower than a threshold frequency, 0.
• At frequencies higher than 0, kinetic energy of ejected electron is h – h0.
• Einstein suggested an explanation by assuming light is a stream of particles called photons.• The energy of each photon is given by Planck’s
equation, E = h.• The minimum energy needed to free an electron is
h0.
• Law of conservation of energy means that the kinetic energy of ejected electron is h – h0.
Photoelectric Effect (cont.)
• Is light a particle, or is it a wave?• Light has both particle and wave
properties, depending on the property.• Particle behavior, wave behavior no
longer considered to be exclusive from each other.
Dual Nature of Light?
• A spectrum is a graph of light intensity as a function of wavelength or frequency.• The light emitted by heated objects is a
continuous spectrum; light of all wavelengths is present.• Gaseous atoms produce a line spectrum
– one that contains light only at specific wavelengths and not at others.
Spectra
Line Spectra of Some Elements
• Study of the spectrum of hydrogen, the simplest element, show that the wavelengths of lines of light can be calculated using the Rydberg equation:
• n1 and n2 are whole numbers and RH = 1.097×107 m-1.
The Rydberg Equation
H 2 21 2
1 1 1R
n n
• Calculate the wavelength (in nm) of the line in the hydrogen atom spectrum for which n1 = 2 and n2 = 3.
Example: Rydberg Equation
• Bohr assumed:• that the electron followed a circular orbit
about the nucleus; and• that the angular momentum of the electron
was quantized.
• Using these assumptions, he found that the energy of the electron was quantized:
The Bohr Model of Hydrogen
2 418
2 2 2
2 1, -2.18 10 Jn
me BE B
h n n
• Assume that when one electron transfers from one orbit to another, energy must be added or removed by a single photon with energy h.• This assumption leads directly to the
Rydberg equation.
Bohr Model and the Rydberg Equation
Hydrogen Atom Energy Diagram
• Louis de Broglie proposed that matter might be viewed as waves as well as particles.• de Broglie suggested that the wavelength of matter is given by
where h is Planck’s constant, p is momentum, m is mass, and v is velocity.
Matter as Waves
h h
p mv
• At room temperature, the average speed of an electron is 1.3×105 m/s. The mass of the electron is about 9.11×10-31 kg. Calculate the wavelength of the electron under these conditions.
• What is the wavelength of a marathon runner moving at a speed of 5 m/s?
(mass of the runner is 52 kg)
Example: de Broglie Wavelength
Uncertainty• Heisenberg showed that the more precisely the
momentum of a particle is known, the less precisely is its position known:
Cannot know precisely where and with what momentum an electron is.
New ideas for determining this information based on probability
Quantum Mechanics was born
(x) (mv) h4
• The vibration of a string is restricted to certain wavelengths because the ends of the string cannot move.
Standing Waves
• The de Broglie wave of an electron in a hydrogen atom must be a standing wave, restricting its wavelength to values of = 2r/n, with n being an integer.• This leads directly to quantized angular
momentum, one of Bohr’s assumptions.
de Broglie Waves in the H Atom
• The wave function () gives the amplitude of the electron wave at any point in space.
• 2 gives the probability of finding the electron at any point in space.
• There are many acceptable wave functions for the electron in a hydrogen (or any other) atom.
• The energy of each wave function can be calculated, and these are identical to the energies from the Bohr model of hydrogen.
Schrödinger Wave Equation
• The solution of the Schrödinger equation produces quantum numbers that describe the characteristics of the electron wave.
• Three quantum numbers, represented by n, , and m, describe the distribution of the electron in three dimensional space.
• An atomic orbital is a wave function of the electron for specific values of n, , and m.
Quantum Numbers in the H Atom
• The principal quantum number, n, provides information about the energy and the distance of the electron from the nucleus.• Allowed value of n are 1, 2, 3, 4, …• The larger the value of n, the greater the average distance of the electron from the nucleus.
• The term principal shell (or just shell) refers to all atomic orbitals that have the same value of n.
The Principal Quantum Number, n
• The angular momentum quantum number, , is associated with the shape of the orbital.• Allowed values: 0 and all positive integers up
to n-1.• The quantum number can never equal or
exceed the value of n.
• A subshell is all possible orbitals that have the same values of both n and .
Angular Momentum Quantum Number,
• To identify a subshell, values for both n and must be assigned, in that order.• The value of is represented by a letter:
0 1 2 3 4 5 etc.
letter s p d f g h etc.• Thus, a 3p subshell has n = 3, = 1.• A 2s subshell has n = 2, = 0.
Notations for Subshells
• The magnetic quantum number, m, indicates the orientation of the atomic orbital in space.• Allowed values: all whole numbers from
– to , including 0.
• A wave function described by all three quantum numbers (n, , m) is called an orbital.
Magnetic Quantum Number, m
Allowed Combinations of n, , m
n m # orbitals
1 0 0 1
2 0
1
0
-1, 0, +1
1
3
3 0
1
2
0
-1, 0, +1
-2, -1, 0, +1, +2
1
3
5
4 0
1
2
3
0
-1, 0, +1
-2, -1, 0, +1, +2
-3, -2, -1, 0, +1, +2, +3
1
3
5
7
• Give the notation for each of the following orbitals if it is allowed. If it is not allowed, explain why.(a) n = 4, = 1, m = 0
(b) n = 2, = 2, m = -1
(c) n = 5, = 3, m = +3
Example: Quantum Numbers
• For each of the following subshells, give the value of the n and the quantum numbers.(a) 2s(b) 3d(c) 4p
Test Your Skill
• An electron behaves as a small magnet that is visualized as coming from the electron spinning.
