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 Schrodinger 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.

 

[DeeNos]

I would approach this problem by first understanding the concept of Particle-in-a-box and its application to the UV/visible spectroscopy of linear conjugated molecules. This concept states that the electrons in a molecule can be treated as particles confined to a box, and their energy levels can be calculated using the Schrodinger equation.

In this case, we are only interested in the pi electrons of 1,3-butadiene, which is a linear conjugated molecule with 2 double bonds. This means that there are 4 pi electrons, with 2 electrons in the lowest energy level (n=1) and 2 in the second energy level (n=2). The energy of the highest energy electron can be calculated using the Schrodinger equation, which is En = n2h2/(8mL2), where n is the energy level, h is Planck's constant, m is the mass of the electron, and L is the length of the box.

For butadiene, L = 3.68 Angstrom, so we can calculate the energy of the highest energy electron using the equation En = (2)2h2/(8m(3.68)2) = 0.56 eV. This means that the highest energy electron is in the second energy level (n=2) with an energy of 0.56 eV.

Now, let's consider the absorption of photon energy by 1,3-butadiene. When a photon of energy hv is absorbed by the molecule, it can promote one of the electrons from the ground state (n=1) to a higher energy level (n=2 or n=3). The lowest energy light that the molecule can absorb is when the energy of the photon is equal to the energy difference between the ground state and the next energy level. In this case, the lowest energy light would be when the energy of the photon is equal to the energy difference between n=1 and n=2, which is 0.56 eV.

In summary, as a scientist, I would approach this problem by understanding the concept of Particle-in-a-box, using the Schrodinger equation to calculate the energy of the highest energy electron, and considering the energy difference between the ground state and the next energy level to determine the lowest energy light that the molecule can absorb.