The bond length in CN⁺ is 0.129 nm. The position of second line in the microwave spectrum is -

Concept:

• Microwave Spectroscopy: Involves the study of the rotational transitions of gas-phase molecules, providing information on molecular structure and bond lengths.

• Rotational Constant (B): Given by the formula , where h is Planck's constant, I is the moment of inertia, and c is the speed of light.

• Moment of Inertia (I): For a diatomic molecule, , where is the reduced mass and r is the bond length.

• Energy Levels and Transitions: The energy associated with rotational levels is given by E_J = B J (J + 1) , and the positions of spectral lines are determined by transitions such as .

Explanation:

To calculate the position of the second line in the microwave spectrum, we need to determine the rotational constant B and use it to find the frequency of the second transition ( and ). 

Moment of inertia ( I ):
where and meters.

Approximate masses are:
m_C = 12 u and m_N = 14 u , where u is the atomic mass unit.
Using atomic mass unit kg:

(approx).

Moment of inertia ( I ):
.

Rotational constant ( B ):

Where
.

Therefore B :

.

Converting  B to wavenumbers ( cm^-1 ):

.

The position of the second line ( J = 1 → J = 2 ) will be 4B :

Conclusion:

The position of the second line in the microwave spectrum is 6.264 cm^-1.