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Chapter 36: Problem 36

Find (a) the energy and (b) the magnitude of the orbital angular momentum for an electron in the \(5 d\) state of hydrogen.

Short Answer

Expert verified
The energy of an electron in the 5d state of hydrogen is -0.544 eV. The magnitude of the orbital angular momentum is \(\sqrt{6} \hbar\).

Step by step solution

01

Calculate the Energy of an Electron in the Hydrogen Atom

Energy of an electron in a hydrogen atom is given by the equation \(E = -13.6/n^2\) eV, where \(n\) is the principal quantum number. For a \(5d\) electron, the principal quantum number \(n=5\). Hence, substituting \(n = 5\) into the equation, we get \(E = -13.6 / 5^2 = -0.544\) eV. The negative sign indicates that the electron is in a bound state.
02

Calculate the Magnitude of the Orbital Angular Momentum

The orbital angular momentum \(L\) of an electron in an atom is given by the equation \(L = \sqrt{l(l+1)} \hbar\), where \(l\) is the azimuthal quantum number, and \(\hbar\) is the reduced Planck's constant. Here, the given state is \(5d\) (n=5, l=2), so we substitute these values into the equation to get \(L = \sqrt{2(2+1)} \hbar = \sqrt{6} \hbar\). Thus, the magnitude of the orbital angular momentum is \(\sqrt{6} \hbar\).

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Most popular questions from this chapter

Using the table below, make a plot of atomic volume versus atomic number, for the elements from \(Z=30\) to \(Z=59\) listed in the table. Comment on the structure of your graph in relation to the periodic table, the electronic structures of atoms, and their chemical properties. (Volumes are in units of \(10^{-30} \mathrm{m}^{3} .\)) $$\begin{array}{|c|c|c|c|c|c|}\hline Z & V & Z & V & Z & V \\\\\hline 30 & 7.99 & 40 & 26.1 & 50 & 11.2 \\\\\hline 31 & 12.5 & 41 & 20.2 & 51 & 8.78 \\\\\hline 32 & 6.54 & 42 & 18.8 & 52 & 6.88 \\\\\hline 33 & 4.99 & 43 & 17.5 & 53 & 5.28 \\\\\hline 34 & 3.71 & 44 & 16.2 & 54 & 4.19 \\\\\hline 35 & 2.85 & 45 & 12.8 & 55 & 95.9 \\\\\hline 36 & 2.57 & 46 & 12.0 & 56 & 51.6 \\\\\hline 37 & 70.3 & 47 & 11.2 & 57 & 49.0 \\\\\hline 38 & 37.2 & 48 & 10.5 & 58 & 46.5 \\\\\hline 39 & 28.3 & 49 & 17.2 & 59 & 44.0 \\\\\hline\end{array}$$

Both the Bohr and Schr枚dinger theories predict the same ground-state energy for hydrogen. Do they agree about the angular momentum in the ground state? Explain.

Which of the following is not a possible value for the magnitude of the orbital angular momentum in hydrogen: (a) \(\sqrt{12} \hbar\) (b) \(\sqrt{20} \hbar ;\) (c) \(\sqrt{30} \hbar ;\) (d) \(\sqrt{40} \hbar ;\) (e) \(\sqrt{56} \hbar ?\)

Find the maximum possible magnitude for the orbital angular momentum of an electron in the \(n=7\) state of hydrogen.

How does the exclusion principle explain the diversity of chemical elements?
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