Three Chapter 9

Wave Mechanical Model of the Hydrogen Atom

• With Bohr's quantum hypothesis, it became possible to combine classical mechanics with quantum wave mechanics, to produce a satisfactory model of atomic structure.
• This model allows us to:
  • explain existing observations.
  • predict new observation.
• Given the proton has a mass of m_p and charge of e⁺ , and the electron has a mass of m_e and charge of e^-, with radius of orbit r_n, and velocity v_n, then if n equals the number of de Broglie wavelengths in the nth orbit, we can write:

F_c = F_e

(m_e(v_n)²)/r_n = (ke²)/(r_n)² equation 1

from 2pr_n = (nh)/(m_ev_n) we can get v_n = (nh)/(2pm_er_n) equation 2

• Substituting v_n (that is equation 2) into equation 1 above and rearranging gives:

r_n = (n²h²)/(4p²m_eke²) equation 3

• r_n = nth radius of the hydrogen atom.
• By substituting in values for h, p , m_e, k, and e gives:

r_n = (5.3 X 10^-11m)n²

• For the smallest orbit, n = 1 and r₁ = 5.3 X 10^-11m (called the Bohr radius).
• The Bohr radius matches closely the value of the measured size of the hydrogen atom.
• By substituting equation 3 into the equation for v_n (equation 2) we find:

v_n = (nh)/(2pm_e((n²h²)/(4p²km_ee²)))

• Rearranging this expression gives:

v_n = (2pke²)/(nh) equation 4

• Substituting in p , k, e, and h gives:

v_n = (2.2 X 10⁶m/s)/n

• This expression allows us to calculate the velocity of the electron in its various orbits.
• The energy in the nth orbit would be the sum of the potential energy and the kinetic energy.

E_n = E_p + E_k

E_n = ((1/2)m_e(v_n)²) + ((-ke²)/r_n) equation 5

• Substituting the general expressions for the radius and speed of the electron (equation 3 and equation 4) into equation 5 gives:

E_n = (1/2)m_e((2pke²)/(nh))² - (ke²)/((n²h²)/(4p²km_ee²))

• Expanding and rearranging gives:

E_n = (-2p²m_ek²e⁴)/(n²h²)

• Substituting for the constants gives:

E_n = (-2.17 X 10^-18J)/n² = (-13.6 eV)/n²

Therefore E₁ = -2.17 X 10^-18J = -13.6 eV

• Remember that E_n = -Rhc/n². Substituting information for the first energy level into this expression allows us to solve for Rydberg's constant.

-2.17 X 10^-18J = -R(6.63 X 10^-34Js)(3.00 X 10⁸m/s)/(1)²

or R = 1.09 X 10⁷m^-1

• This calculation of Rydberg's constant gave Bohr confidence enough to publish his work.
• This wave - mechanical model for the hydrogen atom matches exactly the evidence from the atom's emission spectrum. We can calculate the energies for the various stationary states, and then calculate:

DE = E₂ - E₁

and compare to the emission spectrum.

• We can now draw a complete energy level diagram for hydrogen.

• The electron in the hydrogen atom normally will be in its ground state. When absorbing energy it moves up the energy levels and away from the nucleus. Once raised to a higher energy, it loses this energy and falls down the energy levels and towards the nucleus.
• When an electron loses energy (falls towards the nucleus), light is given off (A photon is emitted).
• When an electron gains energy (moves away from the nucleus), kinetic energy from a particle is absorbed, or energy from a photon is absorbed and the photon disappears.
• The vertical lines in the energy level diagram represent groups of transitions based on the final destination of the electron. They are named after the scientist who discovered them by experiment.
• Bohr's theory was successful in that:
  1. It provided a physical model of the atom, whose internal energy levels matched those of the observed hydrogen spectrum.
  2. It accounted for the stability of atoms.
  3. It applied equally well to other one electron atoms such as a singly ionized helium ion.
• Bohr's theory failed in that:
  1. It broke down when applied to many electron atoms, because it took no account of the interactions between electrons in orbit.
  2. With the development of more precise spectroscope techniques, it became apparent that each of the excited states was not a unique single energy level, but a group of finely separated levels.
• The finely separated levels required explanations which involved the introduction of:
  1. modifications to the shape of the Bohr orbits.
  2. electron spin.
• The wave mechanical model of the atom has been able to resolve all of the problems associated with the Bohr model of the atom.

January 22, 2014