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Chapter 8: Problem 31

For the Bohr hydrogen atom determine (a) the radius of the orbit \(n=4\) (b) whether there is an orbit having a radius of \(4.00 \AA\) (c) the energy level corresponding to \(n=8\) (d) whether there is an energy level at \(-2.5 \times 10^{-17} \mathrm{J}\)

Short Answer

Expert verified
(a) The radius for the \(n=4\) orbit is \(8.464 \, \AA\). (b) No, there is no orbit having a radius of \(4.00 \, \AA\). (c) The energy level corresponding to \(n=8\) is \(-3.425 \times 10^{-20} \, \mathrm{J}\). (d) Yes, there is an energy level at \(-2.5 \times 10^{-17} J\), which corresponds to \(n=3\).

Step by step solution

01

Calculation of radius for \(n=4\)

Use the given formula for radius \(r_n = n^2 \times r_1\). Substitute \(n=4\) and \(r_1 = 0.529 \, \AA\) into the formula to find \(r_4\).
02

Check for an orbit with radius \(4.00 \, \AA\)

Rearrange the radius formula as \(n= \sqrt{\frac{{r}}{{r_1}}}\), where \(r\) is the given radius. Substitute \(r = 4.00 \, \AA\) and \(r_1 = 0.529 \, \AA\) into the formula. If \(n\) is a whole number, then there exists an orbit with that radius.
03

Calculation of energy level for \(n=8\)

Use the given formula for energy levels \(E_n= -\frac{{R_H}}{{n^2}}\). Substitute \(n=8\) and \(R_H = 2.18 \times 10^{-18} \mathrm{J}\) into the formula to find \(E_8\).
04

Check for an energy level at \(-2.5 \times 10^{-17} \, \mathrm{J}\)

Rearrange the energy formula as \(n= \sqrt{\frac{{-R_H}}{{E}}}\), where \(E\) is the given energy. Substitute \(E = -2.5 \times 10^{-17} \, \mathrm{J}\) and \(R_H = 2.18 \times 10^{-18} \mathrm{J}\) into the formula. If \(n\) is a whole number, then there exists an energy level with that value.

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Most popular questions from this chapter

Use appropriate relationships from the chapter to determine the wavelength of the line in the emission spectrum of \(\mathrm{He}^{+}\) produced by an electron transition from \(n=5\) to \(n=2\).

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) \(n=2, \ell=0, m_{\ell}=0\)

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