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Chapter 6: Problem 103

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 that preceded Thomson's work, that the cathode rays are a wave phenomenon?

Short Answer

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Rutherford's experiments on the scattering of 伪 particles by a gold foil demonstrated that atoms have a small, positively charged nucleus containing most of the mass, with electrons orbiting in mostly empty space. This paved the way for Bohr's theory, which proposed quantized energy levels for electrons in hydrogen atoms and explained their discrete emission spectra. De Broglie's hypothesis, which suggests wave-particle duality for all particles, is consistent with J.J. Thomson's conclusion that electrons have mass and the earlier proposals that cathode rays are a wave phenomenon, as it unifies their wave-like and particle-like properties.

Step by step solution

01

Rutherford's Experiments

Rutherford's experiments involved the bombardment of a thin gold foil with 伪 particles (helium nuclei). He observed that most of the 伪 particles went straight through the foil with little or no deflection, but a small fraction of the particles were deflected at large angles. This observation led him to propose a new model where the positive charge and most of the mass of the atom is concentrated in a very small nucleus, and the electrons move around the nucleus in an otherwise mostly empty space.
02

Bohr's Theory of the Hydrogen Atom

Rutherford's nuclear model of the atom paved the way for Bohr's theory, which was focused on explaining the behavior of electrons in hydrogen atoms. Bohr proposed that the electrons in an atom can only exist in well-defined energy levels, and when an electron transitions between these levels, it absorbs or emits a specific amount of energy in the form of light. The quantized energy levels explained the observed discrete emission spectra of the hydrogen atom, which could not be accounted for by classical electromagnetic theory.
03

De Broglie's Hypothesis

De Broglie's hypothesis proposed the wave-particle duality of matter, stating that all particles, including electrons, have both particle-like and wave-like properties. This idea was based on the observation that the behavior of electrons seemed to exhibit characteristics of both particles and waves.
04

Consistency with J.J. Thomson's Conclusion

De Broglie's hypothesis is consistent with J.J. Thomson's conclusion that the electron has mass because it does not deny the particle-like properties of the electron. Instead, it suggests that the electron has both particle-like and wave-like properties, allowing it to have mass as well as other properties associated with waves, such as interference and diffraction.
05

Consistency with Wave Phenomenon Proposals

De Broglie's hypothesis is also consistent with proposals preceding Thomson's work that cathode rays are a wave phenomenon because it unifies the wave-like and particle-like properties of the electron, suggesting that its behavior as a cathode ray can be adequately explained by considering both aspects of its nature.

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Most popular questions from this chapter

The energy from radiation can be used to cause the rupture of chemical bonds. A minimum energy of \(941 \mathrm{~kJ} / \mathrm{mol}\) is required to break the nitrogen-nitrogen bond in \(\mathrm{N}_{2}\). What is the longest wavelength of radiation that possesses the necessary energy to break the bond? What type of electromagnetic radiation is this?

Sodium metal requires a photon with a minimum energy of \(4.41 \times 10^{-19} \mathrm{~J}\) to emit electrons. (a) What is the minimum frequency of light necessary to emit electrons from sodium via the photoelectric effect? (b) What is the wavelength of this light? (c) If sodium is irradiated with light of \(439 \mathrm{~nm}\), what is the maximum possible kinetic energy of the emitted electrons? (d) What is the maximum number of electrons that can be freed by a burst of light whose total energy is \(1.00 \mu \mathrm{J} ?\)

(a) What experimental evidence is there for the electron having a "spin"? (b) Draw an energy-level diagram that shows the relative energetic positions of a \(1 s\) orbital and a 2 s orbital. Put two electrons in the 1 s orbital. (c) Draw an arrow showing the excitation of an electron from the 1s to the 2 s orbital.

Give the numerical values of \(n\) and \(l\) corresponding to each of the following orbital designations: (a) \(3 p,(\) b) \(2 s\), (c) \(4 f\), (d) \(5 d\).

The discovery of hafnium, element number 72, provided a controversial episode in chemistry. G. Urbain, a French chemist, claimed in 1911 to have isolated an element number 72 from a sample of rare earth (elements 58-71) compounds. However, Niels Bohr believed that hafnium was more likely to be found along with zirconium than with the rare earths. D. Coster and \(G\). von Hevesy, working in Bohr's laboratory in Copenhagen, showed in 1922 that element 72 was present in a sample of Norwegian zircon, an ore of zirconium. (The name hafnium comes from the Latin name for Copenhagen, Hafnia). (a) How would you use electron configuration arguments to justify Bohr's prediction? (b) Zirconium, hafnium's neighbor in group \(4 \mathrm{~B}\), can be produced as a metal by reduction of solid \(\mathrm{ZrCl}_{4}\) with molten sodium metal. Write a balanced chemical equation for the reaction. Is this an oxidation- reduction reaction? If yes, what is reduced and what is oxidized? (c) Solid zirconium dioxide, \(\mathrm{ZrO}_{2}\), is reacted with chlorine gas in the presence of carbon. The products of the reaction are \(\mathrm{ZrCl}_{4}\) and two gases, \(\mathrm{CO}_{2}\) and \(\mathrm{CO}\) in the ratio \(1: 2 \mathrm{Write}\) a balanced chemical equation for the reaction. Starting with a 55.4-g sample of \(\mathrm{ZrO}_{2}\), calculate the mass of \(\mathrm{ZrCl}_{4}\) formed, assuming that \(\mathrm{ZrO}_{2}\) is the limiting reagent and assuming \(100 \%\) yield. (d) Using their electron configurations, account for the fact that \(\mathrm{Zr}\) and Hf form chlorides \(\mathrm{MCl}_{4}\) and oxides \(\mathrm{MO}_{2}\).
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