Rutherford, Planck, and Bohr Models

Rutherford, Planck, and Bohr

  • Rutherford (1910):

    • Provided experimental evidence that an atom has a dense, positively charged nucleus. This nucleus accounts for only a small portion of the atom's total volume.

  • Planck:

    • Quantum Theory: Developed the quantum theory proposing that energy emitted as electromagnetic radiation comes in discrete bundles called quanta.
    • Planck Relation: The energy of a quantum is given by E = hf, where:
      • E is the energy.
      • h is Planck's constant, h = 6.626 \times 10^{-34} \text{ Js}.
      • f (or \nu) is the frequency of the radiation.

Bohr Model (1913)

  • Niels Bohr's Model:

    • Used Rutherford's and Planck's work to develop a model for the electronic structure of the hydrogen atom.
    • Assumptions:
      • The hydrogen atom consists of a central proton around which an electron travels in a circular orbit.
      • The centripetal force on the electron is provided by the electrostatic force between the proton and electron.
    • Correction of Classical Physics: Applied Planck's quantum theory to correct classical physics assumptions about electron pathways.

  • Classical Mechanics Postulates:

    • An object revolving in a circle (like an electron) can have an infinite number of values for its radius and velocity.
    • Angular momentum l = mvr and kinetic energy k = \frac{1}{2}mv^2 could take on any value.

  • Bohr's Restrictions:

    • Incorporated Planck's quantum theory to restrict the possible values of angular momentum.
    • Quantized Angular Momentum: l = \frac{nh}{2\pi}, where:
      • n is the principal quantum number (any positive integer).
      • h is Planck's constant.
    • Angular momentum changes only in discrete amounts with respect to the principal quantum number.

  • Energy of the Electron:

    • Related permitted angular momentum values to the energy of the electron: E = \frac{R_H}{n^2}, where:
      • RH is the Rydberg unit of energy, RH = 2.18 \times 10^{-18} \text{ J}.
    • The energy of the electron changes in discrete amounts with respect to the quantum number.
    • Zero energy is assigned when the proton and electron are separated (no attractive force).
    • The negative sign indicates an attractive force between the electron and proton in quantized states.
    • The energy of an electron increases (becomes less negative) as it is located further from the nucleus (increase in n).

  • Quantized Energy Analogy:

    • Analogous to changes in gravitational potential energy when ascending/descending stairs.
    • Unlike a ramp (continuous potential energy changes), a staircase allows only certain discrete, quantized changes in potential energy.

  • Bohr's Description of Hydrogen Atom:

    • A nucleus with one proton around which a single electron revolves in a defined orbit at a discrete energy value.
    • Electrons can jump from one orbit to a higher energy one if an amount of energy exactly equal to the difference between the two orbits is transferred.
    • Orbits have increasing radii; the orbit with the smallest, lowest energy radius is the ground state (n = 1).

  • Ground State: The state of lowest energy in which all electrons are in the lowest possible orbitals.

  • Excited State: When an electron is promoted to an orbit with a larger radius (higher energy).

    • An atom is in an excited state when at least one electron has moved to a subshell of higher than normal energy.

  • Analogy to Planets Orbiting the Sun: Each planet travels along a roughly circular pathway at set distances and energy values from the sun.

  • Reconsideration of Bohr's Model: The model was later reconsidered, but remains an important conceptualization of atomic behavior.

    • Electrons are not restricted to specific pathways but are localized in certain regions of space.

Applications of the Bohr Model

  • Useful for explaining the atomic emission and absorption spectra of hydrogen atoms and other one-electron systems (e.g., \text{He}^+ and \text{Li}^{2+}).

Atomic Emission Spectra

  • Excitation: Electrons can be excited to higher energy levels by heat or other energy forms, yielding excited states.

  • Emission: Electrons rapidly return to the ground state, emitting discrete amounts of energy in the form of photons.

  • Photon Energy: The electromagnetic energy of these photons is determined by E = \frac{hc}{\lambda}, where:

    • h is Planck's constant.
    • c is the speed of light in a vacuum (3.00 \times 10^8 \text{ m/s}).
    • \lambda is the wavelength of the radiation.

  • Combination of Equations: The equation E = \frac{hc}{\lambda} combines E = hf and c = f\lambda.

  • Quantized Transitions: Energy transitions do not form a continuum but are quantized to certain values.

  • Line Spectrum: Each line on the emission spectrum corresponds to a specific electron transition.

  • Unique Spectra: Each element has a unique atomic emission spectrum, which can be used as a fingerprint for the element, because electrons can be excited to a different set of distinct energy levels.

  • Application: Analysis of stars and planets: light from a star is resolved into its component wavelengths and matched to known line spectra of elements.

  • Hydrogen Spectrum: The Bohr model explained the atomic emission spectrum of hydrogen.

  • Lyman Series: Transitions from energy levels n \geq 2 to n = 1. Larger energy transitions with shorter photon wavelengths in the UV region.

  • Balmer Series: Transitions from energy levels n \geq 3 to n = 2. Includes four wavelengths in the visible region.

  • Paschen Series: Transitions from n \geq 4 to n = 3.

  • Energy of Emitted Photon:

    • The energy associated with the change in the principal quantum number from a higher initial value ni to a lower final value nf is equal to the energy of the photon predicted by Planck's quantum theory.
    • E = \frac{hc}{\lambda} = RH \left( \frac{1}{nf^2} - \frac{1}{n_i^2} \right)
    • The energy of the emitted photon corresponds to the difference in energy between the higher energy initial state and the lower energy final state.

Atomic Absorption Spectra

  • Energy Absorption: When an electron is excited to a higher energy level, it must absorb exactly the right amount of energy to make that transition.

  • Unique Absorption Spectra: Every element possesses a characteristic absorption spectrum.

  • Correspondence of Wavelengths: The wavelengths of absorption correspond exactly to the wavelengths of emission because the difference in energy between levels remains unchanged.

    • Identification of elements in the gas phase requires absorption spectra.

  • Key Takeaway: Each element has a characteristic set of energy levels.

    • Electrons must absorb the right amount of energy (in the form of light) to move from a lower energy level to a higher energy level.
    • Electrons emit the same amount of energy (in the form of light) when they move from a higher energy level to a lower energy level.