Quantum Physics and Atomic Models II - Notes

Thomson and Rutherford Models of the Atom

• Early Atomic Views:

  • Around 400 B.C., Greek philosophers viewed atoms as indivisible objects with shapes determining physical properties (e.g., slippery water atoms, solid iron atoms).
  • In 1802, John Dalton proposed elements consist of specific, indivisible atoms that combine and cannot be altered chemically, based on experimental research.

• Thomson's Plum Pudding Model:

  • Thomson's discovery of the electron provided the first evidence of the atom's underlying structure.
  • In 1904, Thomson proposed the "plum pudding" model, where electrons move within a mass of positive charge.
  • The model failed to explain the different wavelengths of light emitted by excited atoms.

• Rutherford Model:

  • Ernest Rutherford, along with Hans Geiger and Ernest Marsden, discovered that positive charge is concentrated in a tiny nucleus of diameter 10^{-15} m.
  • They aimed alpha particles at gold foil and observed some were deflected, even bounced back, which contradicted the plum pudding model.

• Rutherford's Interpretation:

  • Most of the atom is empty space, with the positive charge concentrated in the nucleus.
  • Alpha particles that struck the nucleus were deflected.
  • Rutherford proposed electrons orbit the nucleus like planets around the Sun, drawing an analogy from the force equations for gravity and the electric force.

• Scale of the Atom (Rutherford Model):

  • If the nucleus were the size of a baseball, the atom would have a radius of 4 km.
  • Electrons would be the size of a period at the end of a sentence.
  • Rutherford's model could not explain the different wavelengths of light emitted by excited atoms.

• Review Questions

  • The plum pudding model was important because it was the first model that illustrated the underlying structure of the atom.
  • The gold foil experiment performed under Ernest Rutherford's direction led to the discovery of the atomic nucleus.
  • In the Rutherford Nuclear atom model, the heavy part of the atom is very small and is surrounded by electrons.

Bohr Model

• Bohr's Application of Quantum Concepts:

  • Niels Bohr applied Einstein's concept of quantized photons to the Rutherford model to explain optical spectra for hydrogen-like atoms (atoms with one electron).
  • Bohr's model, introduced in 1913, explained why hydrogen emits many wavelengths of light when excited.

• Optical Spectra:

  • White light passed through a diffraction grating produces a rainbow because it contains all wavelengths.
  • Gases emit light when a high voltage is applied; each element has a unique spectrum of frequencies.
  • The Rutherford model could not explain this phenomenon.

• Hydrogen Spectra:

  • The hydrogen spectrum has been well-studied, and equations were derived experimentally to fit the spectra.
  • Johann Balmer created an equation in 1885 for the visible spectral lines of hydrogen.
  • Johannes Rydberg expanded the equation in 1888 to include ultraviolet and infrared lines.

• Rydberg Constant:

  • The constant in these equations was later named after Rydberg.
  • These equations exactly match the experimentally found hydrogen spectra lines (Balmer, Lyman, Paschen series).

• Bohr's Postulates:

  • In 1913, Bohr proposed electrons orbit the nucleus in specific energy levels.
  • Electrons jump to higher energy orbits when they absorb energy and release energy as light when returning to their original orbit.
  • The energy of the emitted light is related to its frequency by E = hf, where h is Planck's constant and f is the frequency.
  • Orbits are quantized, so only unique light frequencies can be emitted. The emitted light frequency must equal the energy difference between the orbits.
  • Orbits are unique to each element.

• Planetary Orbit Problem:

  • Electrons are kept in orbit by the Coulomb force providing the centripetal force.
  • Classical electromagnetic theory states accelerating charges emit radiation; thus, electrons should lose energy, slow down, and collapse into the nucleus within 10^{-11} s.
  • This collapse does not happen.

• The Bohr Atom:

  • An electron is held in orbit by the Coulomb force.
  • Z represents the number of protons.
  • e represents the electron charge, 1.6 \times 10^{-19} C.
  • +Ze is the charge of the nucleus.

• Bohr's Solution and Assumptions:

  • Bohr's model is semi-classical, combining classical physics with quantum aspects.
  • First Assumption: Electrons revolve around the nucleus in specific circular orbits with fixed angular momenta and energy.
  • Second Assumption: Energy is emitted as light when electrons move between orbits.

• Implications of Bohr's Assumptions:

  • Electrons exist only in specific locations around the nucleus.
  • The further the electron is from the nucleus, the greater its momentum and energy.
  • Electrons in these orbits do not radiate electromagnetic energy.

• Quantization of Energy:

  • Since orbits are fixed, the emitted energy is also fixed and quantized.
  • Only photons of particular energy, representing the energy difference between allowed orbits, are emitted.

• Electron Orbit Model:

  • Electrons can exist in specific orbits (e.g., n=1, 2, 3).
  • When an electron falls from a higher energy level (greater n) to a lower energy level, it emits a photon with frequency f proportional to the energy difference between the levels.

• Review Questions

  • The Hydrogen spectra were first explained by fitting equations to experimental data and Bohr's theory.
  • The Balmer Series of Hydrogen spectral lines are in the visible section of the electromagnetic spectrum.
  • Bohr used a similar explanation for the Hydrogen spectra, that Einstein used for the Photoelectric Effect, due to Excited Hydrogen atoms emitting light of specific frequencies.
  • Bohr assumed that When an electron was in an allowed orbit, it would not radiate electromagnetic energy, violating previous work in electromagnetic theory.

