Physical Chemistry , 1st ed.

where the asterisk is used to indicate an isotopic substitution. When we take

a ratio of the above two equations, such treatments will cancel each other and

we can get



8.652

*

1013 s^1



which reduces to



8.652

*

1013 s^1

 





*



Because the frequency ratio is related to a reduced mass ratio,it doesn’t mat-

ter what units we use to express the reduced mass ratio. Rather than express

the reduced masses in atomic values (on the order of 10^27 kg or so), we can

simply use grams per mole as the unit of mass. The reduced mass of^1 H^35 Cl

is therefore 0.9722 (grams per mole), whereas the reduced mass of^1 H^37 Cl is

0.9737 (grams per mole). The above equation becomes



8.652

*

1013 s^1

 

0

0

.

.

9

9

7

7

2

3

2

(^7)
g
g

/

/

m

m

o

o

l

l



The units cancel, yielding a ratio that has the same value no matter what units

of mass are used. Evaluating:



8.652

*

1013 s^1

0.9992

This rearranges to * 8.659 

1013 s^1 , or 2884 cm^1 (to four significant

figures). This is a relatively insignificant change, although it is easily de-

tectable. However, for^2 H^35 Cl:



8.652

*

1013 s^1

 

0

1

.9

.8

7

9

2

1

2

(^) g
g
/

/

m

m

o

o

(^) l
l


8.652

*

1013 s^1

0.7170 (unitless)

* 6.203 

1013 s^1 or 2069 cm^1 (to four significant figures)

This predicts a shift of over 800 wavenumbers. The measured vibrational fre-

quency of^2 H^35 Cl is 2091 cm^1 , which agrees with the assumption of an ideal

system.

It is not necessary to convert a wavenumber value into a frequency value

when doing an example like the one above, because the two quantities are di-

rectly proportional to each other. Also, even if a molecule isn’t a diatomic mole-

cule, and even though a normal mode consists of the vibrations of all atoms in

the molecule, in many cases for stretching-type motions a “diatomic approxi-

mation” can be made for isotopic substitution. The next example illustrates.

Example 14.13

If the symmetric O–H stretch for water occurs at 3657 cm^1 , predict the fre-

quency of the O–D stretch of D 2 O (D is^2 H) assuming that the O–H stretch

acts as a diatomic species.



2

1



 



k

*







2

1



 



k



486 CHAPTER 14 Rotational and Vibrational Spectroscopy