Chapter 6

Consider the colors of the visible spectrum. (a) Which colors of light involve less energy than green light? (b) Which color of light has photons of greater energy, yellow or blue? (c) Which color of light has the higher frequency, blue or green?

Suppose you are standing \(225 \mathrm{m}\) from a radio transmitter. What is your distance from the transmitter in terms of the number of wavelengths if (a) the station is broadcasting at \(1150 \mathrm{kHz}\) (on the AM radio band)? \(\left(1 \mathrm{kHz}=1 \times 10^{3} \mathrm{Hz}\right)\) (b) the station is broadcasting at \(98.1 \mathrm{MHz}\) (on the FM radio band)? \(\left(1 \mathrm{MHz} \times 10^{6} \mathrm{Hz}\right)\)

Green light has a wavelength of \(5.0 \times 10^{2} \mathrm{nm}\) What is the energy, in joules, of one photon of green light? What is the energy, in joules, of 1.0 mol of photons of green light?

The most prominent line in the emission spectrum of aluminum is at \(396.15 \mathrm{nm}\). What is the frequency of this line? What is the energy of one photon with this wavelength? Of 1.00 mol of these photons?

The most prominent line in the emission spectrum of magnesium is \(285.2 \mathrm{nm} .\) Other lines are found at 383.8 and \(518.4 \mathrm{nm}\). In what region of the electromagnetic spectrum are these lines found? Which is the most energetic line? What is the energy of \(1.00 \mathrm{mol}\) of photons with the wavelength of the most energetic line?

Place the following types of radiation in order of increasing energy per photon: (a) yellow light from a sodium lamp (b) x-rays from an instrument in a dentist's office (c) microwaves in a microwave oven (d) your favorite FM music station at \(91.7 \mathrm{MHz}\)

Place the following types of radiation in order of increasing energy per photon: (a) radiation within a microwave oven (b) your favorite radio station (c) gamma rays from a nuclear reaction (d) red light from a neon sign (e) ultraviolet radiation from a sun lamp

An energy of \(3.3 \times 10^{-19} \mathrm{J} /\) atom is required to cause a cesium atom on a metal surface to lose an electron. Calculate the longest possible wavelength of light that can ionize a cesium atom. In what region of the electromagnetic spectrum is this radiation found?

You are an engineer designing a switch that works by the photoelectric effect. The metal you wish to use in your device requires \(6.7 \times 10^{-19} \mathrm{J} /\) atom to remove an electron. Will the switch work if the light falling on the metal has a wavelength of \(540 \mathrm{nm}\) or greater? Why or why not?

What are the wavelength and frequency of the radiation involved in the least energetic emission line in the Lyman series? What are the values of \(n_{\text {initial }}\) and \(n_{\text {final }} ?\)

The energy emitted when an electron moves from a higher energy state to a lower energy state in any atom can be observed as electromagnetic radiation. (a) Which involves the emission of less energy in the H atom, an electron moving from \(n=4\) to \(n=2\) or an electron moving from \(n=3\) to \(n=2 ?\) (b) Which involves the emission of more energy in the H atom, an electron moving from \(n=4\) to \(n=1\) or an electron moving from \(n=5\) to \(n=2 ?\) Explain fully.

If energy is absorbed by a hydrogen atom in its ground state, the atom is excited to a higher energy state. For example, the excitation of an electron from \(n=1\) to \(n=3\) requires radiation with a wavelength of \(102.6 \mathrm{nm} .\) Which of the following transitions would require radiation of longer wavelength than this? (a) \(n=2\) to \(n=4\) (b) \(n=1\) to \(n=4\) (c) \(n=1\) to \(n=5\) (d) \(n=3\) to \(n=5\)

Calculate the wavelength and frequency of light emitted when an electron changes from \(n=3\) to \(n=1\) in the \(\mathrm{H}\) atom. In what region of the spectrum is this radiation found?

Calculate the wavelength and frequency of light emitted when an electron changes from \(n=4\) to \(n=3\) in the \(\mathrm{H}\) atom. In what region of the spectrum is this radiation found?

An electron moves with a velocity of \(2.5 \times 10^{8} \mathrm{cm} / \mathrm{s}\) What is its wavelength?

A beam of electrons \(\left(m=9.11 \times 10^{-31} \mathrm{kg} / \text { electron }\right)\) has an average speed of \(1.3 \times 10^{8} \mathrm{m} / \mathrm{s}\). What is the wavelength of electrons having this average speed?

