Calculate the wavelength of the electromagnetic radiation required to excite an electron from the ground state to the level with n = 4 in a one-dimensional box 37.7 pm in length.
The formula for the energy of the particle in a 1D box is given by below equation
E=n2h2/8ma2
Where n= energy levels
h= planks constant
m=mass of the particle
a=length of the box
in the given problem
n=4, for this we have to calculate the energy associated with n=6 and ground state that is n=1
then E=E4-E1
E=(n42h2 / 8ma2) –(n12h2 / 8ma2)
= h2(n42-n12) / 8ma2
As we know that
E=hc/ ( where h = planks constant; c= velocity of light and = wavelength
So wavelength = hc/E
= hc / h2(n42-n12) / 8ma2
= 8ma2c / h (n42-n12)
= (8) (9.109 X 10-31 Kg) (0.449 X 10-10 m)2 (3 X 10-8 m/s) / (6.626 X 10-34 J s) (16-1)
= 8 X 9.109 X 0.449 X 0.449 X 3 X 10-43 / 6.626 X 15 X 10-34
= 44.07 X 10-43 / 99.39 X 10-34
= 0.4434 X 10-9 m
= 434.4 pm
answer= 434.4 pm
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