Chapter Questions
Why is the quantum-mechanical model of the atom important for understanding chemistry?
What is light? How fast does it travel in a vacuum?
Define the wavelength and amplitude of a wave.
Define the frequency of electromagnetic radiation. How is frequencyrelated to wavelength?
What determines the color of light? Describe the difference betweenred light and blue light.
What determines the color of a colored object? Explain why grass appears green.
Give an approximate range of wavelengths for each type of electromagnetic radiation and summarize the characteristics and/or the usesof each.$$\begin{array}{l}{\text { a) gamma rays }} \\ {\text { b) } X \text { -rays }} \\ {\text { c) ultraviolet radiation }} \\ {\text { d) visible light }} \\ {\text { e) infrared radiation }} \\ {\text { f) microwave radiation }} \\ {\text { g) radio waves }}\end{array}$$
Explain the wave behavior known as interference. Explain the difference between constructive and destructive interference.
Explain the wave behavior known as diffraction. Draw the diffractionpattern that occurs when light travels through two slits comparable insize and separation to the light's wavelength.
Describe the photoelectric effect. How did experimental observationsof this phenomenon differ from the predictions of classical electromagnetic theory?
How did the photoelectric effect lead Einstein to propose that light isquantized?
What is a photon? How is the energy of a photon related to its wave-length? Its frequency?
What is an emission spectrum? How does an emission spectrum of a gasin a discharge tube differ from a white light spectrum?
Describe the Bohr model for the atom. How did the Bohr model account for the emission spectra of atoms?
Explain electron diffraction.
What is the de Broglie wavelength of an electron? What determines thevalue of the de Broglie wavelength for an electron?
What are complementary properties? How does electron diffractiondemonstrate the complementarity of the wave nature and particle natureof the electron?
Explain Heisenberg's uncertainty principle. What paradox is at least partially solved by the uncertainty principle?
What is a trajectory? What kind of information do you need to predictthe trajectory of a particle?
Why does the uncertainty principle make it impossible to predict a trajectory for the electron?
Newton's laws of motion are deterministic. Explain this statement.
An electron behaves in ways that at least partially indeterminate. Explain this statement.
What is a probability distribution map?
For each solution to the Schrodinger equation, which quantity can beprecisely specified: the electron's energy or its position? Explain.
What is a quantum-mechanical orbital?
What is the Schrodinger equation? What is a wave function? How is awave function related to an orbital?
What are the possible values of the principal quantum number $n$ ? Whatdoes the principal quantum number determine?
What are the possible values of the angular momentum quantum number $l$ ? What does the angular momentum quantum number determine?
What are the possible values of the magnetic quantum number $m_{F}^{>}$ Whatdoes the magnetic quantum number determine?
List all the orbitals in each principal level. Specify the three quantumnumbers for each orbital.$$\begin{array}{ll}{\text { a) } n} & {=1} \\ {\text { b) } n} & {=2} \\ {\text { c) } n} & {=3} \\ {\text { d) }n} & {=4}\end{array}$$
Explain the difference between a plot showing the probability densityfor an orbital and one showing the radial distribution function.
Sketch the general shapes of the $s,$ and $d$ orbitals.
List the four different sublevels. Given that only a maximum of two electrons can occupy an orbital, determine the maximum number of electrons that can exist in each sublevel.
Why are atoms usually portrayed as spheres when most orbitals are notspherically shaped?
The distance from the sun to Earth is $1.496 \times 10^{8} \mathrm{km}$ . How longdoes it take light to travel from the sun to Earth?
The nearest star to our sun is Proxima Centauri, at a distance of 4.3 light-years from the sun. A light-year is the distance that light travels in one year $(365$ days). How far away, in $\mathrm{km}$ , is Proxima Centauri from the sun?
