Electrons in atoms

The quantum theory was used to show how the wavelike behavior of electrons leads to quantized energy states when the electrons are bound or trapped. We can use the quantum theory to explain the origin of spectral lines and to describe the electronic structure of atoms.

Learning Objectives

  • Explain the difference between a continuous spectrum and a line spectrum.
  • Explain the difference between an emission and an absorption spectrum.
  • Use the concept of quantized energy states to explain atomic line spectra.
  • Given an energy level diagram, predict wavelengths in the line spectrum, and vice versa.
  • Define and distinguish between shells, subshells, and orbitals.
  • Explain the relationships between the quantum numbers.
  • Use quantum numbers to label electrons in atoms.
  • Describe and compare atomic orbitals given the n and l quantum numbers.
  • List a set of subshells in order of increasing energy.
  • Write electron configurations for atoms in either the subshell or orbital box notations.
  • Write electron configurations of ions.
  • Use electron configurations to predict the magnetic properties of atoms.

Emission Spectra

  • experimental key to atomic structure: analyze light emitted by high temperature gaseous elements
      • experimental setup: spectroscopy
      • atoms emit a characteristic set of discrete wavelengths- not a continuous spectrum!
         
        • atomic spectrum can be used as a "fingerprint" for an element
    • hypothesis: if atoms emit only discrete wavelengths, maybe atoms can have only discrete energies
    • an analogy
       
      A turtle sitting on a ramp can have any height above the ground- and so, any potential energy
        A turtle sitting on a staircase can take on only certain discrete energies
      • energy is required to move the turtle up the steps (absorption)
      • energy is released when the turtle moves down the steps (emission)
      • only discrete amounts of energy are absorbed or released (energy is said to be quantized)
    • energy staircase diagram for atomic hydrogen
       
      For 1 e- species:

      En = -2.18 x 10-18 Z2 / n2

      Z = atomic number (for H,  Z = 1)

      DE = hn for a transition from ni to nf

       

       
    • summary: line spectra arise from transitions between discrete (quantized) energy states

    The quantum mechanical atom

    • Electrons in atoms have quantized energies
      • Electrons in atoms are bound to the nucleus by electrostatic attraction
      • Electron waves are standing matter waves
      • Standing matter waves have quantized energies, like the Bohr atom
    • Electron standing matter waves are 3 dimensional
      • The Bohr atom is one dimensional; one quantum number is required to describe the state of the electron
      • A 3D model requires three quantum numbers
      • A three-dimensional standing matter wave that describes the state of an electron in an atom is called an atomic orbital
    • The energies and mathematical forms of the orbitals can be computed using the Schrödinger equation
      • quantization is not assumed; it arises naturally in solution of the equation
      • every electron adds 3 variables (x, y, z) to the equation; it is very hard to solve equations with more than 2 variables.
      • energy-level separations computed with the Schrödinger equation agree very closely with those computed from atomic spectral lines

    Quantum numbers

    • Think of the quantum numbers as addresses for electrons
    • The principal quantum number, n
      • determines the size of an orbital (larger n = bigger orbitals)
      • largely determines the energy of the orbital (larger n = higher energy)
      • can take on integer values n = 1, 2, 3, ..., ¥
      • all electrons in an atom with the same value of n are said to belong to the same shell
         
    • the azimuthal quantum number, l  
      • designates the overall shape of the orbital within a shell
      • affects orbital energies (larger l  = higher energy)
      • all electrons in an atom with the same value of l are said to belong to the same subshell
      • only integer values between 0 and n-1 are allowed
      • sometimes called the orbital angular momentum quantum number
      • spectroscopists use the following notation for subshells
         
        Chemist's notation for subshells.
         
        l subshell name
        0 s
        1 p
        2 d
        3 f
         
    • the magnetic quantum number, ml
      • determines the orientation of orbitals within a subshell
      • does not affect orbital energy (except in magnetic fields!)
      • only integer values between -l and +l are allowed
      • the number of mlvalues within a subshell is the number of orbitals within a subshell
         

        The number of possible ml values determines the number of orbitals in a subshell.

         
        l possible values of ml number of orbitals in this subshell
        0 0 1
        1 -1, 0, +1 3
        2 -2, -1, 0, +1, +2 5
        3 -3, -2, -1, 0, +1, +2, +3 7
         
         

    • the spin quantum number, ms
      • several experimental observations can be explained by treating the electron as though it were spinning
      • spin makes the electron behave like a tiny magnet
      • spin can be clockwise or counterclockwise
      • spin quantum number can have values of +1/2 or -1/2

