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a supplemental resource for students

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Lecture20221 Lecture20221 Presentation Transcript

  • Quantum Theory: Atomic Orbitals and Quantum Numbers Lecture 20
  • Quantum mechanics is the study of mechanical systems whose dimensions are close to or below the atomic scale, such as molecules, atoms, electrons, protons and other subatomic particles.
    • Erwin Rudolf Josef Alexander Schrodinger (1887-1961), Austrian-Irish scientist
  • Schrödinger Equation
  • Wave function (ψ) is a mathematical description of the state of the electron’s matter-wave in terms of position in three dimensions.
  • Schrödinger Equation
  • E is the energy of the atom. H is the Hamiltonian operator. It represents a set of mathematical operations that, when carried out on a particular ψ, yields an allowed energy value.
  • Atomic orbital is a given wave function associated with a solution to the Schrödinger equation (i.e. an energy state of the atom).
  • Do not mix orbitals with orbits. No resemblance at all.
  • We cannot know precisely where the electron is at any moment. But we can describe where it probably is, where it is most likely to be found, or where it spends most of its time.
  • The wave function ψ has no direct physical meaning! Its square does have. ψ 2 is the probability density, a measure of the probability that the electron can be found within a particular tiny volume of the atom.
  • Electron density probability
  • The electron probability density decreases with distance from the nucleus along a line.
  • The electron probability density being far from the nucleus is very small but not zero.
  • Electron density probability
  • An atomic orbital is specified by three quantum numbers.
    • The principal quantum number ( n ) is a positive integer (1, 2, 3, asf.).
    • It indicates the relative size of the orbital and therefore the relative distance from the nucleus of the peak in the radial probability distribution plot.
    • It specifies the energy level: the higher the n value, the higher the energy level.
  • An atomic orbital is specified by three quantum numbers.
    • The angular momentum quantum number ( l ) is an integer from 0 to n-1 (0, 1, 2, asf.).
    • It is related to the shape of the orbital and is sometimes called the orbital-shape quantum number.
    • n limits l : for n = 4 l = 0, 1, 2, and 3.
    • Number of possible l values equals the value of n .
  • An atomic orbital is specified by three quantum numbers.
    • The magnetic quantum number ( m l ) is an integer from -1 through 0 to +l .
    • It prescribes the orientation of the orbital in the space around the nucleus and is sometimes called the orbital-orientation quantum number.
    • l sets the possible values of m l : for l = 2 m = -2, -1, 0, +1, and +2.
    • Number of possible m values equals 2l+1 .
  • A sample problem on determining quantum numbers for an energy level.
  • Some specific terms:
    • Level (or shell). The levels are given by the n value: the smaller it is, the lower the energy level.
    • Sublevel (or subshell). The sublevels designate the orbital shape. Each sublevel has a letter designation: s ( l =0), p ( l =1), d ( l =2), f ( l =3).
    • Orbital . The orbitals are specified by allowed combinations of n , l , and m l values.
  • The three quantum numbers describe orbital size (energy), shape and spatial orientation.
  • A sample problem on determining sublevel names and orbital quantum numbers.