List these types of electromagnetic radiation in order of (i) increasingwavelength and (ii) increasing energy per photon:$$\begin{array}{ll}{\text { a. radio waves }} & {\text { b. microwaves }} \\ {\text { c. infrared radiation }} & {\text { d. ultraviolet radiation }}\end{array}$$
List these types of electromagnetic radiation in order of (i) increasingfrequency and (ii) decreasing energy per photon:$$\begin{array}{ll}{\text { a. gamma rays }} & {\text { b. radio waves }} \\ {\text { c. microwaves }} & {\text { d. visible light }}\end{array}$$
Calculate the frequency of each wavelength of electromagneticradiation:
a. 632.8 $\mathrm{nm}$ (wavelength of red light from helium-neon laser)b. 503 $\mathrm{nm}$ (wavelength of maximum solar radiation)c. 0.052 $\mathrm{nm}$ (wavelength contained in medical X-rays)
Calculate the wavelength of each frequency of electromagneticradiation:a. 100.2 $\mathrm{MHz}$ (typical frequency for FM radio broadcasting)b. 1070 $\mathrm{kHz}$ (typical frequency for AM radio broadcasting) (assumefour significant figures)c. 835.6 $\mathrm{MHz}$ (common frequency used for cell phone communication)
Calculate the energy of a photon of electromagnetic radiation at each of the wavelengths indicated in Problem 39.
Calculate the energy of a photon of electromagnetic radiation at each of the frequencies indicated in Problem $40 .$
A laser pulse with wavelength 532 $\mathrm{nm}$ contains 3.85 $\mathrm{mJ}$ of energy. How many photons are in the laser pulse?
A heat lamp produces 32.8 watts of power at a wavelength of 6.5$\mu \mathrm{m} .$ How many photons are emitted per second? $(1$ watt $=1 \mathrm{J} / \mathrm{s})$
Determine the energy of 1 $\mathrm{mol}$ of photons for each kind of light. (Assume three significant figures.)\begin{equation}\begin{array}{l}{\text { a. infrared radiation }(1500 \mathrm{nm})} \\ {\text { b. visible light }(500 \mathrm{nm})} \\ {\text { c. ultraviolet radiation }(150 \mathrm{nm})}\end{array}\end{equation}
How much energy is contained in 1 mol of each?\begin{equation}\begin{array}{l}{\text { a. } X \text { -ray photons with a wavelength of } 0.135 \mathrm{nm}} \\ {\text { b. } \gamma \text { -ray photons with a wavelength of } 2.15 \times 10^{-5} \mathrm{nm}}\end{array}\end{equation}
Sketch the interference pattern that results from the diffraction of electrons passing through two closely spaced slits.
What happens to the interference pattern described in Problem 47 if the rate of electrons going through the slits is decreased to one electron per hour? What happens to the pattern if we try to determine which slit the electron goes through by using a laser placed directly behind the slits?
The resolution limit of a microscope is roughly equal to the wavelength of light used in producing the image. Electron microscopes use an electron beam (in place of photons) to produce much higher resolution images, about 0.20 $\mathrm{nm}$ in modern instruments. Assuming that the resolution of an electron microscope is equal to the de Broglie wavelength of the electrons used, to what speed must the electrons be accelerated to obtain a resolution of 0.20 $\mathrm{nm} ?$
The smallest atoms can themselves exhibit quantum-mechanical behavior. Calculate the de Broglie wavelength (in $\mathrm{pm}$ ) of a hydrogen atom traveling 475 $\mathrm{m} / \mathrm{s}$ .
What is the de Broglie wavelength of an electron traveling at $1.35 \times 10^{5} \mathrm{m} / \mathrm{s}$ ?
A proton in a linear accelerator has a de Broglie wavelength of 122 $\mathrm{pm.}$ What is the speed of the proton?
Calculate the de Broglie wavelength of a $143-$ g baseball traveling at 95 mph. Why is the wave nature of matter not important for a baseball?
A 0.22 -caliber handgun fires a $27-$ bullet at a velocity of 765 $\mathrm{m} / \mathrm{s}$ . Calculate the de Broglie wavelength of the bullet. Is the wave nature of matter significant for bullets?
