Concepts & Calculations

Note: Each of these problems consists of Concept Questions followed by a related quantitative Problem. The Concept Questions involve little or no mathematics. They focus on the concepts with which the problems deal. Recognizing the concepts is the essential initial step in any problem-solving technique.

| 55 | Concept Questions A hydrogen atom is in an excited state. (a) What is meant by the phrase “ionization energy of the excited state”? (b) Is the ionization energy of an excited state greater than, less than, or the same as the ionization energy for the ground state? Why?
Problem (a) What is the ionization energy of a hydrogen atom that is in the n = 4 state? (b) Determine the ratio of the ionization energy for n = 4 to that of the ground state. Check to make sure that your answer is consistent with your answers to the Concept Questions.

| 56 | Concept Questions Answer the following questions using the quantum mechanical model of the atom. (a) Can the orbital quantum number be greater than the principal quantum number n? (b) Can the orbital quantum number have negative values? (c) Is it possible for the magnetic quantum number to be greater than the orbital quantum number? (d) Can the magnetic quantum number have negative values?
Problem The table lists the quantum numbers for five states of the hydrogen atom. Which (if any) of them are not possible?

| n |  |
(a) | 3 | 3 | 0
(b) | 2 | 1 | −1
(c) | 4 | 2 | 3
(d) | 5 | −3 | 2
(e) | 4 | 0 | 0

| 57 | Concept Questions Answer the following questions using the quantum mechanical model of the atom. (a) For a given principal quantum number n, what values of the angular momentum quantum number are possible? (b) For a given angular momentum, what values of the magnetic quantum number are possible?
Problem Which of the following subshell configurations are not allowed? For those that are not allowed, give the reason why. (a) 3s1, (b) 2d2, (c) 3s4, (d) 4p8, (e) 5f12.

| 58 | Concept Questions Suppose that the molybdenum (Z = 42) target in an X-ray tube is replaced by a silver (Z = 47) target. Do (a) the cutoff wavelength and (b) the wavelength of the X-ray photon increase, decrease, or remain the same? Assume that the voltage across the tube is constant and is sufficient to produce characteristic X-rays from both targets. Provide a reason for each answer.
Problem The voltage across the X-ray tube is 35.0 kV. Determine (a) the cutoff wavelength and (b) the wavelengths of the X-ray photons emitted by the molybdenum and silver targets. Verify that your answers are consistent with your answers to the Concept Questions.

| 59 | Concept Questions The drawing shows three energy levels of a laser that are involved in the lasing action. These levels are analogous to the levels in the Ne atoms of a He-Ne laser. The E2 level is a metastable level, and the E0 level is the ground state. (a) What two levels are involved when an external source of energy initially causes the electrons to jump to a higher energy state? (b) Between what two levels does the population inversion occur? (c) Between what two levels does the lasing action occur? Provide a reason for each answer.

Problem The difference between the energy levels of the laser is shown in the drawing. (a) What energy (in eV per electron) must an external source provide to start the lasing action? (b) What is the wavelength of the laser light? (c) In what region of the electromagnetic spectrum (see Figure 24-10) does the laser light lie? Check your answers for consistency with respect to your answers to the Concept Questions.

| 60 | Concept Questions A sodium atom (Z = 11) contains 11 protons in its nucleus. Strictly speaking, the Bohr model does not apply, because the neutral atom contains 11 electrons instead of a single electron. However, we can apply the model to the outermost electron as an approximation, provided that we use an effective value Zeffective rather than 11 for the number of protons in the nucleus. Since electrons are sometimes between the nucleus and the outermost electron, Zeffective is less than 11. The repulsion from these “inner” electrons offsets some of the attractive force that the positive nuclear charge exerts on the outermost electron; this is called “shielding.” If the 10 inner electrons were permanently within the nucleus, we would have Zeffective = 1. The inner electrons are in orbit, however. (a) Since the orbiting inner electrons spend part of the time on the side of the nucleus opposite to the outermost electron, they are not always between the nucleus and the outermost electron. Considering this fact, is Zeffective less than or greater than 1? (b) Do you expect the radius of the outermost Bohr orbit to be greater with or without the effect of shielding to reduce Z to a value smaller than 11? Explain your answers.
Problem (a) The ionization energy for the outermost electron in a sodium atom is 5.1 eV. Use the Bohr model with Z = Zeffective to calculate a value for Zeffective. (b) Using Z = 11 and Z = Zeffective, determine the corresponding two values for the radius of the outermost Bohr orbit. Verify that your answers are consistent with your answers to the Concept Questions.

| 61 | Concept Questions The Bohr model, although not strictly applicable, can be used to estimate the minimum energy Emin that an incoming electron must have in an X-ray tube in order to knock a K-shell electron entirely out of an atom in the metal target. (a) Explain how the Bohr model is used for this purpose, (b) Explain how the Bohr model can be used to estimate the wavelength of the line in the X-ray spectrum of a metal. (c) If the wavelength of metal A (atomic number = ZA) is greater than that of metal B (atomic number = ZB), is ZA greater than, less than, or equal to ZB? (d) If the wavelength of metal A (atomic number = ZA) is greater than that of metal B (atomic number = ZB), is Emin for metal A greater than, less than, or equal to Emin for metal B?
Problem The X-ray wavelength of metal A is twice that of metal B. What is the ratio of Emin for metal A to Emin for metal B? Check to see that your answer is consistent with your answer to Concept Question (d).

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