Chapter Questions
A hypothetical electromagnetic wave is pictured here. What is the wavelength of this radiation?
For the electromagnetic wave described in Exercise 1 what are (a) the frequency, in hertz, and (b) the energy, in joules per photon?
The magnesium spectrum has a line at $266.8 \mathrm{nm}$. Which of these statements about this radiation is (are) correct? Explain.(a) It has a higher frequency than radiation with wavelength $402 \mathrm{nm}$(b) It is visible to the eye.(c) It has a greater speed in a vacuum than does red light of wavelength $652 \mathrm{nm}$(d) Its wavelength is longer than that of X-rays.
The most intense line in the cerium spectrum is at 418.7 nm.(a) Determine the frequency of the radiation producing this line.(b) In what part of the electromagnetic spectrum does this line occur?(c) Is it visible to the eye? If so, what color is it? If not, is this line at higher or lower energy than visible light?
Without doing detailed calculations, determine which of the following wavelengths represents light of the highest frequency: (a) $6.7 \times 10^{-4} \mathrm{cm} ;$ (b) $1.23 \mathrm{mm}$(c) $80 \mathrm{nm} ;$ (d) $6.72 \mu \mathrm{m}$
Without doing detailed calculations, arrange the following electromagnetic radiation sources in order of increasing frequency: (a) a red traffic light,(b) a $91.9 \mathrm{MHz}$ radio transmitter,(c) light with a frequency of $3.0 \times 10^{14} \mathrm{s}^{-1}$(d) light with a wavelength of $49 \mathrm{nm}$.
How long does it take light from the sun, 93 million miles away, to reach Earth?
In astronomy, distances are measured in light-years, the distance that light travels in one year. What is the distance of one light-year expressed in kilometers?
Use the Balmer equation (8.2) to determine(a) the frequency, in $s^{-1}$, of the radiation corresponding to $n=5$(b) the wavelength, in nanometers, of the line in the Balmer series corresponding to $n=7$(c) the value of $n$ corresponding to the Balmer series line at $380 \mathrm{nm}$
How would the Balmer equation (8.2) have to be modified to predict lines in the infrared spectrum of hydrogen? [Hint: Compare equations (8.2) and (8.6).]
Use Planck's equation (8.3) to determine(a) the energy, in joules per photon, of radiation of frequency $7.39 \times 10^{15} \mathrm{s}^{-1}$(b) the energy, in kilojoules per mole, of radiation of frequency $1.97 \times 10^{14} \mathrm{s}^{-1}$
Use Planck's equation (8.3) to determine(a) the frequency, in hertz, of radiation having an energy of $8.62 \times 10^{-21} \mathrm{J} /$ photon;(b) the wavelength, in nanometers, of radiation with$360 \mathrm{kJ} / \mathrm{mol}$ of energy.
What is $\Delta E$ for the transition of an electron from $n=6$ to $n=3$ in a Bohr hydrogen atom? What is the frequency of the spectral line produced?
What is $\Delta E$ for the transition of an electron from $n=5$ to $n=2$ in a Bohr hydrogen atom? What is the frequency of the spectral line produced?
To what value of $n$ in equation (8.2) does the line in the Balmer series at $389 \mathrm{nm}$ correspond?
The Lyman series of the hydrogen spectrum can be represented by the equation$$\nu=3.2881 \times 10^{15} \mathrm{s}^{-1}\left(\frac{1}{1^{2}}-\frac{1}{n^{2}}\right)(\text { where } n=2,3, \ldots)$$(a) Calculate the maximum and minimum wavelength lines, in nanometers, in this series.(b) What value of $n$ corresponds to a spectral line at95.0 nm?(c) Is there a line at $108.5 \mathrm{nm} ?$ Explain.
Calculate the wavelengths, in nanometers, of the first four lines of the Balmer series of the hydrogen spectrum, starting with the longest wavelength component.
A line is detected in the hydrogen spectrum at $1880 \mathrm{nm}$. Is this line in the Balmer series? Explain.
A certain radiation has a wavelength of $574 \mathrm{nm}$. What is the energy, in joules, of (a) one photon; (b) a mole of photons of this radiation?
What is the wavelength, in nanometers, of light with an energy content of $1979 \mathrm{kJ} / \mathrm{mol} ?$ In what portion of the electromagnetic spectrum is this light?
