Chapter 7 electronic structure
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Chapter 7 Electronic Structure. Waves. Waves are periodic disturbances – they repeat at regular intervals of time and distance. Properties of Waves. Wavelength ( l ) is the distance between one peak and the next . Frequency ( n ) is the number of waves that pass a fixed point each second.

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Chapter 7 Electronic Structure

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Chapter 7Electronic Structure


  • Waves are periodic disturbances – they repeat at regular intervals of time and distance.

Properties of Waves

  • Wavelength(l) is the distance between one peak and the next.

  • Frequency(n) is the number of waves that pass a fixed point each second.

Electromagnetic Radiation

  • Light or electromagnetic radiation consists of oscillating electric and magnetic fields.

Speed of Light

  • 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.

Example: Electromagnetic Radiation

  • An FM radio station broadcasts at a frequency of 100.3 MHz (1 Hz = 1 s-1). Calculate the wavelength of this electromagnetic radiation.

Kinds of 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.

Quantization of Energy

  • 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.

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, n0.

    • At frequencies higher than n0, kinetic energy of ejected electron is hn – hn0.

Photoelectric Effect (cont.)

  • 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 = hn.

    • The minimum energy needed to free an electron is hn0.

    • Law of conservation of energy means that the kinetic energy of ejected electron is hn – hn0.

Dual Nature of Light?

  • 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.


  • 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.

Line Spectra of Some Elements

The Rydberg Equation

  • 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.

Example: Rydberg Equation

  • Calculate the wavelength (in nm) of the line in the hydrogen atom spectrum for which n1 = 2 and n2 = 3.

The Bohr Model of Hydrogen

  • 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:

Bohr Model and the Rydberg Equation

  • Assume that when one electron transfers from one orbit to another, energy must be added or removed by a single photon with energy hn.

  • This assumption leads directly to the Rydberg equation.

Hydrogen Atom Energy Diagram

Matter as Waves

  • 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.

Example: de Broglie Wavelength

  • 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)



(x) (mv) 


  • 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

Standing Waves

  • The vibration of a string is restricted to certain wavelengths because the ends of the string cannot move.

de Broglie Waves in the H Atom

  • The de Broglie wave of an electron in a hydrogen atom must be a standing wave, restricting its wavelength to values of l = 2pr/n, with n being an integer.

    • This leads directly to quantized angular momentum, one of Bohr’s assumptions.

Schrödinger Wave Equation

  • The wave function (Y) gives the amplitude of the electron wave at any point in space.

  • Y2 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.

Quantum Numbers in the H Atom

  • The solution of the Schrödinger equation produces quantum numbers that describe the characteristics of the electron wave.

  • Three quantum numbers, represented by n, l, and ml, 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, l, and ml.

The Principal Quantum Number, n

  • 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.

Angular Momentum Quantum Number, l

  • The angular momentum quantum number, l, is associated with the shape of the orbital.

    • Allowed values: 0 and all positive integers up to n-1.

    • The l 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 l.

Notations for Subshells

  • To identify a subshell, values for both n and l must be assigned, in that order.

  • The value of l is represented by a letter:



    • Thus, a 3p subshell has n = 3, l = 1.

    • A 2s subshell has n = 2, l = 0.

Magnetic Quantum Number, ml

  • The magnetic quantum number, ml, indicates the orientation of the atomic orbital in space.

    • Allowed values: all whole numbers from –l to l, including 0.

  • A wave function described by all three quantum numbers (n, l, ml) is called an orbital.

Allowed Combinations of n, l, ml

Example: Quantum Numbers

  • Give the notation for each of the following orbitals if it is allowed. If it is not allowed, explain why.

    (a) n = 4, l = 1, ml = 0

    (b) n = 2, l = 2, ml = -1

    (c) n = 5, l = 3, ml = +3

Test Your Skill

  • For each of the following subshells, give the value of the n and the l quantum numbers.

    (a) 2s

    (b) 3d

    (c) 4p

Electron Spin

  • 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 Density Diagrams

  • Different densities of dots or colors are used to represent the probability of finding the electron in space.

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 (l = 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 ml.

    • 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.

Energies of Hydrogen Atom Orbitals

  • 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.

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.).

Effective Nuclear Charge

  • 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+.

Energy Dependence on l

  • 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.

Multielectron Energy Level Diagram

  • Within any principal shell, the energy increases in the order of the l quantum number: 4s < 4p < 4d < 4f.

Increasing Energy Order

  • 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

Electrons in Multielectron Atoms

  • Each electron in a multielectron atom can be described by hydrogen-like wave functions by assigning values to the four quantum numbers n, l, ml, 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 l.

Pauli Exclusion Principle

  • 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.

The Aufbau 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.

Orbital Diagrams

  • 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.

Electron Configuration

  • 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 Configurations of Elements

  • Hydrogen contains one electron in the 1s subshell.


  • Helium has two electrons in the 1s subshell.


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

  • N1s2 2s2 2p3

  • O 1s2 2s2 2p4

  • F 1s2 2s2 2p5

  • Ne 1s2 2s2 2p6

Electron Configurations of Heavier Atoms

  • 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


Anomalous Electron Configurations

  • 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.

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