Chapter 7

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 greater frequency, blue or green?

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 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 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 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) radar signals (b) radiation within a microwave oven (c) gamma rays from a nuclear reaction (d) red light from a neon sign (c) ultraviolet radiation from a sun lamp

An energy of \(2.0 \times 10^{2} \mathrm{kJ} / \mathrm{mol}\) 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 nm or greater? Why or why not?

Consider only transitions involving the \(n=1\) through \(n=5\) energy levels for the H atom (where the energy level spacings below are not to scale). $$\begin{aligned} &\begin{aligned} & n=5 \\ & n=4 \\ & n=3 \\ & n=2 \end{aligned}\\\ &7\\\ &n=1 \end{aligned}$$ (a) How many emission lines are possible, considering only the five quantum levels? (b) Photons of the highest frequency are emitted in a transition from the level with \(n=\quad\) to a level with \(n=\) (c) The emission line having the longest wavelength corresponds to a transition from the level with \(n=\ldots\) to the level with \(n=\)

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 \(\mathrm{H}\) 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 \(\mathrm{H}\) 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 the level with \(n=1\) to the level with \(n=3\) 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 \(n=4\) (c) \(n=1\) to \(n=5\) (b) \(n=1\) to \(n=4\) (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 \(H\) atom. In what region of the spectrum is this radiation found?

An electron moves with a velocity of \(2.5 \times 10^{8} \mathrm{cm} \cdot \mathrm{s}^{-1}\) 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 \(1.3 \times 10^{8} \mathrm{m} \cdot \mathrm{s}^{-1} .\) What is the wavelength of electrons having this average speed?

Calculate the wavelength, in nanometers, associated with a \(1.0 \times 10^{2}-\mathrm{g}\) golf ball moving at \(30 . \mathrm{m} \cdot \mathrm{s}^{-1}\) (about \(67 \mathrm{mph}\) ). How fast 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 \(7.00 \times\) \(10^{2}\) mph. 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 \(m_{e} ?\) (d) For a 4 forbital, what are the possible values of \(n, \ell\) and \(m_{e} ?\)

(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) If a subshell is labeled \(f\), how many orbitals occur in the subshell? What are the values of \(m_{\ell} ?\)

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

A possible excited state for the \(\mathrm{H}\) 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 bricfly 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) \(n=3, \ell=3, m_{\ell}=0\) (c) \(n=6, \ell=5, m_{\ell}=-1\) (b) \(n=2, \ell=1, m_{\ell}=0\) (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. (c) \(n=7, \ell=5\) (a) \(n=3, \ell=0, m_{\ell}=+1\) (b) \(n=5, \ell=1\) (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) \(n=4, \ell=3\) (c) \(n=2, \ell=2\) (b) \(n=5\) (d) \(n=3, \ell=1, m_{\ell}=-1\)

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 are incorrect designations for orbitals according to the quantum theory: \(3 p, 4 s, 2 f\) and \(1 p .\) Briefly explain your answers.

Write a complete set of quantum numbers \(\left(n, \ell, \text { and } m_{\ell}\right)\) 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 \(\left.m_{\ell}\right)\) 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 \(m_{e}=\) -1. This could not be a (an) (a) forbital (c) \(p\) orbital (b) \(d\) orbital (d) s orbital

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

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

Which of the following are applicable when explaining the photoelectric effect? Correct any statements that are wrong. (a) Light is electromagnetic radiation. (b) The intensity of a light beam is related to its frequency. (c) Light can be thought of as consisting of massless particles whose energy is given by Planck's equation, \(E=h v\)

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 for each orbital type: \(s\) \(p, d,\) and \(f\)

What is the maximum number of sorbitals found in a given electron shell? The maximum number of porbitals? Of \(d\) orbitals? Of forbitals?

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.

Excited \(\mathrm{H}\) atoms have many emission lines. One series of lines, called the Pfund series, occurs in the infrared region. It results when an electron changes from higher energy levels to a level with \(n=5 .\) Calculate the wavelength and frequency of the lowest energy line of this series.

An advertising sign gives off red light and green light. (a) Which light has the 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 nm, what amount of energy, in kilojoules per mole of photons, are you being subjected to?

A cell phone sends signals at about \(850 \mathrm{MHz}\) ( \(1 \mathrm{MHz}=\) \(1 \times 10^{6} \mathrm{Hz}\) or cycles per second). (a) What is the wavelength of this radiation? (b) What is the energy of 1.0 mol of photons with a frequency of \(850 \mathrm{MHz}\) ? (c) Compare the energy in part (b) with the energy of a mole of photons of blue light \((420 \mathrm{nm})\) (d) Comment on the difference in energy between 850 MHz radiation and blue light.

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?

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

Rank the following orbitals in the \(\mathrm{H}\) 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) \(3 p\) (d) \(6 d\) (g) \(n=5\) (b) \(4 p\) (e) \(5 d\) (h) \(7 s\) (c) \(4 p_{x}\) (f) \(5 f\)

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 MeV. (1 MeV = 1 million electron-volts and 1 \(\mathrm{eV}=\) \(9.6485 \times 10^{4} \mathrm{J} / \mathrm{mol} .\). What are the wavelength and frequency of a \(\gamma\) -ray photon with an energy of \(1.173 \mathrm{MeV} ?\)

A Exposure to high doses of microwaves can cause damage. Estimate how many photons, with \(\lambda=12 \mathrm{cm}\) must be absorbed to raise the temperature of your eye by \(3.0^{\circ} \mathrm{C} .\) Assume the mass of an eye is \(11 \mathrm{g}\) and its specific heat capacity is \(4.0 \mathrm{J} / \mathrm{g} \cdot \mathrm{K}\)