An electron is in an infinite square well (a box) that is 2.0 nm wide. The electron makes a transition from the n = 8 to the n = 7 state, what is the wavelength of the emitted photon?(h = 6.626 × 10^-34 J ∙ s, m_el = 9.11 × 10^-31 kg, 1 eV = 1.60 × 10^-19) A) 880 nm B) 750 nm C) 610 nm D) 1000 nm E) 1100 nm

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Quizplus · Accepted Answer

The energy of an electron in an infinite square well is 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 electron, and L is the width of the well.The wavelength of a photon emitted when an electron transitions from a higher energy state to a lower energy state is given by:$$\lambda = \frac{hc}{E_i - E_f}$$where h is Planck's constant, c is the speed of light, $E_i$ is the initial energy of the electron, and $E_f$ is the final energy of the electron.In this problem, the electron transitions from the n = 8 to the n = 7 state. The energy of the electron in the n = 8 state is:$$E_8 = \frac{8^2h^2}{8mL^2}$$The energy of the electron in the n = 7 state is:$$E_7 = \frac{7^2h^2}{8mL^2}$$The wavelength of the emitted photon is:$$\lambda = \frac{hc}{E_8 - E_7}$$$$= \frac{hc}{8^2h^2/8mL^2 - 7^2h^2/8mL^2}$$$$= \frac{hc}{h^2/8mL^2}$$$$= \frac{8mL^2c}{h}$$$$= \frac{8(9.11 	imes 10^{-31} kg)(2.0 	imes 10^{-9} m)^2(3 	imes 10^8 m/s)}{6.626 	imes 10^{-34} J \cdot s}$$$$= 880 nm$$Therefore, the wavelength of the emitted photon is 880 nm.

An Electron Is in an Infinite Square Well (A Box)