    Electron configurations of atoms

    • Electron configuration is a list showing how many electrons are in each orbital or subshell in an atom or ion
    • subshell notation: list subshells of increasing energy, with number of electrons in each subshell as a superscript
    • examples
      • 1s2 2s2 2p5 means "2 electrons in the 1s subshell, 2 electrons in the 2s subshell, and 5 electrons in the 2p subshell"
      • 1s2 2s2 2p6 3s2 3p3 is an electron configuration with 15 electrons total; 2 electrons have n=1 (in the 1s subshell); 8 electrons have n=2 (2 in the 2s subshell, and 6 in the 2p subshell); and 5 electrons have n=3 (2 in the 3s subshell, and 3 in the 3p subshell).
    • ground state configurations fill the lowest energy orbitals first

    Electron configurations of the first 11 elements, in subshell notation. Notice how configurations can be built by adding one electron at a time.

      atom Z ground state electronic configuration
      H 1 1s1
      He 2 1s2
      Li 3 1s2 2s1
      Be 4 1s2 2s2
      B 5 1s2 2s2 2p1
      C 6 1s2 2s2 2p2
      N 7 1s2 2s2 2p3
      O 8 1s2 2s2 2p4
      F 9 1s2 2s2 2p5
      Ne 10 1s2 2s2 2p6
      Na 11 1s2 2s2 2p6 3s1
       

    Writing electron configurations

    • strategy: start with hydrogen, and build the configuration one electron at a time (the Aufbau principle)
      • fill subshells in order by counting across periods, from hydrogen up to the element of interest:
        Filling order of subshells from the periodic table
    • rearrange subshells (if necessary) in order of increasing n & l  
    • examples: Give the ground state electronic configurations for:
      • Al
      • Fe
      • Ba
      • Hg
    • watch out for d & f block elements; orbital interactions cause exceptions to the Aufbau principle
      • half-filled and completely filled d and f subshells have extra stability

    Know these exceptions to the Aufbau principle in the 4th period. (There are many others at the bottom of the table)
        exception configuration predicted by the Aufbau principle true ground state configuration
        Cr 1s2 2s2 2p6 3s2 3p6 4s2 3d4 1s2 2s2 2p6 3s2 3p6 4s1 3d5
        Cu 1s2 2s2 2p6 3s2 3p6 4s2 3d9 1s2 2s2 2p6 3s2 3p6 4s1 3d10

    Electron configurations including spin

    • unpaired electrons give atoms (and molecules) special magnetic and chemical properties
    • when spin is of interest, count unpaired electrons using orbital box (or line) diagrams
       
      Examples of ground state electron configurations in the orbital box notation that shows electron spins.
       
      atom orbital box diagram
      B
      1s
      2s
      2p
      C
      1s
      2s
      2p
      N
      1s
      2s
      2p
      O
      1s
      2s
      2p
      F
      1s
      2s
      2p
      Cl
      1s
      2s
      2p
      3s
      3p      		    
      Mn
      1s
      2s
      2p
      3s
      3p
      4s
      3d
    • drawing orbital box diagrams
      1. write the electron configuration in subshell notation
      2. draw a box (or line) for each orbital.
        •  Remember that s, p, d, and f subshells contain 1, 3, 5, and 7 degenerate orbitals, respectively.
        • Remember that an orbital can hold 0, 1, or 2 electrons only, and if there are two electrons in the orbital, they must have opposite (paired) spins (Pauli exclusion principle)
      1. within a subshell (depicted as a group of boxes), spread the electrons out and line up their spins as much as possible (Hund's rule)
    • the number of unpaired electrons can be counted experimentally
      • configurations with unpaired electrons are attracted to magnetic fields (paramagnetism)
      • configurations with only paired electrons are weakly repelled by magnetic fields (diamagnetism)

    Core and valence electrons

    • chemistry involves mostly the shell with the highest value of n, called the valence shell
    • the noble gas core under the valence shell is chemically inert
    • simplify the notation for electron configurations by replacing the core with a noble gas symbol in square brackets:

      • Examples of electron configurations written with the core/valence notation

      atom full configuration core valence configuration full configuration using core/valence notation
      O 1s2 2s2 2p4 He 2s2 2p4 [He] 2s2 2p4
      Cl 1s2 2s2 2p6 3s2 3p5 Ne 3s2 3p5 [Ne] 3s2 3p5
      Al 1s2 2s2 2p6 3s2 3p1 Ne 3s2 3p1 [Ne] 3s2 3p1