Calculate the wavelength, in nanometers, associated with a \(46-\mathrm{g}\) golf ball moving at \(30 . \mathrm{m} / \mathrm{s}\) (about \(67 \mathrm{mph}) .\) At what speed must the ball travel to have a wavelength of \(5.6 \times 10^{-3} \mathrm{nm} ?\)

A rifle bullet (mass \(=1.50 \mathrm{g}\) ) has a velocity of \(7.00 \times 10^{2} \mathrm{mph}\) (miles per hour). What is the wavelength associated with this bullet?

(a) When \(n=4,\) what are the possible values of \(\ell ?\) (b) When \(\ell\) is \(2,\) what are the possible values of \(m_{\ell} ?\) (c) For a \(4 s\) orbital, what are the possible values of \(n, \ell,\) and \(m_{\ell} ?\) (d) For a \(4 f\) orbital, what are the possible values of \(n, \ell,\) and \(m_{\ell} ?\)

(a) When \(n=4, \ell=2,\) and \(m_{\ell}=-1,\) to what orbital type does this refer? (Give the orbital label, such as \(1 s .\) ) (b) How many orbitals occur in the \(n=5\) electron shell? How many subshells? What are the letter labels of the subshells? (c) How many orbitals occur in an \(f\) subshell? What are the values of \(m_{\ell} ?\)

A possible excited state of the H atom has the electron in a \(4 p\) orbital. List all possible sets of quantum numbers \(n, \ell,\) and \(m_{\ell}\) for this electron.

A possible excited state for the H atom has an electron in a \(5 d\) orbital. List all possible sets of quantum numbers \(n, \ell,\) and \(m_{\ell}\) for this electron.

How many subshells occur in the electron shell with the principal quantum number \(n=4 ?\)

How many subshells occur in the electron shell with the principal quantum number \(n=5 ?\)

Explain briefly why each of the following is not a possible set of quantum numbers for an electron in an atom. (a) \(n=2, \ell=2, m_{\ell}=0\) (b) \(n=3, \ell=0, m_{\ell}=-2\) (c) \(n=6, \ell=0, m_{\ell}=1\)

Which of the following represent valid sets of quantum numbers? For a set that is invalid, explain briefly why it is not correct. (a) \(n=3, \ell=3, m_{\ell}=0\) (b) \(n=2, \ell=1, m_{\ell}=0\) (c) \(n=6, \ell=5, m_{\ell}=-1\) (d) \(n=4, \ell=3, m_{\ell}=-4\)

What is the maximum number of orbitals that can be identified by each of the following sets of quantum numbers? When "none" is the correct answer, explain your reasoning. (a) \(n=3, \ell=0, m_{\ell}=+1\) (b) \(n=5, \ell=1\) (c) \(n=7, \ell=5\) (d) \(n=4, \ell=2, m_{\ell}=-2\)

What is the maximum number of orbitals that can be identified by each of the following sets of quantum numbers? When "none" is the correct answer, explain your reasoning. (a) \(n=4, \ell=3\) (b) \(n=5\) (c) \(n=2, \ell=2\) (d) \(n=3, \ell=1, m_{\ell}=-1\)

Explain briefly why each of the following is not a possible set of quantum numbers for an electron in an atom. In each case, change the incorrect value (or values) to make the set valid. (a) \(n=4, \ell=2, m_{\ell}=0, m_{\mathrm{s}}=0\) (b) \(n=3, \ell=1, m_{\ell}=-3, m_{\mathrm{s}}=-1 / 2\) (c) \(n=3, \ell=3, m_{\ell}=-1, m_{\mathrm{s}}=+1 / 2\)

Explain briefly why each of the following is not a possible set of quantum numbers for an electron in an atom. In each case, change the incorrect value (or values) to make the set valid. (a) \(n=2, \ell=2, m_{\ell}=0, m_{\mathrm{s}}=+1 / 2\) (b) \(n=2, \ell=1, m_{\ell}=-1, m_{\mathrm{s}}=0\) (c) \(n=3, \ell=1, m_{\ell}=-2, m_{\mathrm{s}}=+1 / 2\)

State which of the following orbitals cannot exist according to the quantum theory: \(2 s, 2 d, 3 p, 3 f, 4 f\) and \(5 s .\) Briefly explain your answers.

State which of the following orbitals cannot exist according to the quantum theory: \(3 p, 4 s, 2 f,\) and 1p. Briefly explain your answers.

