Isotope Substitution: Isotopes of an element have different atomic masses but similar chemical properties. When an isotope (e.g., substituting hydrogen with deuterium) is substituted in a diatomic molecule, the reduced mass of the molecule changes.
Vibrational Frequency Shift: The vibrational frequency of a diatomic molecule, which depends on the reduced mass of the molecule and the force constant of the bond, undergoes a shift upon isotopic substitution.
Magnitude of Shift: The isotopic effect typically results in a shift to lower frequencies (red shift) when heavier isotopes are substituted because the reduced mass μ increases. Conversely, substitution with lighter isotopes can lead to a blue shift (higher frequency).
Experimental Observations: Spectroscopic measurements of vibrational bands in diatomic molecules show distinct shifts in peak frequencies when isotopic substitutions are made. This allows experimental determination of the force constants k and provides valuable information about the molecular structure and dynamics.

Significance of Isotopic Effect:

Isotope Identification: The isotopic effect is used in spectroscopy to identify isotopes within molecules. By comparing the vibrational spectra of a molecule with different isotopic substitutions, scientists can identify and quantify the presence of specific isotopes.
Fundamental Constants: Measurement of vibrational frequencies with isotopic substitutions contributes to refining fundamental physical constants such as atomic masses and force constants of chemical bonds.
Astrophysical and Geochemical Applications: Isotopic effects observed in vibrational spectra are crucial for studying molecular compositions in astrophysical environments (e.g., stellar atmospheres) and geochemical processes (e.g., isotopic fractionation in minerals and gases).
Let us consider two isotopic form of same diatomic molecule having reduced mass μ and μ'' and equilibrium frequencies we' and we''. Then for two isotopic forms, the equilibrium frequencies will be

Assuming the anharmonicity constant 'x' is equilibrium frequency . So we can write

x_2 = 𝜌x_1 ....4

From 3, we'' = 𝜌we'....5

We know that frequency of the centre of the fundamental band, first and second overtones are given by

f_1 = ( 1- 2x) w_e

f_2 = ( 1- 3x) 2w_e

f_3 = ( 1- 4x) 3w_e ....6

Thus for first isotropic form

f_1' = ( 1- 2x_1) w_e'

f_2' = ( 1- 3x_1) 2w_e'

f_3' = ( 1- 4x_1) 3w_e' ....7

for second isotropic form

f_1^(2) = ( 1- 2x_2) w_e''

f_2'^(2) = ( 1- 3x_2) 2w_e''

f_3' ^(2) = ( 1- 4x_2) 3w_e'' ....8

Using x_2 and we'' we can write equation 8 as

f_1^(2) = ( 1- 2𝜌x_1) 𝜌w_e'

f_2'^(2) = ( 1- 3𝜌x_1) 2𝜌w_e'

f_3' ^(2) = ( 1- 4𝜌x_1) 3𝜌w_e' ...9

Now the change in frequency of the centre of fundamental band due to isotopic exchange will be

f_1' - f_2^(2) = w_e' - 2x_1 w_e'- 𝜌w_e' + 2𝜌²x_1w_e'

= w_e' ( 1- 𝜌) - 2x_1 w_e' ( 1- 𝜌²)

= ( 1- 𝜌)[ w_e' - 2x_1 w_e' ( 1+𝜌)]

= ( 1- 𝜌) w_e' [ 1- 2x_1( 1+ 𝜌)] ....10

The isotopic shift for second overtone will be

f_2' - f_2^(2) = 2w_e' - 6x_1 w_e'- 2𝜌w_e' + 6𝜌²x_1w_e'

= 2w_e'( 1- 𝜌) - 6x_1 w_e'( 1- 𝜌)²

=2w_e' ( 1- 𝜌)[ 1 - 3x_1( 1+𝜌) ]...11

f_3' - f_3^(2) = 3w_e' - 12x_1 w_e'- 3𝜌w_e' -12𝜌²x_1w_e'

= 3w_e' ( 1- 𝜌)[ 1 - 4x_1( 1+𝜌) ]...12

From equation 10, 11 and 12, it is seen that if anharmonically constant x_1 and equilibrium frequency of first isotopic form are known, then the isotopic shift depends upon the factor ( 1- 𝜌). As the value of 𝜌 = sq root of 𝜇' / 𝜇'' i.e 𝜇'' > 𝜇', ( 1- 𝜌) increases and hence isotopic shift increases and vice versa.

The isotopic effect in vibrational bands of diatomic molecules plays a significant role in spectroscopic analysis, providing insights into molecular structure, dynamics, and isotopic composition. It demonstrates the sensitivity of vibrational frequencies to changes in atomic mass, making it a powerful tool in both fundamental physics and applied sciences.

This note is a part of the Physics Repository.

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