An electron has an uncertainty in its position of 552 $\mathrm{pm} .$ What is the uncertainty in its velocity?
An electron traveling at $3.7 \times 10^{5} \mathrm{m} / \mathrm{s}$ has an uncertainty in its velocityof $1.88 \times 10^{5} \mathrm{m} / \mathrm{s}$ . What is the uncertainty in its position?
Which electron is, on average, closer to the nucleus: an electron in a 2$s$ orbital or an electron in a 3 s orbital?
Which electron is, on average, further from the nucleus: an electron in a 3$p$ orbital or an electron in a 4$p$ orbital?
What are the possible values of $l$ for each given value of $n$ ?$$\begin{array}{llll}{\text { a. } 1} & {\text { b. } 2} & {\text { c. } 3} & {\text { d. } 4}\end{array}$$
What are the possible values of $m_{l}$ for each given value of $l ?$$$\begin{array}{lllll}{\text { a. } 0} & {\text { b. } 1} & {\text { c. } 2} & {\text { d. } 3}\end{array}$$
Which set of quantum numbers cannot occur together to specify an orbital?$$\begin{array}{l}{\text { a. } n=2, l=1, m_{l}=-1} \\ {\text { b. } n=3, l=2, m_{l}=0} \\ {\text { c. } n=3, l=3, m_{l}=2} \\ {\text { d. } n=4, l=3, m_{l}=0}\end{array}$$
Which combinations of $n$ and $l$ represent real orbitals, and which donot exist?$$\begin{array}{lllll}{\text { a. } 1 s} & {\text { b. } 2 p} & {\text { c. } 4 s} & {\text { d. } 2 d}\end{array}$$
Sketch the 1$s$ and 2$p$ orbitals. How do the 2$s$ and 3$p$ orbitals differ from the 1$s$ and 2$p$ orbitals?
Sketch the 3$d$ orbitals. How do the 4$d$ orbitals differ from the 3$d$ orbitals?
An electron in a hydrogen atom is excited with electrical energy to an excited state with $n=2 .$ The atom then emits a photon. What is the value of $n$ for the electron following the emission?
Determine whether each transition in the hydrogen atom correspondsto absorption or emission of energy.$$\begin{array}{ll}{\text { a. }} & {n=3 \longrightarrow n=1} \\ {\text { b. }} & {n=2 \longrightarrow n=4} \\ {\text { c. }} & {n=4 \longrightarrow n=3}\end{array}$$
According to the quantum-mechanical model for the hydrogen atom, which electron transition produces light with the longer wavelength:$$2p \longrightarrow 1 s \quad {\text{or}}\quad 3p \longrightarrow 1 s?$$
According to the quantum-mechanical model for the hydrogen atom,which electron transition produces light with the longer wavelength:$$3p \longrightarrow 2 s \quad {\text{or}} \quad 4p \longrightarrow 3 p ?$$
Calculate the wavelength of the light emitted when an electron in a hydrogen atom makes each transition and indicate the region of the electromagnetic spectrum (infrared, visible, ultraviolet, etc.) where the light isfound.$$\begin{array}{ll}{\text { a. }} & {n=2 \longrightarrow n=1} \\ {\text { b. }} & {n=3 \longrightarrow n=1} \\ {\text { c. }} & {n=4 \longrightarrow n=2} \\ {\text { d. }} & {n=5 \longrightarrow n=2}\end{array}$$
Calculate the frequency of the light emitted when an electron in a hydrogen atom makes each transition:\begin{equation}\begin{array}{ll}{\text { a. } n} & {=4 \longrightarrow n=3} \\ {\text { b. } n} & {=5 \longrightarrow n=1} \\ {\text { c. } n} & {=5 \longrightarrow n=4} \\ {\text { d. } n} & {=6 \longrightarrow n=5}\end{array}\end{equation}
An electron in the $n=7$ level of the hydrogen atom relaxes to a lowerenergy level, emitting light of 397 $\mathrm{nm} .$ What is the value of $n$ for thelevel to which the electron relaxed?
