The energy separation of ¹²C¹⁶O rotational energy levels between J"=3 and J" = 9 is 24 cm^-1. The rotational constant of ¹³C¹⁶O in cm^-1 is closest to

Concept:

• The basic quantum mechanical model of rotational spectroscopy is "particle on a sphere" or "the rigid rotor model".
• Rotational spectroscopy deals with a system consisting of two atoms connected by a bond of fixed length (simply a diatomic molecule). It can rotate to describe the entire 3D space resulting in an infinite number of circular trajectories which eventually describe a sphere.
• The rotational energy level for a particular state (Jth state) is given by

, where J is a quantum number (J=0,1,2...) and I is the moment of inertia (, where  is the reduced mass and R is the bond length).

• As 

 let,  

 where B is the rotational constant.

• The energy difference between any two consecutive levels J and J+1 is given by:

• The spectrum depicted below is an absorption spectrum:

• Thus, these spectral lines are equispaced.

Explanation:

• Now, the energy separation of 12C16O rotational energy levels between J"=3 and J" = 9 is 24 cm-1.

E₃ = B x 3(4)= 12 B
E₉ = B x 9(10) = 90B

 = 90B - 12B= 78B 

Also =24 cm-1.

24 cm-1= 78B 

B = 24/78 = 0.307

• Thus for 12C16O, 

​

• As the reduced mass changes () the value of rotational constant (B) also changes as

​

• ​So, the value of rotational constant B for 13C16O is closest to 0.298 cm^-1. 

Conclusion:

Thus, the rotational constant of 13C16O in cm-1 is closest to 0.298 cm-1.