Without doing detailed calculations, indicate which of the following electromagnetic radiations has the greatest energy per photon and which has the least: (a) $662 \mathrm{nm}$ (b) $2.1 \times 10^{-5} \mathrm{cm} ;$ (c) $3.58 \mu \mathrm{m} ;$ (d) $4.1 \times 10^{-6} \mathrm{m}$.
Without doing detailed calculations, arrange the following forms of electromagnetic radiation in increasing order of energy per mole of photons: (a) radiation with $\nu=3.0 \times 10^{15} \mathrm{s}^{-1},$ (b) an infrared heat lamp, (c) radiation having $\lambda=7000 \AA,$ (d) dental X-rays.
In what region of the electromagnetic spectrum would you expect to find radiation having an energy per photon 100 times that associated with 988 nm radiation?
High-pressure sodium vapor lamps are used in street lighting. The two brightest lines in the sodium spectrum are at 589.00 and $589.59 \mathrm{nm}$. What is the difference in energy per photon of the radiations corresponding to these two lines?
The lowest-frequency light that will produce the photoelectric effect is called the threshold frequency.(a) The threshold frequency for indium is $9.96 \times$ $10^{14} \mathrm{s}^{-1} .$ What is the energy, in joules, of a photon of this radiation?(b) Will indium display the photoelectric effect withUV light? With infrared light? Explain.
The minimum energy required to cause the photoelectric effect in potassium metal is $3.69 \times 10^{-19} \mathrm{J}$ Will photoelectrons be produced when visible light shines on the surface of potassium? If 400 nm radiation is shone on potassium, what is the velocity of the ejected electrons?
Use the description of the Bohr atom given in the text to determine (a) the radius, in nanometers, of the sixth Bohr orbit for hydrogen; (b) the energy, in joules, of the electron when it is in this orbit.
Calculate the increase in (a) distance from the nucleus and (b) energy when an electron is excited from the first to the third Bohr orbit.
What are the (a) frequency, in $s^{-1}$, and (b) wavelength, in nanometers, of the light emitted when the electron in a hydrogen atom drops from the energy level $n=7$ to $n=4 ?$ (c) In what portion of the electromagnetic spectrum is this light?
Without doing detailed calculations, indicate which of the following electron transitions requires the greatest amount of energy to be absorbed by a hydrogen atom:from (a) $n=1$ to $n=2 ;$ (b) $n=2$ to $n=4 ;$ (c) $n=3$ to $n=9 ;$ (d) $n=10$ to $n=1$
For the Bohr hydrogen atom determine(a) the radius of the orbit $n=4$(b) whether there is an orbit having a radius of $4.00 \AA$(c) the energy level corresponding to $n=8$(d) whether there is an energy level at $-2.5 \times 10^{-17} \mathrm{J}$
Without doing detailed calculations, indicate which of the following electron transitions in the hydrogen atom results in the emission of light of the longest wavelength. (a) $n=4$ to $n=3 ;$ (b) $n=1$ to $n=2$(c) $n=1$ to $n=6 ;$ (d) $n=3$ to $n=2$.
What electron transition in a hydrogen atom, starting from the orbit $n=7,$ will produce light of wavelength$410 \mathrm{nm} ?$
What electron transition in a hydrogen atom, ending in the orbit $n=3,$ will produce light of wavelength $1090 \mathrm{nm} ?$
The emission spectrum below for a one-electron (hydrogen-like) species in the gas phase shows all the lines, before they merge together, resulting from transitions to the ground state from higher energy states. Line A has a wavelength of $103 \mathrm{nm}$.(a) What are the upper and lower principal quantum numbers corresponding to the lines labeled A and B?(b) Identify the one-electron species that exhibits the spectrum.
The emission spectrum below for a one-electron (hydrogen-like) species in the gas phase shows all the lines, before they merge together, resulting from transitions to the first excited state from higher energy states. Line A has a wavelength of $434 \mathrm{nm}$.(a) What are the upper and lower principal quantum numbers corresponding to the lines labeled A and B?(b) Identify the one-electron species that exhibits the spectrum.
The emission spectrum below for a one-electron (hydrogen-like) species in the gas phase shows all the lines, before they merge together, resulting from transitions to the first excited state from higher energy states. Line A has a wavelength of $27.1 \mathrm{nm}$.(a) What are the upper and lower principal quantum numbers corresponding to the lines labeled A and B?(b) Identify the one-electron species that exhibits the spectrum.