• The electron spin quantum number, ms, has two allowed values: +1/2 and -1/2.
Electron Spin
• Different densities of dots or colors are used to represent the probability of finding the electron in space.
Electron Density Diagrams
Contour Diagrams• In a contour diagram, a surface is drawn
that encloses some fraction of the electron probability (usually 90%).
Shapes of p Orbitals• p orbitals ( = 1) have two lobes of electron
density on opposite sides of the nucleus.
Orientation of the p Orbitals• There are three p orbitals in each principal
shell with an n of 2 or greater, one for each value of m.
• They are mutually perpendicular, with one each directed along the x, y, and z axes.
Shapes of the d Orbitals• The d orbitals have four lobes where the
electron density is high.• The dz2 orbital is mathematically equivalent to
the other d orbitals, in spite of its different appearance.
• The energies of the hydrogen atom orbitals depend only on the value of the n quantum number.
• The s, p, d, and f orbitals in any principal shell have the same energies.
Energies of Hydrogen Atom Orbitals
Other One-Electron Systems• The energy of a one-electron species also depends
on the value of n, and are given by the equation
where Z is the charge on the nucleus.• This equation applies to all one-electron species
(H, He+, Li2+, etc.).
2 18 2
2 2
2.18 10 joulesn
Z B ZE
n n
• In multielectron atoms, the energy dependence on nuclear charge must be modified to account for interelectronic repulsions.• The effective nuclear charge is a
weighted average of the nuclear charge that affects an electron in the atom, after correction for the shielding by inner electrons and interelectronic repulsions.
Effective Nuclear Charge
• Electron shielding is the result of the influence of inner electrons on the effective nuclear charge.
• The effective nuclear charge that affects the outer electron in a lithium atom is considerably less than the full nuclear charge of 3+.
Effective Nuclear Charge
• The 2s electron penetrates the electron density of the 1s electrons more than the 2p electrons, giving it a higher effective nuclear charge and a lower energy.
Energy Dependence on
• Within any principal shell, the energy increases in the order of the quantum number: 4s < 4p < 4d < 4f.
Multielectron Energy Level Diagram
• Based on experimental observations, subshells are usually occupied in the order
1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p
< 5s < 4d < 5p < 6s < 4f < 5d < 6p < 7s < 5f < 6d
Increasing Energy Order
• Each electron in a multielectron atom can be described by hydrogen-like wave functions by assigning values to the four quantum numbers n, , m, and ms.
• These wavefunctions differ from those in the hydrogen atom because of interelectronic repulsions.• The energy of these wave functions depends
on both n and .
Electrons in Multielectron Atoms
• The Pauli Exclusion Principle: no two electrons in the same atom can have the same set of four quantum numbers.• A difference in only one of the four
quantum numbers means that the sets are different.
Pauli Exclusion Principle
• The aufbau principle: as electrons are added to an atom one at a time, they are assigned the quantum numbers of the lowest energy orbital that is available.• The resulting atom is in its lowest
energy state, called the ground state.
The Aufbau Principle
• An orbital diagram represents each orbital with a box, with orbitals in the same subshell in connected boxes; electrons are shown as arrows in the boxes, pointing up or down to indicate their spins.• Two electrons in the same orbital must
have opposite spins.
Orbital Diagrams
↑↓
• An electron configuration lists the occupied subshells using the usual notation (1s, 2p, etc.). Each subshell is followed by a superscripted number giving the number of electrons present in that subshell.• Two electrons in the 2s subshell would
be 2s2 (spoken as “two-ess-two”).• Four electrons in the 3p subshell would
be 3p4 (“three-pea-four”).
Electron Configuration
• Hydrogen contains one electron in the 1s subshell.
1s1
• Helium has two electrons in the 1s subshell.
1s2
Electron Configurations of Elements
↑
↑↓
Electron Configurations of Elements
• Lithium has three electrons.1s2 2s1
• Beryllium has four electrons.1s2 2s2
• Boron has five electrons.1s2 2s2 2p1
↑↓ ↑
↑↓ ↑↓
↑↓ ↑↓ ↑
Orbital Diagram of Carbon• Carbon, with six electrons, has the
electron configuration of 1s2 2s2 2p2.• The lowest energy arrangement of
electrons in degenerate (same-energy) orbitals is given by Hund’s rule: one electron occupies each degenerate orbital with the same spin before a second electron is placed in an orbital.
↑↓ ↑↓ ↑ ↑
Other Elements in the Second Period
• N 1s2 2s2 2p3
• O 1s2 2s2 2p4
• F 1s2 2s2 2p5
• Ne 1s2 2s2 2p6
↑↓ ↑↓ ↑ ↑ ↑
↑↓ ↑↓ ↑↓ ↑ ↑
↑↓ ↑↓ ↑↓ ↑↓ ↑
↑↓ ↑↓ ↑↓ ↑↓ ↑↓
• Heavier atoms follow aufbau principle in organization of electrons.• Because their electron configurations
can get long, larger atoms can use an abbreviated electron configuration, using a noble gas to represent core electrons.
Fe: 1s2 2s2 2p6 3s2 3p6 4s2 3d6 → [Ar] 4s2 3d6
Ar
Electron Configurations of Heavier Atoms
• The electron configurations for some atoms do not strictly follow the aufbau principle; they are anomalous.• Cannot predict which ones will be
anomalous.• Example: Ag predicted to be
[Kr] 5s2 4d9; instead, it is
[Kr] 5s1 4d10.
Anomalous Electron Configurations
Top Related