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Chapter 12: Problem 1
The nucleus of an ordinary hydrogen atom consists of (A) a neutron. (B) a proton. (C) a proton and a neutron. (D) a proton and two neutrons. (E) two protons and two neutrons.
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In 1923 Arthur Compton observed that \(x\)-rays scattered off free electrons were shifted in wavelength by an amount that could be explained by assuming that the \(\mathrm{x}\)-rays and electrons obeyed the relativistic relationships for energy and momentum. Compton used \(\mathrm{x}\)-rays \((\gamma)\) with a wavelength of approximately \(7.30 \times 10^{-11} \mathrm{~m}\). Assume that such a photon is incident on a stationary electron, as shown below. For purposes of illustration, the photon is reflected off the electron and its wavelength is observed to shift by a magnitude \(|\Delta \lambda|=\left|\lambda_o-\lambda_f\right|=h / m_e c\), where \(\lambda_o\) and \(\lambda_f\) are the initial and final wavelengths, respectively. a) What is the energy of the incoming photon in \(\mathrm{eV}\) ? in joules? b) Does the wavelength of the photon increase or decrease? Explain your reasoning. c) What is the momentum acquired by the electron?
Which of the following statements is true? The existence of the de Broglie wavelength \(\lambda_{d B}\) implies (A) that matter particles should undergo interference. (B) that matter waves travel at the speed of light. (C) that the frequency of matter waves is \(c / \lambda_{d B}\), where \(c\) is the speed of the particle. (D) that matter waves are given off by accelerating charges. (E) that matter waves are polarized.
A hypothetical atom has energy levels at \(-12 \mathrm{eV},-8 \mathrm{eV},-3 \mathrm{eV},-1 \mathrm{eV}\). a) Draw the energy levels of the atom. Label the levels with the principal quantum number. b) An electron with velocity \(v=1.326 \times 10^6 \mathrm{~m} / \mathrm{s}\) is incident on the atom. What is the de Broglie wavelength of the electron? Ignore any relativistic effects. c) Can the electron excite an electron in any of the energy levels to a higher state? If so, which are the two levels involved? d) An electron decays from the \(-3 \mathrm{eV}\) state to the \(-8 \mathrm{eV}\) state and emits a photon. What is its wavelength? What part of the electromagnetic spectrum is it in?
Meteorites created in the early solar system contaned aluminum-26, which is a radioactive isotope of aluminum with a half-life of \(7.2 \times 10^5 \mathrm{yrs}\). Aluminum-26 decays first into an excited state of magnesium-26 via the reaction \({ }_{13}^{26} \mathrm{Al} \rightarrow{ }_{12}^{26} \mathrm{Mg}^*+\mathrm{e}^{+}\), where the \(e^{+}\)has energy \(2.99 \mathrm{MeV}\). (The \(e^{+}\)is a positron; see previous problem. The asterisk (*) indicates "excited.") The \({ }_{12}^{26} \mathrm{Mg}^*\) then decays into the stable isotope magnesium- 26 via the reaction \({ }_{12}^{26} \mathrm{Mg}^* \rightarrow{ }_{12}^{26} \mathrm{Mg}+\gamma\). The \(\gamma\) has energy \(1.8 \mathrm{MeV}\). a) If you were asked to calculate the de Broglie wavelength of the positron, would it be permissible to use Newtonian physics? Justify your answer. b) What is the wavelength of the photon emitted when the excited magnesium-26 decays into its ground state? What is its momentum? c) \({ }_{12}^{26} \mathrm{Mg}\) has an atomic mass of \(25.9826 \mathrm{u}\). What is the speed of the recoiling nucleus when the photon is emitted? d) What is the nucleus' kinetic energy in electron volts? e) Precise measurements indicate that for a certain meteorite \(A\) the present ratio \({ }^{26} \mathrm{Mg} /{ }^{27} \mathrm{Al}=5 \times 10^{-5}\), where \({ }^{27} \mathrm{Al}\) is the common, stable isotope of aluminum. For a meteorite \(B\) the ratio is \({ }^{26} \mathrm{Mg} /{ }^{27} \mathrm{Al}=1.55 \times 10^{-7}\). Assuming that the different ratios are due to the difference in the meteorites' times of creation, how much older is meteorite \(B\) than \(A\) ?
A virus has a mass of about \(10^{-18} \mathrm{~kg}\). The de Broglie wavelength of a virus being blown in the wind at \(10 \mathrm{~m} / \mathrm{s}\) is nearest to (A) one billionth the size of a hydrogen atom. (B) one millionth the size of a hydrogen atom. (C) one thousandth the size of a hydrogen atom. (D) the size of a hydrogen atom. (E) one thousand times the size of a hydrogen atom.
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