Bohr Model Calculations

• Mathematical Foundation:

  • Quantized angular momentum: L = n\hbar, where n is the energy level number.
  • Coulomb force: Provides the centripetal force.

• Radius of Quantum Levels:

  • r = \frac{n^2 h^2}{4 \pi^2 m k Z e^2}

• Energy of Quantum Levels:

  • Calculated using classical equations for kinetic energy, potential energy, angular momentum, Coulomb's Law, Newton's Second Law, and Bohr's quantum assumption.

  • E = -\frac{Z^2 e^4 m k^2}{2 h^2} \frac{1}{n^2}

• Bohr Radius:

  • Combining constants yields the Bohr radius, a_0.
  • a_0 = \frac{h^2}{4 \pi^2 m k e^2} = 5.29 \times 10^{-11} m
  • Represents the electron orbital radius for the ground state (n=1) of Hydrogen (Z=1).
  • Radii of other Hydrogen-like atoms: r = a_0 \frac{n^2}{Z}, where n is the energy level and Z is the number of protons.

• Energy Equation with Constants:

  • E = -13.6 eV \frac{Z^2}{n^2}
  • Energy levels are negative, indicating electrons are bound in the atom.
  • Levels get closer together as n increases.

• Agreement with Optical Spectra:

  • Bohr's calculations agree with observed optical spectra.
  • Transitions between quantum levels are named after the physicists who observed them.
  • The Balmer series corresponds to transitions in the visible spectrum.

• Ground and Excited States:

  • The lowest energy level (n=1) is the ground state; others are excited states.

• Bohr Model's Significance:

  • Critical step in transitioning from classical to quantum world views.
  • Used classical equations to explain quantum phenomena.
  • Required the assumption that accelerating charges in specific orbits do not emit electromagnetic radiation.

• Limitations:

  • Applicable only to hydrogen-like atoms (single electron).
  • Based on the assumption that accelerating charges in specific orbits do not emit electromagnetic radiation.
  • Predicts photon frequencies but not intensities.

• Example Calculations:

  Example 1: What is the radius of a hydrogen atom with an electron in the second energy level?

  Given: n = 2, Z = 1, a_0 = 5.29 \times 10^{-11} m

  Solution: r = a_0 \frac{n^2}{Z} = 5.29 \times 10^{-11} m \cdot \frac{2^2}{1} = 2.12 \times 10^{-10} m

  Example 2: What is the energy of an electron in the second energy level of a hydrogen atom?

  Given: n = 2, Z = 1

  Solution: E = -13.6 eV \frac{Z^2}{n^2} = -13.6 eV \cdot \frac{1^2}{2^2} = -3.4 eV

  Example 3: How much energy must be absorbed by an electron transitioning from the ground state to the fifth energy level?

  Solution: E = -13.6 eV \frac{Z^2}{n^2}. Since we are using hydrogen (Z = 1), we will begin with
  E1 = -13.6 eV (\frac{1^2}{1^2}) = -13.6 eV E5 = -13.6 eV (\frac{1^2}{5^2}) = -0.544 eV
  Subtract E5 from E1 to get the change in energy, or the energy that must be absorbed: E = -0.544 eV - (-13.6 eV)= 13.056 eV

  Example 4: What is the wavelength of the photon emitted when an electron in a hydrogen atom makes the transition from n = 6 to n = 2?

  Solution: First find E2 and E6:
  En = -13.6eV \frac{Z^2}{n^2} E2 = -13.6eV \frac{1^2}{2^2} = -3.4 eV
  E6 = -13.6eV \frac{1^2}{6^2} = -0.378 eV Subtract E6 from E_2 to find the change in energy.

  E = -3.4 eV - (-0.378 eV) = -3.022 eV
  The electron loses energy when it moves to a lower energy state, but the photon gains this energy, so it is positive. Use the energy of the photon and E = \frac{hc}{\lambda} to solve for the wavelength of the photon. Since energy is in eV, be sure to use h in eV also.

  E = \frac{hc}{\lambda} => \lambda = \frac{hc}{E} = \frac{4.14 x 10^{-15} eVs * 3 x 10^8 m/s}{3.022 eV} = 4.11 x 10^{-7} m = 411 nm

  Example 5: A hypothetical atom has energy levels that correspond to the following equation: What are the first four energy levels for this atom?
  Since this problem states "hypothetical atom," we do not have to use the formal equation for the energy levels, we will use the equation given for the energy levels.
  En = -\frac{24 eV}{n^2} E1 = -\frac{24 eV}{1^2} = -24 eV
  E2 = -\frac{24 eV}{2^2} = -6 eV E3 = -\frac{24 eV}{3^2} = -2.67 eV
  E_4 = -\frac{24 eV}{4^2} = -1.5 eV

  The energy of the second excited state (n=3) of the Hydrogen atom is -1.51 eV.

  The energy of the fifth excited state (n=6) of the Hydrogen atom? is -0.378 eV.

  The frequency of the photon emitted when an electron falls from the n=6 level to the n=3 level is 2.73x10^{14} Hz.