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