9.21.Integrate Planck’s law (equation 9.23) from the wave-
length limits 0 to to get equation 9.24. You will
have to rewrite the expression by redefining the variable (and
its infinitesimal) and use the following integral:
0exx
3
1 dx 1
54
(9.43)9.22.Calculate the power of light in the wavelength range
350–351 nm (that is, let d be 1 nm in Planck’s law,
and let be 350.5 nm) at temperatures of 1000 K, 3000 K,
and 10,000 K.
9.23.Verify that the collection of constants in equation 9.24
reproduces the correct (or close to it) value of the Stefan-
Boltzmann constant.
9.24.Work functions are typically given in units of electron
volts, or eV. 1 eV equals 1.602
10 ^19 J. Determine the min-
imum wavelength of light necessary to overcome the work
function of the following metals (“minimum” implies that the
excess kinetic energy, ^12 mv^2 , is zero): Li, 2.90 eV; Cs, 2.14 eV;
Ge, 5.00 eV.
9.25.Determine the speed of an electron being emitted by
rubidium (2.16 eV) when light of the following wave-
lengths is shined on the metal in vacuum: (a)550 nm,
(b)450 nm, (c)350 nm.
9.26.The photoelectric effect is used today to make light-
sensitive detectors; when light hits a sample of metal in a
sealed compartment, a current of electrons may flow if the
light has the proper wavelength. Cesium is a desirable com-
ponent for such detectors. Why?
9.27.Calculate the energy of a single photon in joules and
the energy of a mole of photons in J/mol for light having
wavelengths of 10 m (radio and TV waves), 10.0 cm (mi-
crowaves), 10 microns (infrared range), 550 nm (green light),
300 nm (ultraviolet), and 1.00 Å (X rays). Do these numbers
explain the relative danger of electromagnetic radiation of dif-
fering wavelengths?
9.9 Bohr’s Theory of Hydrogen
9.28.Show that both sides of equation 9.27 reduce to units
of force, or N.
9.29.Use equation 9.34 to determine the radii, in meters and
angstroms, of the fourth, fifth, and sixth energy levels of the
Bohr hydrogen atom.
9.30.Calculate the energies of an electron in the fourth, fifth,
and sixth energy levels of the Bohr hydrogen atom.
9.31.Calculate the angular momenta of an electron in the
fourth, fifth, and sixth energy levels of the Bohr hydrogen
atom.
9.32.Show that the collection of constants given in equation
9.40 gives the correct numerical value of the Rydberg
constant.
9.33.Equations 9.33 and 9.34 can be combined and re-
arranged to find the quantized velocity of an electron in the
Bohr hydrogen atom. (a)Determine the expression for the ve-
locity of an electron. (b)From your expression, calculate the
velocity of an electron in the lowest quantized state. How does
it compare to the speed of light? (c2.9979
108 m/s)
(c) Calculate the angular momentum Lmvrof the electron
in the lowest energy state of the Bohr hydrogen atom. How
does this compare with the assumed value of the angular mo-
mentum from equation 9.33?
9.34. (a)Compare equations 9.31, 9.34, and 9.41 and pro-
pose a formula for the radius of a hydrogen-like atom that has
atomic charge Z. (b)What is the radius of a U^91 ion if the
electron has a quantum number of 100? Ignore any possible
relativistic effects.9.10 The de Broglie Equation
9.35.The de Broglie equation for a particle can be applied to
an electron orbiting a nucleus if one assumes that the electron
must have an exact integral number of wavelengths as it cov-
ers the circumference of the orbit having radius r: n 2 r.
From this, derive Bohr’s quantized angular momentum
postulate.
9.36.What is the wavelength of a baseball having mass
100.0 g traveling at a speed of 160 km/hr? What is the wave-
length of an electron traveling at the same speed?
9.37.What velocity must an electron have in order to have a
de Broglie wavelength of 1.00 Å? What velocity must a pro-
ton have in order to have the same de Broglie wavelength?9.38.Plot Planck’s law of energy density versus wavelength at
various temperatures. Integrate it to show that you can get the
Stefan-Boltzmann law and constant.
9.39.Determine under what conditions of temperature and
wavelength the Rayleigh-Jeans law approximates Planck’s law.
9.40.Using the second-order differential equation for the
motion of a harmonic oscillator, find solutions to the equation
and plot them versus time.
9.41.Construct a table of the first 50 lines of the first six se-
ries of the hydrogen atom spectrum. Can you predict the se-
ries limit in each case?272 Exercises for Chapter 9
Symbolic Math Exercises