Physical Chemistry Third Edition

930 22 Translational, Rotational, and Vibrational States of Atoms and Molecules

22.16a. From the vibrational frequency in Table A.22, find the
value of the force constant for the HF molecule.

b. From the vibrational frequency in Table A.22, find the

value of the force constant for the HCl molecule.

c.From the vibrational frequency in Table A.22, find the

value of the force constant for the H 2 molecule.

22.17When electromagnetic radiation is emitted or absorbed by
a diatomic molecule in making a transition between
different values of the rotational quantum numberJ, the
value ofJchanges by±1.

a. Find the frequency and wavelength of radiation

absorbed in theJ0toJ1 transition for NO.

b.Find the frequency and wavelength of radiation

absorbed in theJ1toJ2 transition for NO.

22.18When electromagnetic radiation is emitted or absorbed by

a diatomic molecule in making a transition between

different values of the rotational quantum numbersJand

ν, the value ofJchanges by±1 and the value ofν

changes by±1.

a.Find the frequency and wavelength of radiation

absorbed in theJ0toJ1,ν0toν 1

transition for HCl.

b.Find the frequency and wavelength of radiation

absorbed in theJ1toJ2,ν0toν 1

transition for NO.

22.3 Nuclear Spins and Wave Function Symmetry

If the nuclei in a diatomic molecule have nonzero spins, the wave function in

Eq. (22.2-35) must be multiplied by a nuclear spin wave function to be a complete

wave function.

ψtotψtrψrotψvibψelψnucspin (22.3-1)

In the case of homonuclear diatomic molecules (molecules with two nuclei of the same

isotope of the same element), the wave function must not pretend to distinguish between

the nuclei, which are indistinguishable from each other. The wave function must be

symmetric with respect to interchange of the nuclei if they are bosons and must be

antisymmetric with respect to interchange of the nuclei if they are fermions.

For our purposes, a nucleus can be considered to be made up of protons and neutrons,

collectively callednucleons. The mass number given as a left superscript on the symbol

for a given isotope is equal to the number of nucleons. Protons and neutrons have a

spin quantum number of 1/2, as do electrons, and are therefore fermions. If a nucleus

contains an odd number of nucleons, it is a fermion, because exchanging two such nuclei

changes the sign of the wave function once for each nucleon. If a nucleus contains an

even number of nucleons, it is a boson, because exchanging two such nuclei changes

the sign of the wave function an even number of times, leaving the original sign.

The spin angular momentum of a nucleus is denoted byIand has the same general

properties as any angular momentum. Its magnitude takes on the values

|I|h ̄

√

I(I+ 1 ) (22.3-2)

whereIis the spin quantum number, equal to an integer for a boson nucleus and equal

to a half-integer for a fermion nucleus. The projection of the spin angular momentum

on thezaxis takes on the values

IzhM ̄ I (22.3-3)

whereMIis a quantum number ranging from+Ito−Iin integral steps. IfIis an

integer, so isMI, and ifIis a half-integer, so isMI.

The angular momentum vectorIis the vector sum of the spin angular momenta of

all of the nucleons in the nucleus. A given nucleus can have different spin states, just as