Explanation:
The positions of the first three lines in the microwave spectrum of CN can be predicted using the formula:

ΔE = hcB(J+1) - hcB(J)

where ΔE is the energy difference between two adjacent rotational levels, h is Planck's constant, c is the speed of light, B is the rotational constant, and J is the quantum number for rotational energy levels.

Calculation of Rotational Constant:
The rotational constant B can be calculated using the following formula:

B = h/(8π²I)

where I is the moment of inertia of the molecule.

The moment of inertia of the molecule can be calculated by adding the moments of inertia of the individual atoms. The moment of inertia of an atom can be calculated using the formula:

I = mr²

where m is the mass of the atom and r is the distance of the atom from the axis of rotation.

For CN molecule, the moment of inertia can be calculated as follows:

I = (mC × rC²) + (mN × rN²)

where mC and mN are the masses of carbon and nitrogen atoms respectively, and rC and rN are the distances of carbon and nitrogen atoms from the axis of rotation respectively.

Calculation of Energy Difference:
Once the rotational constant B is calculated, we can use it to calculate the energy difference ΔE between two adjacent rotational levels for different values of J.

The positions of the first three lines in the microwave spectrum of CN can be predicted by calculating the energy differences for J = 0, 1, and 2.

Final Answer:
Using the above calculations, the positions of the first three lines in the microwave spectrum of CN can be predicted to be at the following frequencies:

- Line 1: ΔE for J = 1 - J = 0 = 2hcB ≈ 19.7 GHz
- Line 2: ΔE for J = 2 - J = 1 = 4hcB ≈ 39.4 GHz
- Line 3: ΔE for J = 3 - J = 2 = 6hcB ≈ 59.1 GHz

Therefore, the first three lines in the microwave spectrum of CN are expected to be observed at frequencies of approximately 19.7 GHz, 39.4 GHz, and 59.1 GHz.