Assume that an electron is confined to a one dimensional of 1.50 nm in length. Calculate the lowest three energy levels for this electron and calculate the wavelength of light necessary to promote the electron from the ground state to the first excited state.
To calculate the lowest three energy levels for an electron confined to a one-dimensional box, we can use the formula:
En = (n^2 * h^2) / (8mL^2)
where En is the energy level, n is the quantum number (1, 2, 3...), h is the Planck's constant (6.626 x 10^-34 m^2 kg / s), m is the mass of the electron (9.11 x 10^-31 kg), and L is the length of the box (1.50 x 10^-9 m).
For n = 1:
E1 = (1^2 * (6.626 x 10^-34)^2) / (8 * 9.11 x 10^-31 * (1.50 x 10^-9)^2)
E1 = 1.941 x 10^-17 J
For n = 2:
E2 = (2^2 * (6.626 x 10^-34)^2) / (8 * 9.11 x 10^-31 * (1.50 x 10^-9)^2)
E2 = 7.765 x 10^-17 J
For n = 3:
E3 = (3^2 * (6.626 x 10^-34)^2) / (8 * 9.11 x 10^-31 * (1.50 x 10^-9)^2)
E3 = 1.739 x 10^-16 J
Now, to calculate the wavelength of light necessary to promote the electron from the ground state (n = 1) to the first excited state (n = 2), we can use the formula:
ΔE = E2 - E1
ΔE = (7.765 x 10^-17 J) - (1.941 x 10^-17 J)
ΔE = 5.824 x 10^-17 J
Now, we can use the formula E = hc / λ to calculate the wavelength of light:
ΔE = hc / λ
λ = hc / ΔE
λ = (6.626 x 10^-34 * 3 x 10^8) / (5.824 x 10^-17)
λ = 1.02 x 10^-6 m
λ = 1020 nm
Therefore, the wavelength of light necessary to promote the electron from the ground state to the first excited state is 1020 nm.