The emission spectrum below for a one-electron (hydrogen-like) species in the gas phase shows all the lines, before they merge together, resulting from transitions to the ground state from higher energy states. Line A has a wavelength of $10.8 \mathrm{nm}$.(a) What are the upper and lower principal quantum numbers corresponding to the lines labeled A and B?(b) Identify the one-electron species that exhibits the spectrum.
Which must possess a greater velocity to produce matter waves of the same wavelength (such as $1 \mathrm{nm}$ ), protons or electrons? Explain your reasoning.
What must be the velocity, in meters per second, of a beam of electrons if they are to display a de Broglie wavelength of $1 \mu \mathrm{m} ?$
Calculate the de Broglie wavelength, in nanometers, associated with a $145 \mathrm{g}$ baseball traveling at a speed of $168 \mathrm{km} / \mathrm{h} .$ How does this wavelength compare with typical nuclear or atomic dimensions?
What is the wavelength, in nanometers, associated with a $1000 \mathrm{kg}$ automobile traveling at a speed of $25 \mathrm{m} \mathrm{s}^{-1},$ that is, considering the automobile to be a matter wave? Comment on the feasibility of an experimental measurement of this wavelength.
Describe how the Bohr model of the hydrogen atom appears to violate the Heisenberg uncertainty principle.
Although Einstein made some early contributions to quantum theory, he was never able to accept the Heisenberg uncertainty principle. He stated, "God does not play dice with the Universe." What do you suppose Einstein meant by this remark? In reply to Einstein's remark, Niels Bohr is supposed to have said, "Albert, stop telling God what to do." What do you suppose Bohr meant by this remark?
A proton is accelerated to one-tenth the velocity of light, and this velocity can be measured with a precision of $1 \% .$ What is the uncertainty in the position of this proton?
Show that the uncertainty principle is not significant when applied to large objects such as automobiles. Assume that $m$ is precisely known; assign a reasonable value to either the uncertainty in position or the uncertainty in velocity, and estimate a value of the other.
What must be the velocity of electrons if their associated wavelength is to equal the radius of the first Bohr orbit of the hydrogen atom?
What must be the velocity of electrons if their associated wavelength is to equal the longest wavelength line in the Lyman series? [Hint: Refer to Figure 8-14.]
A standing wave in a string $42 \mathrm{cm}$ long has a total of six nodes (including those at the ends). What is the wavelength, in centimeters, of this standing wave?
What is the length of a string that has a standing wave with four nodes (including those at the ends) and $\lambda=17 \mathrm{cm} ?$
Calculate the wavelength of the electromagnetic radiation required to excite an electron from the ground state to the level with $n=4$ in a one-dimensional box 50. pm long.
An electron in a one-dimensional box requires a wavelength of $618 \mathrm{nm}$ to excite an electron from the $n=2$ level to the $n=4$ level. Calculate the length of the box.
An electron in a $20.0 \mathrm{nm}$ box is excited from the ground state into a higher energy state by absorbing a photon of wavelength $8.60 \times 10^{-5} \mathrm{m}$. Determine the final energy state.
Calculate the wavelength of the electromagnetic radiation required to excite a proton from the ground state to the level with $n=4$ in a one-dimensional box 50. pm long.
Describe some of the differences between the orbits of the Bohr atom and the orbitals of the wave mechanical atom. Are there any similarities?
The greatest probability of finding the electron in a small-volume element of the 1 s orbital of the hydrogen atom is at the nucleus. Yet the most probable distance of the electron from the nucleus is $53 \mathrm{pm}$. How can you reconcile these two statements?
Select the correct answer and explain your reasoning. An electron having $n=3$ and $m_{\ell}=0$ (a) must have$m_{s}=+\frac{1}{2} ;(\mathbf{b})$ must have $\ell=1 ;(\mathbf{c})$ may have $\ell=0,1$or $2 ;$ (d) must have $\ell=2$.
