22.1 The Translational States of Atoms 917
EXAMPLE22.1
The energy difference between the ground state and first excited electronic level of a
hydrogen atom is 10.2 eV. Compare this energy difference with the spacing between the
ground state and first excited translational level of a hydrogen atom in a cubical box
0.100 m on a side.
Solution
The ground translational state corresponds tonxnynz1, which we denote by (111).
The first excited level consists of the states denoted by (112), (121), and (211), so the energy
difference is∆EE 112 −E 111
h^2
8 ma^2(6−3)(
6. 6261 × 10 −^34 Js) 28(
1. 674 × 10 −^27 kg)
( 0 .100 m)^2( 6 − 3 )(
1eV
1. 6022 × 10 −^19 J) 6. 14 × 10 −^20 eVwhich is smaller than the electronic excitation energy by a factor of approximately 10−^20.Exercise 22.1
Calculate the difference in energy between the ground state and the first excited translational
level of a xenon atom in a box 0.100 m on each side. Express it in joules and in electron
volts. Compare it with the corresponding value for a hydrogen atom in Example 22.1, and
also compare it with the excitation energy to the first excited electronic level of the xenon atom,
8.315 eV.Since the translational energy levels are very closely spaced, the translational quan-
tum numbers of the states occupied at room temperature can be very large.EXAMPLE22.2
According to gas kinetic theory, the average translational energy of the atoms of a gas is
equal to〈Etr〉
3 kBT
2(22.1-5)wherekBis Boltzmann’s constant, 1. 3807 × 10 −^23 JK−^1 , and whereTis the absolute
temperature.
a.For a hydrogen atom in the box of Example 22.1 with translational energy equal to 3kBT/ 2
at 298 K, find the value of the three translational quantum numbers, assuming them to be
equal to each other.
b.Find the change in energy if one of the translational quantum numbers is increased by
unity from its value in part a. Find the ratio of this change to the translational energy and
to the energy of the first excited electronic state, 10.2 eV.