An electron is confined in a one-dimensional box (an infinite well). Two adjacent allowed energies of the electron are 1.068 × 10^-18 J and 1.352 × 10^-18 J. What is the length of the box? (h = 6.626 × 10-³⁴ J ∙ s, ^mel = 9.11 × 10^-31 kg) A) 1.9 nm B) 0.93 nm C) 1.1 nm D) 2.3 nm

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Quizplus · Accepted Answer

The energy levels of an electron in a one-dimensional box are given by:$$E_n = \frac{n^2h^2}{8mL^2}$$where:* $E_n$ is the energy of the electron in the nth energy level* $h$ is Planck's constant* $m$ is the mass of the electron* $L$ is the length of the boxThe difference in energy between two adjacent energy levels is given by:$$\Delta E = E_{n+1} - E_n = \frac{(n+1)^2h^2}{8mL^2} - \frac{n^2h^2}{8mL^2} = \frac{h^2}{8mL^2}$$We are given that the difference in energy between two adjacent energy levels is 1.068 × 10^-18 J and 1.352 × 10^-18 J. We can use this to solve for the length of the box:$$\Delta E = \frac{h^2}{8mL^2}$$$$L = \sqrt{\frac{h^2}{8m\Delta E}}$$$$L = \sqrt{\frac{(6.626 imes 10^{-34} ext{ J s})^2}{8(9.11 imes 10^{-31} ext{ kg})(1.068 imes 10^{-18} ext{ J})}}$$$$L = 1.9 ext{ nm}$$Therefore, the length of the box is 1.9 nm.

An Electron Is Confined in a One-Dimensional Box (An Infinite