14.4 The Old Quantum Theory 651
14.24a.Derive an expression for the period of the electronic
motion in the Bohr theory (the time required for an
electron to make one circuit around a Bohr orbit) as a
function ofn.
b.Find the value of the period and the frequency (the
reciprocal of the period) forn1 and for
n1,000,000.
14.25Find the radius of then 1. 00 × 106 orbit of the
electron in the hydrogen atom according to the Bohr
theory.
14.26The Balmer series of hydrogen atom spectral lines
corresponds to transitions from higher values ofnto
n2 in the Bohr energy expression. Find the
wavelengths in vacuum of all lines in the Balmer series
that lie in the visible region.
14.27Find the wavelengths of the first six lines in the hydrogen
atom spectrum corresponding to transitions ton1 (the
Lyman series). In what region of the electromagnetic
spectrum do these lines lie?
14.28Find the wavelengths of the first six lines in the hydrogen
atom spectrum corresponding to transitions ton3. In
what region of the electromagnetic spectrum do these
lines lie?
14.29In order to visualize the approximation that the nucleus
is considered stationary while the electron moves around
it, assume that a hydrogen atom is moving through space
at a speed of 2500 m s−^1 , roughly the average speed
predicted by gas kinetic theory for a hydrogen atom at
room temperature. Calculate how many times the
electron in then1 Bohr orbit goes around the nucleus
in the time required for the atom to move a distance
equal to the Bohr radius.
14.30We can correct for the assumption that the nucleus of a
hydrogen atom is stationary by replacing the mass of the
electron by the reduced mass of the electron and nucleus
defined byμmnme
mn+me
wheremnis the mass of the nucleus. See Appendix E for
details.
a.Find the reduced mass of the nucleus and electron.
b.Find the value of the Bohr radius with this correction.
c.Find the expression for the energy of a hydrogen
atom in joules and in electron volts with this
correction.d.Find the value of Rydberg’s constant with this
correction.
14.31A positronium atom is a hydrogen-like atom with a
nucleus consisting of a positron (an antiparticle with
chargeeand mass equal to that of the electron).
a.Find the value of the reduced mass of the two
particles in a positronium atom and find the ratio of
this reduced mass to the mass of an electron.
b.Find the value of the Bohr radius for positronium.
c.Find the energy of then1 state of positronium,
and find the ratio of this energy to that of a hydrogen
atom.
d.Find the radius of the circle in which each particle
moves around the center of mass for then1 state
according to the Bohr theory.
14.32 a.Find the reduced mass of the two particles in a
deuterium atom (^2 H). The mass of a deuterium atom
in amu is listed in the appendix.
b.Find the energy of a deuterium atom in the 1s state
and find the ratio of this energy to that of a hydrogen
atom.
c.Find the energy of a He+ion in the 1s state and find
the ratio of this energy to that of a hydrogen
atom.
14.33The gravitational potential energy between two objects is
equal toVgG
m 1 m 2
r 12whereGis the gravitational constant, equal to
6. 673 × 10 −^11 m^3 kg−^1 s−^2 and wherem 1 andm 2 are
the masses of the two objects. Since this has the same
dependence onr 12 as does the Coulomb potential energy,
the Bohr theory can be transcribed to apply to a
hydrogen atom held together by gravity instead of
electrostatic attraction.
a.Pretend that the charges on a proton and electron can
be somewhat “turned off.” Find the Bohr radius for a
hydrogen atom with only the gravitational attraction
between the proton and the electron, assuming the
actual proton and electron masses.
b.Find the energy of then1,n2, andn3 states
of such an atom. Express your answer in joules and in
electron volts.