Module 9: Quantum Theory and The Electronic Structure of Atoms

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Module 9 Quantum Theory and The Electronic
Structure of Atoms

• By Alyssa Jean-Mary
• Source Modular Study Guide for First Semester
  Chemistry by Anthony J. Papaps and Marta E.
  Goicoechea-Pappas

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Electromagnetic (EM) Radiation

• Electronic transitions - associated with
  absorption OR emission of electromagnetic
  radiation, most of which cannot be seen, with
  only the radiation that is in the visible region
  of the electromagnetic spectrum being seen.

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Electromagnetic (EM) Radiation Wave Nature

• Wave nature of EM radiation
• EM radiation travels in waves through space at
  the speed of light (c), where c 3.00 x 108m/s
  OR 3.00 x 1010cm/sec
• Wave components
• Crest top part of the wave
• Trough bottom part of the wave
• Wavelength (?) distance between peaks of
  consecutive waves (i.e. distance from crest to
  crest or trough to trough)
• Units m, cm, OR Å (Angstrom), where 1 Å
  10-10m 10-8 cm
• Frequency (v) the number of waves that pass a
  particular point in 1 second
• Units Hertz (Hz) cycles/sec 1/sec sec-1
• Wave equation v c / ?

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Electromagnetic (EM) Radiation Particle Nature

• Particle nature of EM radiation
• Einstein discovered that the photoelectric
  effect, which is when electrons are ejected from
  the surface of a metal when light shines on it,
  can be explained by postulating that light has
  both wave and particle properties. When light is
  a particle, it is known as a photon. Thus, since
  light has both a wave nature and a particle
  nature, it has what is called wave-particle
  duality
• Plancks Equation, which gives the energy of a
  photon
• E hv h (c / ?)
• In this equation
• E energy, with units of J (Joule) OR erg
  (gcm2/sec2)
• h Plancks constant 6.63 x 10-34 Jsec OR
  6.63 x 10-27 ergsec
• v frequency, with units of sec-1
• c speed of light 3.00 x 108m/s OR 3.00 x
  1010cm/sec
• ? wavelength, with units or m, cm, OR Å

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Examples Using the Wave Equations

• Example 1 If the wavelength of a light wave is
  2335Å,
• What is its frequency?
• What is its energy in J?
• What is its energy in erg?
• Example 2 If the energy of a light wave is 6.66
  x 10-20J,
• What is its frequency?
• What is its wavelength in cm?
• What is its wavelength in Å?

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Electromagnetic Spectrum

• Electromagnetic spectrum different forms of
  light arranged in a particular order according to
  their energy, frequency, and wavelength
• The visible region of the EM spectrum (violet to
  red) 4000-7000 Å, OR 400-700 nm, OR 4.00 x
  10-77.00 x 10-7 m

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The Wave Nature of Electrons

• L. de Broglie predicted that a particle that has
  a mass (m) and a velocity (v) should have a
  certain wavelength associated with it
• ? h / (mv), where h Plancks constant
• This equation shows that wavelength and mass
  wavelength and velocity are inversely
  proportional to each other, so, if either the
  mass or the velocity of a particle increases, its
  wavelength decreases

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An Example Using L. de Broglies Equation

• Example 1 If a particle with a mass of 6.33 x
  10-26 is travelling at 3.00 x 108 cm/sec, what is
  its wavelength in cm?

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The Bohr Atom, Quantization of Energy, Atomic
Spectra 1

• Bohr Model of the atom assumes that electrons
  revolve around the nucleus of an atom in circular
  orbits
• Quantization of energy When an atom is excited,
  an electron is promoted from a lower energy level
  (i.e. orbit) to a higher energy level (i.e.
  orbit). This electron absorbs a definite (i.e.
  quantized) amount of energy. When the electron
  falls back to its original, lower energy level,
  exactly the same amount of energy that it
  absorbed by moving from the lower energy level to
  the higher energy level is emitted.
• This idea is illustrated in the picture below.
  Here, n is the principal energy level or orbit.
  As n increases, the energy levels or orbits get
  closer to each other.
• the maximum number of electrons per level max
  e-/level 2n2
• For n 1, max e- 2(1)2 2 for n 2, max e-
  2(2)2 8 for n 3, max e- 2(3)2 18 for
  n 4, max e- 2(4)2 32
• The Bohr Model of the atom only predicts the
  correct electronic structure for the first 18
  elements.

