22.2 The Nonelectronic States of Diatomic Molecules 929
c.
1
λ
∆E
hc
E 11
hc−
E 00
hc ̃νe− 2 ̃Be 2169 .8cm−^1 +2(1.931 cm−^1 ) 2173 .7cm−^1
λ 4. 6005 × 10 −^4 cm 4. 6005 μmν
c
λ
2. 9979 × 1010 cm s−^1
4. 6005 × 10 −^4 cm 6. 5164 × 1013 s−^1PROBLEMS
Section 22.2: The Nonelectronic States of Diatomic
Molecules
22.6 Compare the energy level spacings for a Cl 2 molecule in
a cubical box 0.200 m on a side. Its vibration frequency is
1. 694 × 1013 s−^1 and its equilibrium bond length for the
ground electronic state is 1. 988 × 10 −^10 m. The energy of
the first excited electronic state observed in the gas phase
is 2.162 eV above that of the ground electronic state.22.7a.Calculate the reduced mass of the H 2 molecule.b.From data in Table A.22 of the appendix, calculate the
value of the force constant for the H 2 molecule.c.Make an accurate graph of the vibrational wave
function for the ground state of the H 2 molecule in the
rigid-rotor-harmonic-oscillator approximation. Use
picometers (pm) for the scale on the horizontal axis
and use arbitrary units on the vertical axis. Comment
on the difference between this graph and that of a
harmonic oscillator wave function.22.8 a.Calculate the rotational energy of a hydrogen
molecule in theν0,J1 state and in the
ν0,J2 state. Make the calculation once without
correction terms and once with correction terms.b.Repeat the calculation of part a for HD(^1 H^2 H) without
the correction terms.22.9 Find the value of the rotational quantum numberJif a
nitrogen molecule has a rotational energy equal tokBTat
298.15 K. Find the degeneracy of this level.22.10Find a formula for the rotational frequency (number of
revolutions per second) of a rigid diatomic molecule
assuming that classical mechanics holds, but that the
angular momentum has the magnitudeh ̄
√
J(J+1).Compare this with the frequency of a photon absorbed
when a quantum-mechanical molecule makes a transition
fromJtoJ+1. Show that the two frequencies are
nearly equal for large values ofJ.
22.11Using information on the normal H 2 molecule, find the
frequency of vibration of the HD molecule, where D is
deuterium,^2 H. Compare with the vibrational frequency
of normal H 2.
22.12Using information on the normal H 2 molecule, find the
frequency of vibration of the D 2 molecule, where D is
deuterium,^2 H. Compare with the vibrational frequency
of normal H 2.
22.13a.Using the expression of Eq. (22.2-50), find the
wavelength and frequency of the light absorbed
when carbon monoxide molecules make the
transition from theν0,J1 state to the
ν1,J0 state.
b.Find the wavelength for the same transition,
neglecting the terms in ̃αand ̃D.
c.Find the wavelength for the same transition,
neglecting the terms inxe, ̃αand ̃D.
22.14Calculate the percent change in the rotational and
vibrational energies of an HCl molecule ifa.^35 Cl is replaced by^37 Cl.
b.^1 H is replaced by^2 H.
22.15The dissociation energy is sometimes approximated by
determining the point at which two successive vibrational
energy levels have the same energy when thexe
correction is included. Estimate the value ofDefor the
HCl molecule using this approach. Compare your value
with that in Table A.22.