Question

8. Consider the structure of beta-carotene:

It is a linear polyene in which 21 bonds, 10 single and 11 double, alternate along a chain of 22 carbon atoms. Assume that beta-carotene can be modeled as a one-dimensional box or pipe with each of the 21 bonds having a length of 1.4x10^-10 m. Assume that there is one freely moving electron in the n=11 energy level that can be promoted to the n=12 energy level. Calculate the energy separation between the two energy levels. Calculate the frequency of radiation that can bring about the n=11 to n=12 transition.

ΔE = ________ J

ν = ________ Hz

Answer 1

Bond length = 1.4 x 10^-10 m

Total length of moleucle = 21 x 1.4 x 10^-10 = 2.94 x 10^-9 m   ( since we had 21 bonds)

Energy of particle in box = n^2 h^2 / ( 8ma^2)

when n = 11 we have E = 11^2 ( h^2 / 8ma^2) ,

E(n=12) - E(n=1) = ( 12^2 - 11^2)   ( h^2 / 8ma^2)

         = 23 ( 6.625 x 10^-34Js)^2 / ( 8 x 9.1 x 10^-31 kg x (2.94x10^-9m)^2 )

        = 6.975 x 10^-21 J

Hence dE = 6.975 x 10^-21 J

now we have relation dE = h x v     where v is frequency

      6.975 x 10^-21 J = 6.625 x 10^-34 Js x v

   v = 1.05 x 10^13 Hz                           ( 1s-1 = 1Hz)