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The Bohr Atom, Quantization of Energy, Atomic
Spectra 2

• Emission (i.e. bright line) spectra When the
  electrons in an atom are exited to higher energy
  levels by passing an electric current through a
  gas in a vacuum tube at a very low pressure, the
  light that the gas emits, which is due to the
  electrons falling back to their original energy
  levels, can be dispersed by a prism into distinct
  lines. Each line corresponds to light with a
  specific wavelength, frequency, and energy, an
  thus, a specific color. The lines obtained are
  referred to the emission spectrum of the element,
  and each element has its own unique emission
  spectrum.
• In addition to using a prism to separate the
  light into different wavelengths, a diffraction
  grating can be used. In a diffraction grating,
  the wavelengths that are the least energetic are
  bent the most, which is opposite from a prism.

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The Bohr Atom, Quantization of Energy, Atomic
Spectra 3

• J. Balmer discovered that the wavelengths of
  various lines in the hydrogen spectrum can be
  related mathematically by the following empirical
  equation
• 1 / ? (R)(Z2)((1/ni2) - (1/nO2))
• Here, ? wavelength in m or cm R Rydberg
  constant 1.097 x 107 m- 1.097 x 105 cm- Z
  atomic number ni inner energy level no
  outer energy level
• This equation can only be used for any
  hydrogen-like species (i.e. species that contain
  only one electron (He, Li2, etc.)
• 1 / ? (R)(Z2)((1/ni2) - (1/nO2))
• Here, ? wavelength in m or cm R Rydberg
  constant 1.097 x 107 m- 1.097 x 105 cm- Z
  atomic number ni inner energy level no
  outer energy level

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An Example Using the Bohr Atom, Quantization of
Energy, and Atomic Spectra

• Example 1 Draw a diagram for the structure of
  Aluminum-27 which indicates the number of
  protons, neutrons, and electrons present, with
  the electrons arranged in their principal energy
  levels.
• Example 2 If an electron in a sample of Li2
  goes from energy level 3 to 1,
• What is its wavelength in cm?
• What is its wavelength in Å?
• Is this transition in the visible portion of the
  EM spectrum?
• What is its frequency in sec-1?
• What is its energy in J?
• What is its energy in erg?

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Quantum Mechanical Picture of the Atom

• Bohr made significant contributions to the
  understanding of atoms, including the still
  unchallenged idea that the energy of an electron
  in an atom is quantized, but, he did not give a
  complete description of the electronic behavior
  in atoms, such as the emission spectra of atoms
  with more than one electron or the dual nature of
  matter.
• The Heisenberg Uncertainty Principle is one
  important consequence of this dual nature matter.
  It states that it is impossible to determine
  accurately both the momentum, which is mass
  multiplied by velocity, and the position of an
  electron (or other very small particles)
  simultaneously. To extend this principle, both
  the energy and position of an electron cant be
  known accurately at the same time. When this
  principle is applied, it shows that electrons do
  not orbit the nucleus in a well-defined path as
  Bohr thought, because if they did, the position
  of electrons could be more precisely determined
  using the radius of the orbit as well as their
  momentum (i.e. their kinetic energy) at the same
  time.
• Quantum mechanics is a highly mathematical branch
  of chemistry/physics that treats small particles
  as waves. It came out of a search to describe the
  behavior of submicroscopic particles, since these
  particles were found to not obey the laws of
  classical mechanics. Using a complex mathematical
  treatment, Schrödinger formulated an equation
  that describes the behavior and energies of
  submicroscopic particles. The Schrödinger
  equation requires advanced calculus to solve.
  This equation includes both the particle behavior
  (in terms of mass) and the wave behavior (in
  terms of a wave function that depends on the
  location in space of the small particles) of
  these small particles.

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Quantum Mechanical Picture of the Atom Basic
Ideas of Quantum Mechanics

• Atoms and molecules can exist only in certain
  energy states. If an atom or molecule changes its
  energy state, it will emit or absorb just enough
  energy to bring it into the new energy state (the
  quantum condition).
• The energy lost, or gained, by an atom as it goes
  from higher to lower, or lower to higher, energy
  states is equal to the energy of the photon that
  is emitted, or absorbed, during the transition.
• Since a particle can behave as a wave, many of
  the concepts and mathematical equations from wave
  theory are used.
• The allowed energy states, which are called
  orbitals of atoms and molecules, can be described
  by sets of four numbers, which are called quantum
  numbers.
• Since the exact energy of these orbitals can be
  calculated, the exact position of the electron is
  not known, a consequence of the Heisenberg
  Uncertainty Principle.

