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Chapter 8: Problem 54

Calculate the wavelength of the electromagnetic radiation required to excite a proton from the ground state to the level with \(n=4\) in a one-dimensional box 50. pm long.

Short Answer

Expert verified
Using the above steps, one can calculate the wavelength (\(\lambda\)) of the electromagnetic radiation required to excite a proton from the ground state to the level with \(n=4\) in a one-dimensional box 50. pm long.

Step by step solution

01

Calculate the Energy Levels

First, calculate the energy levels for \(n=1\) and \(n=4\) using the formula \(E = n^2h^2/(8mL^2)\), where \(n\) is the energy level, \(h\) is the Planck's constant equal to \(6.63 x 10^{-34}\) Js, \(m\) is the mass of the proton, about \(1.672 x 10^{-27}\) kg, and \(L\) is the length of the box, \(50 x 10^{-12}\).
02

Find the Difference in Energy Levels

Next, find the difference between the energy levels for \(n=4\) and \(n=1\) to get the energy of the radiation needed to excite the proton from the ground state to \(n=4\).
03

Calculate the Wavelength

Finally, find the wavelength using Einstein's relation \(E=h\nu = hc/ \lambda\), where \(E\) is the energy of the radiation, \(h\) is the Planck's constant, \(c\) is the speed of light at \(3 x 10^{8}\) m/s, and \(\lambda\) is the wavelength. Solving for \(\lambda\) yields \(\lambda = hc/E\).

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Most popular questions from this chapter

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.

What type of orbital (i.e., \(3 s, 4 p, \ldots)\) is designated by these quantum numbers? (a) \(n=5, \ell=1, m_{\ell}=0\) (b) \(n=4, \ell=2, m_{\ell}=-2\) (c) \(n=2, \ell=0, m_{\ell}=0\)

Explain the important distinctions between each pair of terms: (a) frequency and wavelength; (b) ultraviolet and infrared light; (c) continuous and discontinuous spectra; (d) traveling and standing waves; (e) quantum number and orbital; (f) spd f notation and orbital diagram; (g) \(s\) block and \(p\) block; (h) main group and transition element; (i) the ground state and excited state of a hydrogen atom.

A molecule of chlorine can be dissociated into atoms by absorbing a photon of sufficiently high energy. Any excess energy is translated into kinetic energy as the atoms recoil from one another. If a molecule of chlorine at rest absorbs a photon of 300 nm wavelength, what will be the velocity of the two recoiling atoms? Assume that the excess energy is equally divided between the two atoms. The bond energy of \(\mathrm{Cl}_{2}\) is \(242.6 \mathrm{kJ} \mathrm{mol}^{-1}\)

Use the Balmer equation (8.2) to determine (a) the frequency, in \(s^{-1}\), of the radiation corresponding to \(n=5\) (b) the wavelength, in nanometers, of the line in the Balmer series corresponding to \(n=7\) (c) the value of \(n\) corresponding to the Balmer series line at \(380 \mathrm{nm}\)
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