Rigid rotators

the classical and mechanical treatment of rigid rotator

1. Rigid Rotator Fatima Syed

2. Rigid Rotators Definition: Two rotating atoms having fixed bond length are called rigid rotators.

3. Why we study rigid rotator: Rigid rotator is studied because it is a model for the diatomic molecule. It helps us to calculate rotational energy of diatomic molecule.

4. Classical treatment; Let us consider two atoms having masses m 1 and m 2 and fixed bond length r. The center of gravity is fixed. m 1 r 1 ═ m 2 r 2 equation (1) For Rigid rotators r ═ r 1 + r 2 equation (2) From equation (1) Put this value in equation 2; By taking LCM;  

5. Similarly; As we know that moment of inertia I= m 1 r 1 2 + m 2 r 2 2 equation (3) Putting the values of r 1 and r 2 in equation (3)  

6. I =µr 2 Where µ= reduced mass Derivation of energy; Total energy= K.E + P.E In case of rigid rotator P.E=0 Total energy = K.E By angular velocity formula Put this value in the above equation  

7. As angular momentum is given as  

8. Quantum Treatment; Schrodinger wave equation in polar coordinates is given as; For rigid rotators r=constant And P.E =0 So; As Putting the value of in above equation  

9. Solution of the above equation is As For rigid rotator Putting value of H For rigid rotator l=j Where; J= Rotational quantum number.  

10. As; B=rotational constant So; It is the relation of rotational energy for rigid rotators having fixed bond length.  

11. ENERGY TRANSITION; By selection rule;  

12. Application of rigid rotator; Calculation of bond length; Where; µ=reduced mass For one molecule Or = B’=