The rotational-vibrational spectrum of the 0 rightarrow 1 vibrational transition

Question

The rotational-vibrational spectrum of the 0 rightarrow 1 vibrational transition in HBr is given below. Each line in this spectrum corresponds to a transit from a state with n = 0 and J = J_initial to a state with n = 1 and J = J_final. What are the allowed values of Delta J = J_final - J_initial in the P and R branches., respectively, of this spectrum? What are the values of J_initial and J_final for, lines A, B, C, D, E, and F in this spectrum? in the harmonic oscillator-rigid rotor approximation, the rotational-vibrational energy of a diatomic molecule is given by E_n, J = (n + 1/2) hv + BJ (J + 1) n = 0, 1, 2, ..., J = 0, 1, 2, ... where v = 1/2 pi squareroot k/mu and B = h^2.21. in units of wave numbers, this becomes E_n, J = E_n, J/hc = (n + 1/2) v_rib + BJ (J + 1) Using the latter equation, derive expressions for the wave numbers v for lines A, B. C, D, E, and F in terms of the pure vibrational wavenumber = (2 pi c)^-1 squareroot k/mu and the rotational constant B = h^2/2hcl = h/8 pi^2 cl.

Explanation / Answer

Solution of part c) (As students needs only part c answer)

Given

E n, j = (n+1/2) hv + BJ(J+1)

Here n = 0, 1, 2,….and J = 0, 1, 2..

Here v = (½*Pi()) * SQRT ( k /)

And B = h^2/2I

In units of wave numbers, it is given that the above equation can be represented as

E’ (n,J) = E n, j/hc

= (n+1/2) v’(vib) + B’J(J+1)

It is also given in the question that v’ (vib) = (2*pi()*c)^-1 * Sqrt ( k/)

And B* = h^2/2hcI = h/8*pi()^2*c*I

We know that v’ = 1/= v/c

So v’ = (½*Pi()) * SQRT ( k /)/ c

Hence v’ = (½*Pi())* c * SQRT ( k /)

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