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Quantum Numbers (n, l, ml, ms)

• There are 4 quantum numbers. These numbers
  describe the electronic arrangement, or
  electronic configuration, of atoms in space
• Principal Quantum Number (n) main energy level
  that an electron occupies
• n is a positive integer (i.e. 1, 2, 3, 4, etc.),
  and it represents shells in the Bohr atom (i.e. 1
  K, 2 L, 3 M, 4 N, etc.)
• Angular Momentum, Subsidiary, or Azimuthal
  Quantum Number (l) specific kind of atomic
  orbital (sublevel) that an electron occupies
• l is a whole number (i.e. 0, 1, 2, 3, etc.), and
  it is always 1 less than n. Also, it represents
  different kinds of sublevels (i.e. 0 2, 1 p,
  2 d, 3 f, etc.).
• For example
• if n 1, l can only 0, which corresponds to
  sublevel s, so there is only one sublevel
  (sublevel s) in the first energy level
• If n 2, l can be 0 or 1, so it corresponds to
  sublevels s and p, so there are two sublevels
  (sublevels s and p) in the second energy level
• Magnetic Quantum Number (ml) spatial
  orientation of atomic orbitals i.e. how many
  distinct regions of space (atomic orbitals) are
  associated with a specific sublevel
• ml is whole number that ranges from -l,0,l.
• For example, if l 1, which corresponds to the
  sublevel p, ml -1, 0, 1, which means there are
  three distinct regions of space (atomic orbitals)
  associated with the sublevel p
• Electron Spin Quantum Number (ms) the spin of
  an electron (either spin up or spin down) and the
  orientation of the magnetic field produced by
  this spin.
• ms ½. This shows that each atomic orbital can
  have only 2 electrons, one with ms ½ and the
  other one with ms -½.
• Pauli Exclusion Principle no two electrons in
  an atom can have the same 4 quantum numbers

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Permissible Values of the Quantum Numbers (n, l,
ml, ms) n 1 to 4
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Examples Using Quantum Numbers

• Are the following permissible or not permissible
  sets of the four quantum numbers?
• (0, 1, 1, -½)
• (3, 2, -2, ½)

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Orbital Diagrams Quantum Numbers 1

• In orbital diagrams, orbitals are represented by
  lines and electrons are represented by arrows,
  with the orbitals being labeled using a number,
  which represents the principal quantum number
  (n), and a letter, which represents the
  subsidiary quantum number (l).
• See below for an example using 2p orbitals
• The 2 represents n 2
• Since n 2, l can be 1 or 0, but since it is p,
  l 1
• Since l 1, ml can be -1, 0, or 1, which means
  there are three kinds of p orbitals, so there are
  three lines, with, by convension, the first line
  representing the most negative value of ml and
  then decreasing the negative value until the last
  line, which represents the least negative value
  of ml
• Since each orbital (i.e. line) can hold a maximum
  of 2 electrons, two arrows are drawn on each
  line, one facing up (ms 1/2) and one facing
  down (ms -1/2)

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Orbital Diagrams for n 1 to 4
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Atomic Orbitals

• An orbital is a region in which there is a high
  probability of finding the electron. Remember
  that each orbital in an atom can hold a maximum
  of two electrons. If all of these atomic orbitals
  are taken together, they appear as a diffuse
  cloud of electrons.

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Electronic Configuration

• Electronic configuration electronic arrangement
  (i.e. the arrangement of electrons) of an atom
• Every electron in an atom has a position is space
  that is defined by the principal energy
  level/shell the sublevel/subshell or orbital
  that it occupies
• For example, 1s2, where
• the number in front (1 in this case) represents
  the principal energy level/shell, which is always
  a whole number (i.e. 1, 2, 3, 4, 5, 6, or 7)
• the letter (s in this case) represents the
  sublevel/subshell or orbital, which can be s, p,
  d, or f
• the superscript number (2 in this case)
  represents the number of electrons present in
  this principal energy level/shell and
  sublevel/subshell or orbital.
• Each orbital can hold a maximum of 2 electrons
• Since orbital s has 1 different type, it has a
  max. of 2 x 1 2 electrons
• Since orbital p has 3 different types, it has a
  max. of 2 x 3 6 electrons
• Since orbital d has 5 different types, it has a
  max. of 2 x 5 10 electrons
• Since orbital f has 7 different types, it has a
  max. of 2 x 7 14 electrons

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Electronic Configuration The Energies of
Orbitals 1

• Aufbau Principle electrons will fill the lowest
  energy atomic orbital available first
• The below picture illustrates this order

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Electronic Configuration The Energies of
Orbitals 2

• The below picture gives the electronic
  configuration, or electronic arrangement, of the
  type of orbital occupied last

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Electronic Configuration The Energies of
Orbitals 3

• The larger the principle energy level, the
  further away the electron is from the nucleus. If
  electrons have the same principal quantum number,
  their proximity to the nucleus follows the order
  s, p, d, f, meaning that those in s are closest
  to the nucleus, and those in f are further away
  from the nucleus. The closer an electron is to
  the nucleus, the less energy it has, and thus,
  the harder it is to be removed from the atom.
• For example, to remove a 3s electron from an
  atom, more energy is needed, than to remove a 3p
  electron.
• Orbitals that have the same n and l, but
  different ml and ms are degenerate, meaning that
  they have the same amount of energy.

