Particle in a box: 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, m_el = 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 a particle in a one-dimensional box are given by:$$E_n = \frac{n^2h^2}{8mL^2}$$where n is the principal quantum number, h is Planck's constant, m is the mass of the particle, and L is the length of the box.The difference in energy between two adjacent levels is:$$\Delta E = E_{n+1} - E_n = \frac{(n+1)^2h^2}{8mL^2} - \frac{n^2h^2}{8mL^2} = \frac{2nh^2}{8mL^2}$$Substituting the given values, we get:$$\Delta E = 1.352 	imes 10^{-18} 	ext{ J} - 1.068 	imes 10^{-18} 	ext{ J} = 2.84 	imes 10^{-19} 	ext{ J}$$$$2.84 	imes 10^{-19} 	ext{ J} = \frac{2nh^2}{8mL^2}$$$$L^2 = \frac{2nh^2}{8m\Delta E}$$$$L = \sqrt{\frac{2nh^2}{8m\Delta E}}$$$$L = \sqrt{\frac{2(1)(6.626 	imes 10^{-34} 	ext{ J s})^2}{8(9.11 	imes 10^{-31} 	ext{ kg})(2.84 	imes 10^{-19} 	ext{ J})}}$$$$L = 1.9 	imes 10^{-9} 	ext{ m} = 1.9 	ext{ nm}$$Therefore, the length of the box is 1.9 nm.

Particle in a Box: an Electron Is Confined in a One-Dimensional