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Chapter 8: Problem 52

An electron in a one-dimensional box requires a wavelength of \(618 \mathrm{nm}\) to excite an electron from the \(n=2\) level to the \(n=4\) level. Calculate the length of the box.

Short Answer

Expert verified
The length of the box is calculated to be approximately \(3.20 \times 10^{-10}\) meters, or 320 picometers.

Step by step solution

01

Calculate the photon's frequency

First, need to calculate the frequency of the photon. This can be done using the equation \(\Omega = c/\lambda\), where \(c\) is the speed of light. But before substituting the values we need to convert the wavelength from nm to m, which means \(\lambda = 618 \times 10^{-9} m\). Now substitute the speed of light \(c=3.0 \times 10^{8} m/s\) and the converted wavelength in the equation and find the frequency.
02

Calculate the energy of the photon

Next, the energy of the photon is calculated using the equation \(\Delta E = h\Omega\), where \(h\) is Planck's constant (\(h=6.626 \times 10^{-34} Js\)). Substitute the calculated frequency and Planck's constant in the equation to find \(\Delta E\).
03

Apply particle in a box model

Now, apply the particle in a box model's equation for energy difference, which is \(\Delta E = h^2/8mL^2 (n_f^2 - n_i^2)\), where \(n_i\) and \(n_f\) are the initial and final energy levels, and \(m\) is the mass of the electron (\(9.11 \times 10^{-31} kg\)), and \(L\) is the length of the box we need to find. Substitute all known values and solve this equation for \(L\).

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Most popular questions from this chapter

Without doing detailed calculations, indicate which of the following electron transitions in the hydrogen atom results in the emission of light of the longest wavelength. (a) \(n=4\) to \(n=3 ;\) (b) \(n=1\) to \(n=2\) (c) \(n=1\) to \(n=6 ;\) (d) \(n=3\) to \(n=2\).

What is the expected ground-state electron configuration for each of the following elements? (a) mercury; (b) calcium; (c) polonium; (d) tin; (e) tantalum; (f) iodine.

How long does it take light from the sun, 93 million miles away, to reach Earth?

What must be the velocity, in meters per second, of a beam of electrons if they are to display a de Broglie wavelength of \(1 \mu \mathrm{m} ?\)

Write an acceptable value for each of the missing quantum numbers. (a) \(n=3, \ell=?, m_{\ell}=2, m_{s}=+\frac{1}{2}\) (b) \(n=?, \ell=2, m_{\ell}=1, m_{s}=-\frac{1}{2}\) (c) \(n=4, \ell=2, m_{\ell}=0, m_{s}=?\) (d) \(n=?, \ell=0, m_{\ell}=?, m_{s}=?\)
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