30.3 Bohr’s theory of the hydrogen atom  (Page 6/14)

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But there are limits to Bohr’s theory. It cannot be applied to multielectron atoms, even one as simple as a two-electron helium atom. Bohr’s model is what we call semiclassical . The orbits are quantized (nonclassical) but are assumed to be simple circular paths (classical). As quantum mechanics was developed, it became clear that there are no well-defined orbits; rather, there are clouds of probability. Bohr’s theory also did not explain that some spectral lines are doublets (split into two) when examined closely. We shall examine many of these aspects of quantum mechanics in more detail, but it should be kept in mind that Bohr did not fail. Rather, he made very important steps along the path to greater knowledge and laid the foundation for all of atomic physics that has since evolved.

Phet explorations: models of the hydrogen atom

How did scientists figure out the structure of atoms without looking at them? Try out different models by shooting light at the atom. Check how the prediction of the model matches the experimental results.

Section summary

• The planetary model of the atom pictures electrons orbiting the nucleus in the way that planets orbit the sun. Bohr used the planetary model to develop the first reasonable theory of hydrogen, the simplest atom. Atomic and molecular spectra are quantized, with hydrogen spectrum wavelengths given by the formula
$\frac{1}{\lambda }=R\left(\frac{1}{{n}_{\text{f}}^{2}}-\frac{1}{{n}_{\text{i}}^{2}}\right),$
where $\lambda$ is the wavelength of the emitted EM radiation and $R$ is the Rydberg constant, which has the value
$R=\text{1.097}×{\text{10}}^{7}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{-1}\text{.}$
• The constants ${n}_{i}$ and ${n}_{f}$ are positive integers, and ${n}_{i}$ must be greater than ${n}_{f}$ .
• Bohr correctly proposed that the energy and radii of the orbits of electrons in atoms are quantized, with energy for transitions between orbits given by
$\Delta E=\text{hf}={E}_{\text{i}}-{E}_{\text{f}},$
where $\Delta E$ is the change in energy between the initial and final orbits and $\text{hf}$ is the energy of an absorbed or emitted photon. It is useful to plot orbital energies on a vertical graph called an energy-level diagram.
• Bohr proposed that the allowed orbits are circular and must have quantized orbital angular momentum given by
$L={m}_{e}{\text{vr}}_{n}=n\frac{h}{2\pi }\left(n=1, 2, 3 \dots \right),$
where $L$ is the angular momentum, ${r}_{n}$ is the radius of the $n\text{th}$ orbit, and $h$ is Planck’s constant. For all one-electron (hydrogen-like) atoms, the radius of an orbit is given by
${r}_{n}=\frac{{n}^{2}}{Z}{a}_{\text{B}}\text{(allowed orbits}\phantom{\rule{0.25em}{0ex}}n=1, 2, 3, ...\right),$
$Z$ is the atomic number of an element (the number of electrons is has when neutral) and ${a}_{\text{B}}$ is defined to be the Bohr radius, which is
${a}_{\text{B}}=\frac{{h}^{2}}{{4\pi }^{2}{m}_{e}{\text{kq}}_{e}^{2}}=\text{0.529}×{\text{10}}^{-\text{10}}\phantom{\rule{0.25em}{0ex}}\text{m}\text{.}$
• Furthermore, the energies of hydrogen-like atoms are given by
${E}_{n}=-\frac{{Z}^{2}}{{n}^{2}}{E}_{0}\left(n=1, 2, 3 ...\right)\text{,}$
where ${E}_{0}$ is the ground-state energy and is given by
${E}_{0}=\frac{{2\pi }^{2}{q}_{e}^{4}{m}_{e}{k}^{2}}{{h}^{2}}=\text{13.6 eV.}$
Thus, for hydrogen,
${E}_{n}=-\frac{\text{13.6 eV}}{{n}^{2}}\left(n,=,1, 2, 3 ...\right)\text{.}$
• The Bohr Theory gives accurate values for the energy levels in hydrogen-like atoms, but it has been improved upon in several respects.

Conceptual questions

How do the allowed orbits for electrons in atoms differ from the allowed orbits for planets around the sun? Explain how the correspondence principle applies here.

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