Write a complete set of quantum numbers \((n, \ell,\) and \(m_{\ell}\) ) that quantum theory allows for each of the following orbitals: (a) \(2 p,\) (b) \(3 d,\) and (c) \(4 f\)

Write a complete set of quantum numbers \((n, \ell,\) and \(m_{\ell}\) ) for each of the following orbitals: (a) \(5 f\) (b) \(4 d,\) and \((c) 2 s\)

A particular orbital has \(n=4\) and \(\ell=2 .\) What must this orbital be: \((a) 3 p,(b) 4 p,(c) 5 d,\) or (d) \(4 d ?\)

A given orbital has a magnetic quantum number of \(m_{\ell}=-1 .\) This could not be a(n) (a) \(f\) orbital (c) \(p\) orbital (b) \(d\) orbital (d) \(s\) orbital

How many nodal surfaces through the nucleus (planar nodes) are associated with each of the following orbitals? (a) \(2 s\) (b) \(5 d\) (c) \(5 f\)

How many nodal surfaces through the nucleus (planar nodes) are associated with each of the following atomic orbitals? (a) \(4 f\) (b) \(2 p\) (c) \(6 s\)

In what region of the electromagnetic spectrum for hydrogen is the Lyman series of lines found? The Balmer series?

Give the number of nodal surfaces through the nucleus (planar nodes) for each orbital type: \(s, p, d\) and \(f\)

What is the maximum number of \(s\) orbitals found in a given electron shell? The maximum number of \(p\) orbitals? Of \(d\) orbitals? Of \(f\) orbitals?

Sketch a picture of the \(90 \%\) boundary surface of an \(s\) orbital and the \(p_{x}\) orbital. Be sure the latter drawing shows why the \(p\) orbital is labeled \(p_{x}\) and not \(p_{y},\) for example.

An advertising sign gives off red light and green light. (a) Which light has higher-energy photons? (b) One of the colors has a wavelength of \(680 \mathrm{nm}\) and the other has a wavelength of \(500 \mathrm{nm}\). Which color has which wavelength? (c) Which light has the higher frequency?

Radiation in the ultraviolet region of the electromagnetic spectrum is quite energetic. It is this radiation that causes dyes to fade and your skin to develop a sunburn. If you are bombarded with 1.00 mol of photons with a wavelength of \(375 \mathrm{nm}\) what amount of energy, in kilojoules per mole of photons, are you being subjected to?

Assume your eyes receive a signal consisting of blue light, \(\lambda=470 \mathrm{nm} .\) The energy of the signal is \(2.50 \times 10^{-14} \mathrm{J} .\) How many photons reach your eyes?

If sufficient energy is absorbed by an atom, an electron can be lost by the atom and a positive ion formed. The amount of energy required is called the ionization energy. In the H atom, the ionization energy is that required to change the electron from \(n=1\) to \(n=\) infinity. Calculate the ionization energy for the He \(^{+}\) ion. Is the ionization energy of the He \(^{+}\) more or less than that of H? (Bohr's theory applies to He \(^{+}\) because it, like the H atom, has a single electron. The electron energy, however, is now given by \(E=-Z^{2} R h c / n^{2},\) where \(Z\) is the atomic number of helium.)

Suppose hydrogen atoms absorb energy so that electrons are excited to the \(n=7\) energy level. Electrons then undergo these transitions, among others: (a) \(n=7 \rightarrow n=1 ;\) (b) \(n=7 \rightarrow n=6 ;\) and (c) \(n=2 \rightarrow n=1 .\) Which of these transitions produces a photon with (i) the smallest energy, (ii) the highest frequency, and (iii) the shortest wavelength?

Rank the following orbitals in the H atom in order of increasing energy: \(3 s, 2 s, 2 p, 4 s, 3 p, 1 s,\) and \(3 d\)

How many orbitals correspond to each of the following designations? (a) \(3 p\) (b) \(4 p\) (c) \(4 p_{x}\) (d) \(6 d\) (e) \(5 d\) (f) \(5 f\) (g) \(n=5\) (h) \(7 s\)

Cobalt-60 is a radioactive isotope used in medicine for the treatment of certain cancers. It produces \(\beta\) particles and \(\gamma\) rays, the latter having energies of 1.173 and \(1.332 \mathrm{MeV} .\left(1 \mathrm{MeV}=10^{6}\) electron-volts \right. and \(\left.1 \mathrm{eV}=1.6022 \times 10^{-19} \mathrm{J} .\right)\) What are the wavelength and frequency of a \(\gamma\) -ray photon with an energy of \(1.173 \mathrm{MeV} ?\)