Question

The first three stokes lines in the rotational Raman spectrum of O2 are separated by 14.4 , 25.8, 37.4 cm from the exciting radiation. Using the rigid rotor model obtain the value of r0.

a) 1.209Aº b) 1.4A0 c) 1.4m d)1.209m

Answer 1

The bond length r_0 of O2 calculated using the rigid rotor model in the rotational Raman spectrum is 1.209 angstroms, corresponding to option a).

Step-by-step explanation:

The question pertains to calculating the bond length of an oxygen molecule (O₂) using the information provided from the rotational Raman spectrum and the rigid rotor model. The first three Stoke's lines are given as being separated from the exciting radiation by 14.4, 25.8, and 37.4 cm-1. In the rigid rotor model for diatomic molecules, the rotational energy levels are given by EJ = B⋅J(J+1), where J is the rotational quantum number and B is the rotational constant. The separations correspond to transitions where ΔJ = ± 1, and therefore the differences between consecutive energy levels (the Raman shifts) are given by Δ3E = 2B(3), Δ2E = 2B(2), and Δ1E = 2B(1) for the first, second, and third lines, respectively.

Using these equations, we can determine B and then the bond length r0 using the formula B = h/(8π²⋅I), where I is the moment of inertia, I = μ⋅r02, and μ is the reduced mass of the molecule. Since oxygen is a homonuclear diatomic molecule, μ = mO/2. Knowing the mass of an oxygen atom (mO) and Planck's constant (h), we can calculate r0. From the options given, it can be determined that the bond length r0 is 1.209 Å (angstroms), which corresponds to option a).