Write an acceptable value for each of the missing quantum numbers.(a) $n=3, \ell=?, m_{\ell}=2, m_{s}=+\frac{1}{2}$(b) $n=?, \ell=2, m_{\ell}=1, m_{s}=-\frac{1}{2}$(c) $n=4, \ell=2, m_{\ell}=0, m_{s}=?$(d) $n=?, \ell=0, m_{\ell}=?, m_{s}=?$
What type of orbital (i.e., $3 s, 4 p, \ldots)$ is designated by these quantum numbers?(a) $n=5, \ell=1, m_{\ell}=0$(b) $n=4, \ell=2, m_{\ell}=-2$(c) $n=2, \ell=0, m_{\ell}=0$
Which of the following statements is (are) correct for an electron with $n=4$ and $m_{\ell}=2 ?$ Explain.(a) The electron is in the fourth principal shell.(b) The electron may be in a $d$ orbital.(c) The electron may be in a $p$ orbital.(d) The electron must have $m_{s}=+\frac{1}{2}$
Concerning the electrons in the shells, subshells, and orbitals of an atom, how many can have(a) $n=4, \ell=2, m_{\ell}=1,$ and $m_{s}=+\frac{1}{2} ?$(b) $n=4, \ell=2,$ and $m_{\ell}=1 ?$(c) $n=4$ and $\ell=2 ?$(d) $n=4 ?$(e) $n=4, \ell=2,$ and $m_{s}=+\frac{1}{2} ?$
Concerning the concept of subshells and orbitals,(a) How many subshells are found in the $n=3$ level?(b) What are the names of the subshells in the $n=3$ level?(c) How many orbitals have the values $n=4$ and $\ell=3 ?$(d) How many orbitals have the values $n=3, \ell=2$ and $m_{\ell}=-2 ?$(e) What is the total number of orbitals in the $n=4$ level?
Calculate the finite value of $r,$ in terms of $a_{0},$ at which the node occurs in the wave function of the $2 s$ orbital of a hydrogen atom.
Calculate the finite value of $r,$ in terms of $a_{0},$ at which the node occurs in the wave function of the 2 s orbital of a $\mathrm{Li}^{2+}$ ion.
Show that the probability of finding a $2 p_{y}$ electron in the $x z$ plane is zero.
Show that the probability of finding a $3 d_{x z}$ electron in the $x y$ plane is zero.
Prepare a two-dimensional plot of $Y(\theta, \phi)$ for the $p_{y}$ orbital in the $x y$ plane.
Prepare a two-dimensional plot of $Y^{2}(\theta, \phi)$ for the $p_{y}$ orbital in the $x y$ plane.
Using a graphical method, show that in a hydrogen atom the radius at which there is a maximum probability of finding an electron is $a_{0}(53 \mathrm{pm})$.
Use a graphical method or some other means to show that in a $\mathrm{Li}^{2+}$ ion, the radius at which there is a maximum probability of finding an electron is $\frac{a_{0}}{3}(18 \mathrm{pm})$.
Identify the orbital that has (a) one radial node and one angular node; (b) no radial nodes and two angular nodes; (c) two radial nodes and three angular nodes.
Identify the orbital that has (a) two radial nodes and one angular node; (b) five radial nodes and zero angular nodes; (c) one radial node and four angular nodes.
A contour map for an atomic orbital of hydrogen is shown at the top of page 355 for the $x y$ and $x z$ planes. Identify the orbital.
A contour map for an atomic orbital of hydrogen is shown below for the $x y$ and $x z$ planes. Identify the type $(s, p, d, f, g \ldots)$ of orbital.
On the basis of the periodic table and rules for electron configurations, indicate the number of (a) $2 p$ electrons in $\mathrm{N} ;$ (b) $4 \mathrm{s}$ electrons in $\mathrm{Rb} ;$ (c) $4 \mathrm{d}$ electrons in As; (d) $4 f$ electrons in $\mathrm{Au} ;$ (e) unpaired electrons in $\mathrm{Pb} ;$ (f) elements in group 14 of the periodic table; (g) elements in the sixth period of the periodic table.
Based on the relationship between electron configurations and the periodic table, give the number of(a) outer-shell electrons in an atom of $\mathrm{Sb} ;$ (b) electrons in the fourth principal electronic shell of $\mathrm{Pt} ;$ (c) elements whose atoms have six outer-shell electrons; (d) unpaired electrons in an atom of Te; (e) transition elements in the sixth period.
Which of the following is the correct orbital diagram for the ground-state electron configuration of phosphorus? Explain what is wrong with each of the others.