An electron in a hydrogen atom relaxes to the $n=4$ level, emitting light of 114 THz. What is the value of $n$ for the level in which the electron originated?
Ultraviolet radiation and radiation of shorter wavelengths can damage biological molecules because they carry enough energy to break bonds within the molecules. A typical carbon-carbon bond requires348 $\mathrm{k} / \mathrm{mol}$ to break. What is the longest wavelength of radiation with enough energy to break carbon-carbon bonds?
The human eye contains a molecule called 11-$cis$-retinal that changes shape when struck with light of sufficient energy. The change in shape triggers a series of events that results in an electrical signal being sent to the brain. The minimum energy required to change the conformation of 11-$ cis $-retinal within the eye is about 164 kJ/mol. Calculate the longest wavelength visible to the human eye.
An argon ion laser puts out 5.0 $\mathrm{W}$ of continuous power at a wave- length of 532 $\mathrm{nm} .$ The diameter of the laser beam is 5.5 $\mathrm{mm}$ . If the laser is pointed toward a pinhole with a diameter of $1.2 \mathrm{mm},$ how many photons will travel through the pinhole per second? Assume that the light intensity is equally distributed throughout the entire cross-sectional area of the beam. $(1 \mathrm{W}=1 \mathrm{J} / \mathrm{s})$
A green leaf has a surface area of 2.50 $\mathrm{cm}^{2} .$ If solar radiation is$1000 \mathrm{W} / \mathrm{m}^{2},$ how many photons strike the leaf every second? Assumethree significant figures and an average wavelength of 504 $\mathrm{nm}$ forsolar radiation.
In a technique used for surface analysis called auger electron spectroscopy (AES), electrons are accelerated toward a metal surface. These electrons cause the emissions of secondary electrons called auger electrons $-$ from the metal surface. The kinetic energy of the auger electrons depends on the composition of the surface. The presence of oxygen atoms on the surface results in auger electrons with a kinetic energy of approximately 506 $\mathrm{eV} .$ What is the de Broglie wavelength of one ofthese electrons?$$\left[\mathrm{KE}=\frac{1}{2} m v^{2} ; 1\quad {\text {electron}}\quad {\text {volt}} \quad(\mathrm{eV})=1.602 \times 10^{-19} \mathrm{J}\right]$$
An X-ray photon of wavelength 0.989 $\mathrm{nm}$ strikes a surface. The emittedelectron has a kinetic energy of 969 $\mathrm{eV} .$ What is the binding energy ofthe electron in $\mathrm{k} \mathrm{j} / \mathrm{mol}$ ?$$\left[\mathrm{KE}=\frac{1}{2} m \nu^{2} ; 1 \text { electron volt }(\mathrm{eV})=1.602 \times 10^{-19} \mathrm{J}\right]$$
Ionization involves completely removing an electron from an atom. Howmuch energy is required to ionize a hydrogen atom in its ground (or lowest energy state? What wavelength of light contains enough energy in asingle photon to ionize a hydrogen atom?
The energy required to ionize sodium is 496 $\mathrm{kJ} / \mathrm{mol} .$ What minimumfrequency of light is required to ionize sodium?
Suppose that in an alternate universe, the possible values of $l$ are the integer values from 0 to $n($ instead of 0 to $n-1) .$ Assuming no other differences between this imaginary universe and ours, how many orbitalswould exist in each level?$$\begin{array}{ll}{\text { a. }} & {n=1} \\ {\text { b. }} & {n=2} \\ {\text { c. }} & {n=3}\end{array}$$
Suppose that, in an alternate universe, the possible values of $m_{l}$ are theinteger values including 0 ranging from $-l-1$ to $I+1$ (instead of simply $-l$ to $+1$ . How many orbitals exist in each sublevel?a. $s$ sublevelb. $p$ sublevelc. $d$ sublevel
An atomic emission spectrum of hydrogen shows three wavelengths:$1875 \mathrm{nm}, 1282 \mathrm{nm},$ and 1093 $\mathrm{nm} .$ Assign these wavelengths to transitions in the hydrogen atom.
