Atomic Forces, Electron Potential Energy, & Photoelectron Spectroscopy

Coulomb’s Law

  • Governs the electrostatic force between two charged particles (e.g., a nucleus and an electron)
    • Quantitative form: F = k\frac{q1 q2}{r^2}
    • F = magnitude of electrostatic force (N)
    • k = Coulomb’s constant \approx 8.99 \times 10^9\, \text{N·m}^2\text{/C}^2
    • q1,\,q2 = charges of interacting particles (C)
    • r = distance between charge centers (m)
  • Key qualitative take-aways (no calculator required in this course):
    • Force is directly proportional to the product of the two charges.
    • Bigger nuclear charge (+Z) ⇒ larger attractive force on every electron.
    • Force is inversely proportional to the square of the separation.
    • Electrons closer to the nucleus experience dramatically stronger attraction.
  • Predictive shortcuts:
    • Doubling the nuclear charge (e.g., comparing \text{He}^{+} and \text{H} at the same radius) doubles the force.
    • Doubling the distance cuts the force to one-quarter (inverse-square relationship).
  • Conceptual relevance to periodic trends:
    • Higher effective nuclear charge (Z_{\text{eff}}) across a period pulls electrons in more tightly, decreasing atomic radius and increasing ionization energy.
    • Additional shells down a group increase r, weakening the pull and lowering ionization energy.

Potential Energy of Electrons in Atoms

  • Because energy and force are directly related, Coulomb’s Law also predicts potential energy (PE) of an electron.
    • Qualitatively: |PE| \propto \frac{Z}{r} (larger nuclear charge, smaller radius ⇒ more extreme potential energy).
  • Sign conventions:
    • Electron’s charge is negative; therefore the potential energy of an electron bound to a positive nucleus is negative.
    • More negative value = lower (more stable) energy state.
    • Electrons in the 1s shell have a large-magnitude negative PE (most stable, hardest to remove).
  • Binding energy (a.k.a. ionization energy for that electron):
    • Defined as the positive amount of energy needed to overcome the attractive PE.
    • Numerically equal to |PE| (the absolute value).
    • Units: electronvolts for single electrons, \text{MJ·mol}^{-1} for macroscopic samples.
  • Practical hierarchy:
    • Inner (core) electrons → very negative PE → very high binding energy.
    • Valence electrons → less negative PE → lower binding energy → chemically relevant.

Photoelectron Spectroscopy (PES)

  • Objective: Map the binding energies of all electrons in an atom to reveal its electronic structure.
  • Principle of operation:
    1. Irradiate a gaseous sample of atoms with photons of known energy E_{\text{photon}}.
    • Photon energy formula (not in original transcript but fundamental): E_{\text{photon}} = h\nu = \frac{hc}{\lambda}.
      • h = Planck’s constant 6.626 \times 10^{-34}\,\text{J·s}.
      • \nu = frequency (Hz); \lambda = wavelength (m).
      • Shorter \lambda ⇒ higher \nu ⇒ higher photon energy.
    1. If E_{\text{photon}} exceeds the binding energy (IE) of a particular electron, that electron is ejected (photoelectric effect).
    2. Measure the kinetic energy (KE) of the ejected electron.
  • Conservation of energy relationship used in analysis: E{\text{photon}} = KE{e^-} + IE
    • Rearranged to find the ionization energy:
      IE = E{\text{photon}} - KE{e^-}
    • Faster ejected electron (higher KE) means the starting electron required less binding energy ⇒ it came from further away (larger initial r).
  • Experimental outputs:
    • Plot of “Relative # of Electrons” vs. “Ionization Energy.”
    • Each peak corresponds to electrons removed from a specific shell or subshell.
    • Area/height of a peak ∝ number of electrons in that subshell.

Reading a Typical PES Spectrum (Example: Oxygen)

  • Two distinct binding-energy regions (large gap between them) ⇒ two principal energy levels (n=1 and n=2 for oxygen).
  • Within the n = 2 region, two separate peaks appear:
    • Lower IE peak (fewer eV) → 2p subshell.
    • Slightly higher IE peak → 2s subshell.
    • Evidence for existence of subshells (s vs. p) even before modern quantum theory.
  • Peak heights convey electron count:
    • 1s peak height represents 2 electrons.
    • 2s peak height also represents 2 electrons.
    • 2p peak is about twice as tall, corresponding to 4 electrons in oxygen’s 2p.
    • Total electron count from all peak areas must match atomic number (8 for O).
  • Practical insights:
    • Comparing elements: shifting of peaks toward higher IE signals increasing nuclear charge without extra shielding (across a period).
    • Core peak positions change little across a period (shielded by same inner shells), whereas valence peaks shift significantly.

Conservation of Energy in PES Experiments

  • Energy bookkeeping for each photon-electron event:
    • Input: E_{\text{photon}} (controlled by the experimenter via light source).
    • Outputs: KE_{e^-} (measured) + IE (calculated).
  • Units:
    • KE often reported in eV; convert with 1\,\text{eV} = 1.6 \times 10^{-19}\,\text{J} if SI is required.
    • For moles of electrons, multiply by Avogadro’s number to obtain \text{MJ·mol}^{-1}.
  • Distance inference:
    • Greater KE (after subtracting IE) isn’t physically possible—so changes in KE directly reflect differences in IE (i.e., binding energy) arising from different starting radii.

Connections & Broader Context

  • Historical significance:
    • PES offered experimental confirmation of quantum shells/subshells before high-resolution quantum calculations were accessible.
  • Relation to other periodic trends:
    • Data parallels first ionization energy values found in macroscopic experiments (e.g., flame tests, electrical discharge).
    • Supports concepts of effective nuclear charge, shielding, and penetration.
  • Real-world applications:
    • Surface analysis in materials science (X-ray PES/Auger spectroscopy) to determine elemental composition and chemical states.
    • Semiconductor research—probing valence band structures.
  • Ethical / safety note:
    • High-energy photons (UV, X-ray) used in PES can be hazardous; proper shielding and protocols are essential.

Quick Reference & Formulas

  • Coulomb’s law: F = k\frac{q1 q2}{r^2}
  • Photon energy: E = h\nu = \frac{hc}{\lambda}
  • PES energy balance: E{\text{photon}} = KE{e^-} + IE
  • Electronvolt conversion: 1\,\text{eV} = 1.6 \times 10^{-19}\,\text{J}
  • Binding energy ≡ |PE| for that electron.

Study Tips

  • Practice qualitative predictions: Given two electrons (different Z or r), decide which has the higher |PE| or IE.
  • Interpret PES plots: Identify number of peaks → shells; relative heights → electron counts; energy positions → relative binding energies.
  • Connect PES data to periodic table: Track movement of valence peaks across periods and down groups.
  • Remember sign conventions: Potential energies of bound electrons are negative; binding/ionization energies are positive magnitudes.