How Does 1,3-Butadiene Absorb UV Light in Terms of Particle-in-a-Box Theory?

[terp.asessed]

Homework Statement

The UV/visible spectroscopy of linear conjugated molecules, particularly 1,3-butadiene in this problem, can often be modeled with the Particle-in-a-box of the electrons. Assume that we are interested in the pi electrons ONLY. A molecule with N double bonds = 2N pi electrons.

(1) To obtain the ground state, put 2N electrons in the LOWEST possible energy levels. Assume the box is of length L, what is the energy of the highest energy electron, as function of L and N?

(2) When 1,3-butadiene absorbs photon energy (hv), it uses the energy to promote 1 electron to a higher level. What is the lowest energy light the molecule can absorb, again, as a function of L and N?

Homework Equations

C-C bond length = 1.54 Angstrom
C=C length = 1.35 Angstrom
The angle between the bonds = 120 degree

L(in Angstrom) = (2.50)N - 1.32

The Attempt at a Solution

For butadiene, it has 2 double bonds...so N = 2 and 2(2) = 4 pi electrons, where 2 pi electrons compose energy level n=1 and second couple at n=2, with the possibility of one electron jumping to n=3 level with energy...

L = 2.50(2) - 1.32 = 3.68 Angstrom

...Could someone hint me as how to progress from here? I am stuck. Also, I know from the Schrödinger Equation that E_n = n²h²/(8mL²), from H wavefunction(x) = E wavefunction (x), where V(x) = 0 b/c of confinement inside a box...so, I wonder if I should use this equation too?

 

[Borek]

I wonder if I should use this equation too?

That would be my approach - it is just a matter of finding appropriate values of n (you already did part of the job earlier).

Disclaimer: QM is not something I feel confident about (but this particular problem looks rather straghtforward).

 

[terp.asessed]

Okay...but, I am stuck as what the problem requests...especially as a function of N and L...Still, thank you.

 

[Borek]

Not sure what your problem is - looks to me like you have correctly listed all the necessary information in your attempts at solving the problem.