An electron in a 10.0-nm one-dimensional box is excited from the ground state into a higher energy state by absorbing a photon with wavelength 1.374 *10^-⁵ m. Determine the final energy level for this transition. A) n = 6 B) n = 4 C) n = 2 D) n = 5 E) n = 3

Question

Quizplus · Accepted Answer

The energy of a photon is given by:
E = hc/λ
where h is Planck's constant, c is the speed of light, and λ is the wavelength of the photon.

We can use this equation to calculate the energy of the absorbed photon:
E = hc/λ = (6.626 x 10^-34 J s)(3.00 x 10^8 m/s)/(1.374 x 10^-10 m) = 1.372 x 10^-18 J

The energy of a particle in a one-dimensional box is given by:
En = (n^2h^2)/(8mL^2)
where n is the quantum number, m is the mass of the particle, and L is the length of the box.

We can use this equation to find the final energy level of the electron:
En = (1^2)(6.626 x 10^-34 J s)^2/(8(9.11 x 10^-31 kg)(10.0 x 10^-9 m)^2) = 4.34 x 10^-20 J
Efinal = Einitial + Eabsorbed = (1^2)(6.626 x 10^-34 J s)^2/(8(9.11 x 10^-31 kg)(10.0 x 10^-9 m)^2) + 1.372 x 10^-18 J = 5.17 x 10^-20 J

We can solve for the final quantum number using the energy equation:
En = (n^2h^2)/(8mL^2)
n^2 = 8mL^2En/h^2
n^2 = 8(9.11 x 10^-31 kg)(10.0 x 10^-9 m)^2(5.17 x 10^-20 J)/(6.626 x 10^-34 J s)^2
n = 5

Therefore, the final energy level for this transition is n = 5, and the correct answer is (D).

An Electron in a 10