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Chapter 6: Problem 109

The first 25 years of the twentieth century were momentous for the rapid pace of change in scientists' understanding of the nature of matter. (a) How did Rutherford's experiments on the scattering of \(\alpha\) particles by a gold foil set the stage for Bohr's theory of the hydrogen atom? (b) In what ways is de Broglie's hypothesis, as it applies to electrons, consistent with J. J. Thomson's conclusion that the electron has mass? In what sense is it consistent with proposals preceding Thomson's work that the cathode rays are a wave phenomenon?

Short Answer

Expert verified
Rutherford's gold foil experiment revealed that atoms have a concentrated positive charge in their nucleus, leading to Bohr's atomic model where electrons orbit the nucleus in discrete energy levels. de Broglie's hypothesis, which proposes wave-particle duality in electrons, is consistent with both J.J. Thomson's conclusion that electrons have mass and the earlier idea that cathode rays are a wave phenomenon by accounting for both the particle and wave experience.

Step by step solution

01

Introduction to Rutherford's Experiment

In Rutherford's experiment, a beam of alpha particles (helium nuclei) was directed at a thin gold foil, and the scattered alpha particles were detected on a fluorescent screen. The experiment showed that the alpha particles were deflected at various angles, suggesting that the gold atoms' positive charge was not distributed evenly but instead concentrated in a small central region called the nucleus. This set the stage for the development of the planetary model of the atom, with negatively charged electrons orbiting the central nucleus.
02

Bohr's Theory of the Hydrogen Atom

Bohr's theory was inspired by Rutherford's discovery of the nucleus. In Bohr's atomic model, negatively charged electrons orbit the positively charged nucleus in discrete energy levels, called orbitals. The electron can only exist in specific, quantized energy states, and absorbs or emits radiation (quanta of energy) when transitioning between these states. This theory successfully predicted the observed spectrum of hydrogen, providing a compelling explanation for the behavior of the simplest atom.
03

de Broglie's Hypothesis

Louis de Broglie's hypothesis proposed that all particles, including electrons, have both particle-like and wave-like properties, meaning they exhibit wave-particle duality. This was significant because it suggested that electrons in orbit around a nucleus could be treated as standing waves, with specific wavelengths corresponding to specific energy levels in atoms.
04

Consistency with Thomson's Conclusion

de Broglie's hypothesis is consistent with J.J. Thomson's conclusion that electrons have mass because it does not deny the particle aspect of the electron, but rather introduces wave-like properties to the existing particle model. The dual nature of electrons allows them to maintain their mass while also exhibiting wave-like behavior as required by de Broglie's hypothesis.
05

Consistency with Wave Phenomenon of Cathode Rays

Before Thomson's discovery of the electron, some scientists believed that cathode rays were a wave phenomenon. de Broglie's hypothesis is consistent with this idea because the wave-like aspect of electrons' behavior allows them to be considered as both particles and components of a wave-like phenomenon, thus reconciling both perspectives. By introducing wave-particle duality, de Broglie's hypothesis encompassed and explained both the particle and wave experience.

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Most popular questions from this chapter

Scientists have speculated that element 126 might have a moderate stability, allowing it to be synthesized and characterized. Predict what the condensed electron configuration of this element might be.

Bohr's model can be used for hydrogen-like ions-ions that have only one electron, such as \(\mathrm{He}^{+}\) and \(\mathrm{Li}^{2+} .\) (a) Why is the Bohr model applicable to \(\mathrm{Li}^{2+}\) ions but not to neutral Li atoms? (b) The ground-state energies of \(\mathrm{B}^{4+}, \mathrm{C}^{5+},\) and \(\mathrm{N}^{6+}\) are tabulated as follows: By examining these numbers, propose a relationship between the ground-state energy of hydrogen-like systems and the nuclear charge, \(Z\). (Hint: Divide by the ground-state energy of hydrogen $\left.-2.18 \times 10^{-18} \mathrm{~J}\right)$ (c) Use the relationship you derive in part (b) to predict the ground-state energy of the \(\mathrm{Be}^{3+}\) ion.

The visible emission lines observed by Balmer all involved $n_{\mathrm{f}}=2 .$ (a) Which of the following is the best explanation of why the lines with \(n_{\mathrm{f}}=3\) are not observed in the visible portion of the spectrum: (i) Transitions to \(n_{\mathrm{f}}=3\) are not allowed to happen, (ii) transitions to \(n_{\mathrm{f}}=3\) emit photons in the infrared portion of the spectrum, (iii) transitions to \(n_{\mathrm{f}}=3\) emit photons in the ultraviolet portion of the spectrum, or (iv) transitions to \(n_{\mathrm{f}}=3\) emit photons that are at exactly the same wavelengths as those to \(n_{\mathrm{f}}=2 .\) (b) Calculate the wavelengths of the first three lines in the Balmer series-those for which \(n_{1}=3,4\), and 5 -and identify these lines in the emission spectrum shown in Figure 6.11

The series of emission lines of the hydrogen atom for which \(n_{f}=4\) is called the Brackett series. (a) Determine the region of the electromagnetic spectrum in which the lines of the Brackett series are observed. (b) Calculate the wavelengths of the first three lines in the Brackett series - those for which \(n_{i}=5,6,\) and 7.

For orbitals that are symmetric but not spherical, the contour representations (as in Figures 6.23 and 6.24 ) suggest where nodal planes exist (that is, where the electron density is zero). For example, the \(p_{x}\) orbital has a node wherever \(x=0\). This equation is satisfied by all points on the \(y z\) plane, so this plane is called a nodal plane of the \(p_{x}\) orbital. (a) Determine the nodal plane of the \(p_{z}\) orbital. (b) What are the two nodal planes of the \(d_{x y}\) orbital? (c) What are the two nodal planes of the \(d_{x^{2}-y^{2}}\) orbital?
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