Which of the following is the correct orbital diagram for the ground-state electron configuration of molybdenum? Explain what is wrong with each of the others.
Use the basic rules for electron configurations to indicate the number of (a) unpaired electrons in an atom of $\mathrm{P} ;$ (b) $3 d$ electrons in an atom of $\mathrm{Br} ;$ (c) $4 p$ electrons in an atom of $\mathrm{Ge} ;$ (d) $6 \mathrm{s}$ electrons in an atom of $\mathrm{Ba}$ (e) $4 f$ electrons in an atom of Au.
Use orbital diagrams to show the distribution of electrons among the orbitals in (a) the $4 p$ subshell of Br;(b) the $3 d$ subshell of $\mathrm{Co}^{2+},$ given that the two electrons lost are $4 s ;$ (c) the $5 d$ subshell of $\mathrm{Pb}$.
The recently discovered element 114 should most closely resemble Pb.(a) Write the electron configuration of $\mathrm{Pb}$.(b) Propose a plausible electron configuration for $8 \equiv$ element 114.
Without referring to any tables or listings in the text, mark an appropriate location in the blank periodic table provided for each of the following: (a) the fifthperiod noble gas; (b) a sixth-period element whose atoms have three unpaired $p$ electrons; (c) a $d$ -block element having one $4 \mathrm{s}$ electron; (d) a $p$ -block element that is a metal.
Which of the following electron configurations corresponds to the ground state and which to an excited state?
To what neutral atom do the following valence-shell configurations correspond? Indicate whether the configuration corresponds to the ground state or an excited state.
What is the expected ground-state electron configuration for each of the following elements? (a) mercury;(b) calcium;(c) polonium; (d) tin; (e) tantalum;(f) iodine.
What is the expected ground-state electron configuration for each of the following elements? (a) tellurium;(b) cesium; (c) selenium; (d) platinum; (e) osmium; (f) chromium.
The following electron configurations correspond to the ground states of certain elements. Name each element. (a) $[\mathrm{Rn}] 7 s^{2} 6 d^{2} ;$ (b) $[\mathrm{He}] 2 s^{2} 2 p^{2} ;$ (c) $[\mathrm{Ar}] 3 d^{3} 4 s^{2}$(d) $[\mathrm{Kr}] 4 d^{10} 5 s^{2} 5 p^{4} ;$ (e) $[\mathrm{Xe}] 4 f^{2} 6 s^{2} 6 p^{1}$
The following electron configurations correspond to the ground states of certain elements. Name each element.(a) $[\mathrm{Ar}] 3 d^{10} 4 s^{2} 4 p^{3} ;$ (b) $[\mathrm{Ne}] 3 s^{2} 3 p^{4} ;$ (c) $[\mathrm{Ar}] 3 d^{1} 4 s^{2}$(d) $[\mathrm{Kr}] 4 d^{6} 5 s^{2} ;$ (e) $[\mathrm{Xe}] 4 f^{12} 6 s^{2}$
Derive the Balmer equation from equation (8.6)
Electromagnetic radiation can be transmitted through a vacuum or empty space. Can heat be similarly transferred? Explain.
The work function is the energy that must be supplied to cause the release of an electron from a photoelectric material. The corresponding photon frequency is the threshold frequency. The higher the energy of the incident light, the more kinetic energy the electrons have in moving away from the surface. The work function for mercury is equivalent to $435 \mathrm{kJ} / \mathrm{mol}$ photons.(a) Can the photoelectric effect be obtained with mercury by using visible light? Explain.(b) What is the kinetic energy, in joules, of the ejected electrons when light of 215 nm strikes a mercury surface?(c) What is the velocity, in meters per second, of the ejected electrons in part (b)?
Infrared lamps are used in cafeterias to keep food warm. How many photons per second are produced by an infrared lamp that consumes energy at the rate of $95 \mathrm{W}$ and is $14 \%$ efficient in converting this energy to infrared radiation? Assume that the radiation has a wavelength of $1525 \mathrm{nm}$.
In $5.0 \mathrm{s},$ a 75 watt light source emits $9.91 \times 10^{20} \mathrm{pho-}$ tons of a monochromatic (single wavelength) radiation. What is the color of the emitted light?
In everyday usage, the term "quantum jump" describes a change of a very significant magnitude compared to more gradual, incremental changes; it is similar in meaning to the term "a sea change." Does quantum jump have the same meaning when applied to events at the atomic or molecular level? Explain.
