Predict the first three rotational lines in a microwave spectrum of ¹⁴N¹⁶O if the bond length of NO is 121 pm.

Question

Quizplus · Accepted Answer

The rotational energy levels of a diatomic molecule are given by the expression:
E (J) = B J(J+1) 
where E is the energy of the level, B is the rotational constant and J is the rotational quantum number.

The rotational constant B can be calculated using the formula:
B = h/8π2I 
where h is the Planck’s constant, and I is the moment of inertia of the molecule.

For a diatomic molecule with masses m1 and m2 separated by a distance r, the moment of inertia is given by the expression:
I = μ r2 
where μ = (m1m2)/(m1+m2) is the reduced mass of the molecule.

In this problem, the molecule is NO. The masses of N and O are 14 u and 16 u, respectively. The bond length is given as 121 pm = 1.21 Å. Using these values, we can calculate the reduced mass and the moment of inertia:
μ = (14 u x 16 u)/(14 u + 16 u) = 7.56 u
I = μ r2 = 7.56 u x (1.21 x 10^-10 m)^2 = 1.136 x 10^-46 kg m^2

Using this moment of inertia value, we can calculate the rotational constant B:
B = h/8π2I = (6.626 x 10^-34 J s)/(8π^2 x 1.136 x 10^-46 kg m^2) = 1.943 x 10^-23 J

The frequency of microwave radiation required to excite a rotational transition from J to J+1 is given by the expression:
ΔE/h = 2B(J+1)

Substituting the value of B, we get:
ΔE/h = 3.886 x 10^-23 (J+1)

To convert this energy to frequency, we use the relation:
ΔE = hν 
where ν is the frequency of the microwave radiation.

Combining the above equations, we get:
ν = ΔE/h = 3.886 x 10^-23 (J+1)

We are asked to predict the first three rotational lines in the microwave spectrum. These correspond to the transitions J=0 to J=1, J=1 to J=2, and J=2 to J=3. Substituting these values in the above equation, we get:

ν(0 → 1) = 3.886 x 10^-23 x 1 = 3.886 x 10^9 Hz = 3.886 GHz
ν(1 → 2) = 3.886 x 10^-23 x 2 = 7.772 x 10^9 Hz = 7.772 GHz
ν(2 → 3) = 3.886 x 10^-23 x 3 = 1.166 x 10^10 Hz = 11.66 GHz

These frequencies correspond to wavelengths of:
λ(0 → 1) = c/ν(0 → 1) = 7.74 mm
λ(1 → 2) = c/ν(1 → 2) = 3.86 mm
λ(2 → 3) = c/ν(2 → 3) = 2.57 mm

The answer is therefore B: 3.08, 6.16, 9.24 cm^(-1)S_(1)U_(1)P_(1→1) S_(1)S_(1)P_(0) because the frequencies are in ascending order and match the formula of the rotational energy levels.

Predict the First Three Rotational Lines in a Microwave Spectrum