Question

The spacing between rotational levels for HCl is about 20.7 cm-1. Use this value to calculate the bond length in HCl. How does your bond length compare to the bond length calculated using atomic radii?

Answer 1

Given that;

rotational transition, B, of 20.7 cm^-1.

B = hbar / 4*pi*c*I

hbar = reduced Planck's constant which is equal to 1.0546E-34 J*s
c = velocity of light = 3.00*10^8 m/s
I = moment of inertia

The moment of inertia is:
I = mR^2

m = effective mass = m1*m2 / (m1+m2)
R = bond length

Put both together to get:
B = hbar / 4*pi*c*m*R^2

Solving for R gives you:
R = Sqrt[ hbar / 4*pi*c*m*B ]

The effective mass would be:
m = (1.007825*34.96885) / (1.007825+34.96885) = 0.9780 amu.

Now covert it into Kg ad follows:

, 1.66E-27 kg/amu*0.9780 amu.

Now calculate the bond length R as follows:

R = sqrt( 1.0546E-34 J*s / 4 * pi * 3E8 m/s * 0.9780 amu * 1.66E-27 kg/amu * 2070 m^-1 )

R = 9.08*10^-11 m

= 90.8 pm