Question

Given that the spacing of the lines in the microwave spectrum of 27Al1H is

constant at 12.60 cm1

. Calculate the moment of inertia and bond length of

the molecule.​

Answer 1

The calculation of the moment of inertia and bond length of the molecule is attached as shown in the two images.

Answer 2

Centrifugal distortion will be negligible since spacing of lines is constant.

F(J) - F(J-1) = 2BJ

For J = 1,

12.60 = 2B

B = 6.3 cm⁻¹ =  6.3 x 10² m⁻¹

l = h / 4Π cB = Meff R²

h / 4Π c = 1.0546 x 10 ⁻³⁴/  4Π x 2.99 x 10⁸ = 2.8 x 10⁻¹⁴ Kg.m

l = 2.8 x 10⁻¹⁴ /  6.3 x 10² = 4.4 x 10 ⁻⁴⁷ kg.m²

Meff = M(A1).M(AH)/ (M(A1) + M(AH))

M(A1) - 26.98, M(AH) - 1.008

Meff = (26.98 x 1.008) /(26.98 + 1.008) u x 1.61 x 10⁻²⁷

R = (1/ Meff)^(1/2) = (4.4 x 10 ⁻⁴⁷/ 1.6 x 10⁻²⁷) = 1.65 x 10⁻¹⁰ m