Question

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.

Answer 1

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 $\Delta$E=E₄-E₁

$\Delta$E=(n₄²h² / 8ma²) –(n₁²h² / 8ma²)

      = h²(n₄²-n₁²) / 8ma²

As we know that

E=hc/$\lambda$ ( where h = planks constant; c= velocity of light and $\lambda$ = 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