Electrons-in-a-Box: Consider an electron that can move freely throughout the aro

Question

Electrons-in-a-Box: Consider an electron that can move freely throughout the aromatic orbitals of benzene. Model the electron as a particle in a two-dimensional box 4 Å × 4 Å. a) Compute , the energy change from the ground state to the first excited state, nx = ny = 2. b) Compute the wavelength of light that would be absorbed in this transition, if = hc/, where h is Planck’s constant and c is the speed of light. c) Will this transition be in the visible part of the electromagnetic spectrum (i.e., is liquid benzene colored or transparent), according to this simple model?

Explanation / Answer

For two dimensional box,

           E = (nx2 + ny2) h2 / 8ml2

For ground state , nx=1,ny=1

E1= 2h2/8ml2

For first excited state , nx=2,ny=2

E2= 4h2/8ml2

E = E2-E1 = 4h2/8ml2 - 2h2/8ml2

E = 2h2/8ml2 = h2/4ml2

m= mass of electron = 9.1 x 10-31 kg

l = length of box = 4 Ao = 4 x 10-10 m

a) E = h2/4ml2

b) E = h2/4ml2

hc/ =  h2/4ml2

= c x 4ml2 / h

   = 3 x108 m/s x 4 x 9.1 x 10-31 kg x (4 x 10-10 m)2 / 6.626 x 10-34 J.s

= 263.6 x 10-9 m

= 263.6 nm

= 263.6 nm

c) This transition not in the visible part of the electromagnetic spectrum.

       visible region = 380 nm - 700 nm

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