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.