Answer :
Answer : The wavelength of the light emitted is, [tex]5.66\times 10^{-24}m[/tex]
Explanation :
The energy level of quantum particle in a one-dimensional box is given as:
[tex]E_n=\frac{n^2h^2}{8mL^2}[/tex]
or,
[tex]\Delta E=E_9-E_8=\frac{n_9^2h^2}{8mL^2}-\frac{n_8^2h^2}{8mL^2}[/tex]
[tex]\Delta E=E_9-E_8=\frac{h^2}{8mL^2}\times (n_9^2-n_8^2)[/tex]
where,
[tex]E_n[/tex] = change in energy
n = energy level
h = Planck's constant = [tex]6.626\times 10^{-34}Js[/tex]
m = mass of electron = [tex]9.109\times 10^{-31}kg[/tex]
L = length of a one-dimensional box = [tex]5.4nm=5.4\times 10^{-9}m[/tex]
Now put all the given values in the above formula, we get:
[tex]\Delta E=E_9-E_8=\frac{(6.626\times 10^{-34}Js)^2}{8\times (9.109\times 10^{-31}kg)\times (5.4\times 10^{-9}m)^2}\times [(9)^2-(8)^2][/tex]
[tex]\Delta E=E_9-E_8=3.5124\times 10^{-2}J[/tex]
Now we have to calculate the wavelength of the light emitted.
[tex]\Delta E=\frac{hc}{\lambda}[/tex]
where,
h = Planck's constant = [tex]6.626\times 10^{-34}Js[/tex]
c = speed of light = [tex]3\times 10^{8}m/s[/tex]
[tex]\lambda[/tex] = wavelength of the light
Now put all the given values in the above formula, we get:
[tex]3.5124\times 10^{-2}J=\frac{(6.626\times 10^{-34}Js)\times (3\times 10^{8}m/s)}{\lambda}[/tex]
[tex]\lambda=5.66\times 10^{-24}m[/tex]
Thus, the wavelength of the light emitted is, [tex]5.66\times 10^{-24}m[/tex]