Q$2$ Butadiene $-$ Particle in a box: The model of a particle in a one$-$dimensional

box can been applied to the $$ electrons in linear conjugated hydrocarbons.

Consider butadiene, $H_{2}C=CH-CH=C{H}_2,$ which has four $$ electrons.

Although butadiene, like all polyenes, is not a linear molecule, we will as$-$

sume for simplicity that the $$ electrons in butadiene move along a straight

line whose length can be estimated as equal to two $C=C$ bond lengths

$(2135pm)$ plus one $C-C$ bond $(154pm)$ plus the distance of a carbon

atom radius at each end $(277pm=154pm),$ giving a total distance of

$578pm.(1pm={10}^{-12}(m)).$ Treating this as a four free electron system in

a one$-$dimensional box, illustrate the ground state and first excited state

configuration of this system using an appropriate energy level diagram.

Calculate the numerical values of the ground state energy and the first

excited state energy in electron volts. Butadiene has an absorption band

at $4\mathrm{.}61{10}^{4}c{m}^{-1}.$ Account for this absorption band within the context

of this model. Show this absorption band on your diagram.