An electron confined in a one-dimensional infinite potential well has an energy of 180 eV.What is its wavelength?
A)65 pm
B)91 pm
C)130 pm
D)370 pm
E)520 pm

Question

Quizplus · Accepted Answer

The energy of an electron in a one-dimensional infinite potential well is given by:
$E = \frac{n^2h^2}{8ma^2}$, where n is the quantum number, h is Planck's constant, m is the mass of the electron, and a is the width of the well.

We can rearrange this equation to solve for the wavelength, λ:
$\lambda = \frac{h}{p}$, where p is the momentum of the electron, given by:
$p = \sqrt{2mE}$

Substituting the given values:
$E = 180$ eV $= 2.88 	imes 10^{-17}$ J
$m = 9.11 	imes 10^{-31}$ kg
$a$ is not given, but it is not needed to solve for λ.
First, we convert the energy to joules, and then solve for the momentum and wavelength:

$p = \sqrt{2mE} = \sqrt{2(9.11 	imes 10^{-31}	ext{ kg})(2.88 	imes 10^{-17}	ext{ J})} \approx 1.27 	imes 10^{-24}	ext{ kg}\cdot	ext{m/s}$

$\lambda = \frac{h}{p} = \frac{6.63 	imes 10^{-34}	ext{ J}\cdot	ext{s}}{1.27 	imes 10^{-24}	ext{ kg}\cdot	ext{m/s}} \approx 91	ext{ pm}$

Therefore, the answer is B.

An Electron Confined in a One-Dimensional Infinite Potential Well Has