Question

The spacing between absorption lines in the far infrared spectrum of H³⁵Cl is 6.350 x 10¹¹Hz. Assuming the Rigid Rotor Model, calculate the values of the moment of inertia and the bond length in H³⁵Cl.

Answer 1

From the given data: the spacing between absorption lines in Infrared spectrum of HCl³⁵ is 6.350 x 10¹¹ Hz.

here the spacing is given in Hz, It is frequency unit. To solve our problem we have to convert this value into wave number units.

frequency V = C/ = C V^-

        v= frequency, C = speed of light 3 x 10¹⁰ cm s^-1, = wave length , V^- = wave number in cm^-1

Rotational wave number for rigid rotor is V^- = 2B(J+1),

thus the rotational absorption frequency beomes V = 2cB(J+1).

spacing between absorption lines becomes V = 2CB((J+2)-(J+1)) =2CB

Thus B = V/2C =  6.350 x 10¹¹ s^-1/ 2 (3 x 10¹⁰cm s^-1) = 10.58 cm^-1

this B value can be used to calculate moment of Inertia (I) = h/8²CB =6.625x10^-34/ 8(3.14)²(3x10¹⁰)(10.58)

The required Moment of Inertia (I) = 2.64 x10^-47kgm²

The reduced mass of HCl³⁵ = m₁ m₂/(m₁+m₂) =(1x35/(1+35)) x (1.66 x10^-27) kg/amu

= 1.61x10^-27 kg

moment of inertia (I) = R²

The bond length R = I/ =2.64 X10-47/(1.61 X10-27) = 1.28 X10^-10  m = 128 pm (1 pm = 10^-12 m)

The bond length of HCl³⁵ is 128 pm