An atomic emission spectrum of hydrogen shows three wavelengths:$121.5 \mathrm{nm}, 102.6 \mathrm{nm},$ and 97.23 $\mathrm{nm}$ . Assign these wavelengths to transitions in the hydrogen atom.
The binding energy of electrons in a metal is 193 $\mathrm{kJ} / \mathrm{mol} .$ Find thethreshold frequency of the metal.
In order for a thermonuclear fusion reaction of two deuterons $\left(_{1}^{2} \mathrm{H}^{+}\right)$to take place, the deuterons must collide with each deuteron traveling at $1 \times 10^{6} \mathrm{m} / \mathrm{s}$ . Find the wavelength of such a deuteron.
The speed of sound in air is 344 $\mathrm{m} / \mathrm{s}$ at room temperature. The lowestfrequency of a large organ pipe is 30 $\mathrm{s}^{-1}$ and the highest frequency of a piccolo is $1.5 \times 10^{4} \mathrm{s}^{-1} .$ Determine the difference in wavelength between these two sounds.
The distance from Earth to the sun is $1.5 \times 10^{8} \mathrm{km} .$ Find the numberof crests in a light wave of frequency $1.0 \times 10^{14} \mathrm{s}^{-1}$ traveling from thesun to the Earth.
The iodine molecule can be photodissociated (broken apart with light)into iodine atoms in the gas phase with light of wavelengths shorter thanabout 792 nm. A glass tube contains $1.80 \times 10^{17}$ iodine molecules.What minimum amount of light energy must be absorbed by the iodinein the tube to dissociate 15.0$\%$ of the molecules?
An ampule of napthalene in hexane contains $5.00 \times 10^{-4}$ mol naptha-lene. The napthalene is excited with a flash of light and then emits 15.5 $\mathrm{J}$of energy at an average wavelength of 349 $\mathrm{nm} .$ What percentage of thenaphthalene molecules emitted a photon?
A laser produces 20.0 $\mathrm{mW}$ of red light. In 1.00 $\mathrm{hr}$ , the laser emits$2.29 \times 10^{20} \mathrm{photons.}$ What is the wavelength of the laser?
A particular laser consumes 150.0 $\mathrm{Watts}$ of electrical power and producesa stream of $1.33 \times 10^{19} 1064 \mathrm{nm}$ photons per second. What is thepercent efficiency of the laser in converting electrical power to light?
An electron confined to a one-dimensional box has energy levels givenby the equation$$E_{n}=n^{2} b^{2} / 8 m L^{2}$$where $n$ is a quantum number with possible values of $1,2,3, \ldots, m$ isthe mass of the particle, and $L$ is the length of the box.a. Calculate the energies of the $n=1, n=2,$ and $n=3$ levels for anelectron in a box with a length of 155 $\mathrm{pm} .$b. Calculate the wavelength of light required to make a transition from$n=1 \longrightarrow n=2$ and from $n=2 \longrightarrow n=3 .$ In what region ofthe electromagnetic spectrum do these wavelengths lie?
The energy of a vibrating molecule is quantized much like the energyof an electron in the hydrogen atom. The energy levels of a vibratingmolecule are given by the equation$$E_{n}=\left(n+\frac{1}{2}\right) b \nu$$where $n$ is a quantum number with possible values of $1,2, \ldots,$ and $\nu$ isthe frequency of vibration. The vibration frequency of HCl is approxi-mately $8.85 \times 10^{13} \mathrm{s}^{-1} .$ What minimum energy is required to excite avibration in HCl? What wavelength of light is required to excite thisvibration?