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Electronic Configuration Writing Electronic
Configurations

• To write electronic configurations, just follow
  the periodic table, until the element is reached

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Examples of Writing Electronic Configurations

• Write the electronic configuration for each of
  the following elements
• Na
• C

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Electronic Configuration Writing Abbreviated
Electronic Configurations

• When writing abbreviated electronic
  configurations, the electronic configuration of
  the noble gas (group VIIIA) previous to the
  element is replaced by the symbol of the noble
  gas enclosed in brackets. The rest of the
  electronic configuration remains the same.

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Examples of Writing Abbreviated Electronic
Configurations

• Write the abbreviated electronic configuration
  for each of the following elements
• P
• V

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Electronic Configuration Orbital Diagram for the
Elements

• Remember that in orbital diagrams, orbitals are
  represented by lines and electrons are
  represented by arrows, with the orbitals being
  labeled using a number, which represents the
  principal quantum number (n), and a letter, which
  represents the subsidiary quantum number (l).
• Since there is only 1 type of s orbital, it is
  represented by 1 line
• Since there are 3 types of p orbitals, they are
  represented by 3 lines
• Since there are 5 types of d orbital, they are
  represented by 5 lines
• Since there are 7 types of f orbitals, they are
  represented by 7 lines
• The following rules also need to be followed when
  writing orbital diagrams
• An orbital can have a maximum of 2 electrons,
  with each electron having an opposite spin. This
  is Paulis Exclusion Principle, which, stated
  another way, is that no 2 electrons can have the
  same set of 4 quantum numbers.
• All orbitals of a given sublevel (i.e. degenerate
  orbitals, orbitals with the same n and l) must be
  occupied by a single electron, having the same
  spin (spin up), before being paired.

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Examples of Writing Orbital Diagrams

• Write the orbital diagram for each of the
  following elements
• Si
• Co

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Electronic Configuration Exceptions to
Electronic Configurations and Orbital Diagrams

• The elements in groups VIB (Cr, Mo, and W, but W
  is actually not an exception to the rules) and IB
  (Cu, Ag, and Au) of the periodic table dont
  follow the rules given above.
• Elements in VIB prefer to have sublevel d halfway
  filled, so one of the two electrons in 4s is
  taken to partially fill the last 3d orbital,
  leaving the 4s orbital with only 1 electron (i.e.
  partially filled).
• Elements in IB prefer to have sublevel d
  completely filled, so one of the two electrons in
  4s is taken to completely fill the last 3d
  orbital, leaving the 4s orbital with only 1
  electron (i.e. partially filled).

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Examples of Finding Electronic Configurations for
Elements with Exceptions to the Rules

• Write the electronic configuration for each of
  the following elements
• Cr
• Cu

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Electronic Configuration Electronic
Configuration of Ions

• To write the electronic configuration of an ion,
  first write the electronic configuration for the
  neutral atom. Then,
• if the ion is negative (an anion), add as many
  electrons to the last orbital(s) being filled as
  the charge on the ion indicates.
• if the ion is positive (a cation), remove as many
  electrons from the least stable orbital(s) (i.e.
  the orbital(s) with the highest energy the last
  orbital(s)) as the charge on the ion indicates.
• If the outermost electrons are in d orbitals,
  electrons are removed from the s orbital first
  and then from the d orbitals.
• If the outermost electrons are in p orbitals,
  electrons are removed from the p orbitals first,
  then from the s orbital, and finally from the d
  orbitals.

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Examples of Finding the Electronic Configuration
of Ions

• Write the electronic configuration for each of
  the following ions
• O2-
• Ca2
• Fe3
• Ga3

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Electronic Configuration Paramagnetism and
Diamagnetism

• Paramagnetic an atom or ion that has one or
  more unpaired electrons
• Diamagnetic an atom or ion that has no unpaired
  electrons

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Examples of Determining Paramagnetism and
Diamagnetism

• How many unpaired electrons do the following
  species have? Are they paramagnetic or
  diamagnetic?
• Cl-
• N
• Cr
• Ga3

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Electronic Configuration Isoelectronic Species

• Species with the same electronic configuration
  are isoelectronic.

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Examples of Finding Isoelectronic Species

• Is Sc3 isoelectronic with Ca2?

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THE END