The particle in a 2D box is useful for modeling the motion of electrons around t

Question

The particle in a 2D box is useful for modeling the motion of electrons around the indole ring (fig. 1). We may approximate indole as a rectangular potential barrier with side a = 280 pm and side b = 450 pm and having 10 electrons in the conjugated system. Complete figure 2 by filling in the rest of the energy levels for the ground and excited states indicating what the quantum numbers nx, ny should be. Calculate the wavelength of radiation that will excite indole from its ground state to the first excited state (i.e. HOMO to LUMO). Indole's maximum wavelength of absorption is lambda max = 272 nm. Compare your answer with this experimental value and comment on its accuracy and give an explanation for any discrepancy you may find.

Explanation / Answer

Q1(a)

side Lx = 280 pm, Ly = 450 pm

For 10 pi electrons
E(nx,ny) = nx^2*h^2/(8*m*Lx^2) + ny^2*h^2/(8*m*Ly^2)
nx = 3, ny =1
h = 6.62618E-34 J*s
m=9.10953E-31 (kg)
Lx = 2.8E-10 m
Ly = 4.5E-10 m

h^2/(8*m) = 0.6025E-37 (J*s)^2/kg

So E(nx,ny) = [nx^2*h^2/((2.8E-10)^2) + ny^2/((4.5E-10)^2)]*0.6025E-37 =
nx^2*7.685E-19 + ny^2*2.975E-19

E(nx,ny) = nx^2*7.685E-19 + ny^2*2.975E-19
E(1,1) = 10.66E-19J
E(1,2) = 14.7E-19 J
E(2,1) = 33.7E-19J
E(2,2) = 42.6E-19J
E(1,3) = 34.5E-19J = E(HOMO)
E(3,1) = 72.14E-19J = E(LUMO)

Q1(b)

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