The wave functions for the 1 s and 2 s orbitals are as follows:$$\begin{aligned} 1 s \psi &=(1 / \pi)^{1 /} \\ 2 s v_{r} &=(1 / 32 \pi\end{aligned}$$where $a_{0}$ is a constant $\left(a_{0}=53 \mathrm{pm}\right)$ and $r$ is the distance from the nucleus. Use a spreadsheet to make a plot of each of these wave functionsfor values of $r$ ranging from 0 pm to 200 pm. Describe the differencesin the plots and identify the node in the 2$s$ wave function.
Before quantum mechanics was developed, Johannes Rydberg developedan equation that predicted the wavelengths $(\lambda)$ in the atomic spectrum ofhydrogen:$$1 / \lambda=R\left(1 / m^{2}-1 / n^{2}\right)$$In this equation $R$ is a constant and $m$ and $n$ are integers. Use the quantum-mechanical model for the hydrogen atom to derive the Rydbergequation.
Find the velocity of an electron emitted by a metal whose threshold frequency is $2.25 \times 10^{14} \mathrm{s}^{-1}$ when it is exposed to visible light of wave-length $5.00 \times 10^{-7} \mathrm{m}$ .
Water is exposed to infrared radiation of wavelength $2.8 \times 10^{-4} \mathrm{cm} . $assume that all the radiation is absorbed and converted to heat. How manyphotons are required for the sample to absorb 16.72${J}$ of heat?
The 2005 Nobel Prize in Physics was given, in part, to scientists who had made ultra short pulses of light. These pulses are important in making measurements involving very short time periods. One challenge in making such pulses is the uncertainty principle, which can be stated with respect to energy and time as $\Delta E \cdot \Delta t \geq b / 4 \pi$ . What is the energy uncertainty $(\Delta E)$ associated with a short pulse of laser light that lasts for only 5.0 femtoseconds (fs)? Suppose the low energy end of the pulse had a wavelength of 722 nm. What is the wavelength of the high-energy end of the pulse that is limited only by the uncertainty principle?
A metal with a threshold frequency of $6.71 \times 10^{14} \mathrm{s}^{-1}$ emits an electronwith a velocity of $6.95 \times 10^{5} \mathrm{m} / \mathrm{s}$ when radiation of $1.01 \times 10^{15} \mathrm{s}^{-1}$strikes the metal. Calculate the mass of the electron.
Find the longest wavelength of a wave that can travel around in a circularorbit of radius 1.8 $\mathrm{m} .$
The amount of heat to melt ice 0.333 is kJ/g. Find the number ofphotons of wavelength $=6.42 \times 10^{-6} \mathrm{m}$ that must be absorbed to melt$5.55 \times 10^{-2}$ mol of ice.
Explain the difference between the Bohr model for the hydrogen atomand the quantum-mechanical model. Is the Bohr model consistentwith Heisenberg's uncertainty principle?
The light emitted from one of the following electronic transitions$(n=4 \longrightarrow n=3$ or $n=3 \longrightarrow n=2)$ in the hydrogen atomcauses the phototoelectric effect in a particular metal while light fromthe other transition does not. Which transition causes the photoelectric effect and why?
Determine whether an interference pattern is observed on the otherside of the slits in each experiment.a. An electron beam is aimed at two closely spaced slits. The beam isattenuated (made dimmer) to produce only 1 electron per minute.b. An electron beam is aimed at two closely spaced slits. A light beam isplaced at each slit to determine when an electron goes through the slit.c. A high-intensity light beam is aimed at two closely spaced slits.d. A gun is fired at a solid wall containing two closely spaced slits.the bullets that pass through the slits form an interference pattern onthe other side of the solid wall?)
Which transition in the hydrogen atom results in emitted light with thelongest wavelength?$$\begin{array}{ll}{\text { a. }} & {n=4 \longrightarrow n=3} \\ {\text { b. }} & {n=2 \longrightarrow n=1} \\ {\text { c. }} & {n=3 \longrightarrow n=2}\end{array}$$