The Pfund series of the hydrogen spectrum has as its longest wavelength component a line at $7400 \mathrm{nm}$ Describe the electron transitions that produce this series. That is, give a Bohr quantum number that is common to this series.
Between which two orbits of the Bohr hydrogen atom must an electron fall to produce light of wavelength $1876 \mathrm{nm} ?$
Use appropriate relationships from the chapter to determine the wavelength of the line in the emission spectrum of $\mathrm{He}^{+}$ produced by an electron transition from $n=5$ to $n=2$.
Draw an energy-level diagram that represents all the possible lines in the emission spectrum of hydrogen atoms produced by electron transitions, in one or more steps, from $n=5$ to $n=1$.
An atom in which just one of the outer-shell electrons is excited to a very high quantum level $n$ is called a "high Rydberg" atom. In some ways, all these atoms resemble a Bohr hydrogen atom with its electron in a high-numbered orbit. Explain why you might expect this to be the case.
If all other rules governing electron configurations were valid, what would be the electron configuration of cesium if (a) there were three possibilities for electron spin; (b) the quantum number $\ell$ could have the value $n ?$
Ozone, $\mathrm{O}_{3},$ absorbs ultraviolet radiation and dissociates into $\mathrm{O}_{2}$ molecules and $\mathrm{O}$ atoms: $\mathrm{O}_{3}+h \nu \longrightarrow$ $\mathrm{O}_{2}+\mathrm{O} . \mathrm{A} 1.00 \mathrm{L}$ sample of air at $22^{\circ} \mathrm{C}$ and $748 \mathrm{mmHg}$ contains $0.25 \mathrm{ppm}$ of $\mathrm{O}_{3}$. How much energy, in joules, must be absorbed if all the $\mathrm{O}_{3}$ molecules in the sample of air are to dissociate? Assume that each photon absorbed causes one $\mathrm{O}_{3}$ molecule to dissociate, and that the wavelength of the radiation is $254 \mathrm{nm}$.
Radio signals from Voyager 1 in the 1970 s were broadcast at a frequency of 8.4 GHz. On Earth, this radiation was received by an antenna able to detect signals as weak as $4 \times 10^{-21} \mathrm{W}$. How many photons per second does this detection limit represent?
Certain metal compounds impart colors to flames sodium compounds, yellow; lithium, red; barium, green-and flame tests can be used to detect these elements. (a) At a flame temperature of $800^{\circ} \mathrm{C}$, can collisions between gaseous atoms with average kinetic energies supply the energies required for the emission of visible light? (b) If not, how do you account for the excitation energy?
The angular momentum of an electron in the Bohr hydrogen atom is mur , where $m$ is the mass of the electron, $u,$ its velocity, and $r,$ the radius of the Bohr orbit. The angular momentum can have only the values nh/2 $\pi$, where $n$ is an integer (the number of the Bohr orbit). Show that the circum frences of the various Bohr orbits are integral multiples of the de Broglie wavelengths of the electron treated as a matter wave.
A molecule of chlorine can be dissociated into atoms by absorbing a photon of sufficiently high energy. Any excess energy is translated into kinetic energy as the atoms recoil from one another. If a molecule of chlorine at rest absorbs a photon of 300 nm wavelength, what will be the velocity of the two recoiling atoms? Assume that the excess energy is equally divided between the two atoms. The bond energy of $\mathrm{Cl}_{2}$ is $242.6 \mathrm{kJ} \mathrm{mol}^{-1}$
Refer to the Integrative Example. Determine whether or not $n=138$ is a bound state. If it is, what sort of state is it? What is the radius of the orbit and how many revolutions per second does the electron make about the nucleus?
Using the relationships given in Table $8.1,$ find the finite values of $r,$ in terms of $a_{0},$ of the nodes for a 3 s orbital.
Use a graphical method or some other means to determine the radius at which the probability of finding a $2 s$ orbital is maximum.
Using the relationships in Table $8.1,$ prepare a sketch of the $95 \%$ probability surface of a $4 p_{x}$ orbital.
Show that the volume of a spherical shell of radius $r$ and thickness $d r$ is $4 \pi r^{2} d r .$ [Hint: This exercise requires calculus.]
In the ground state of a hydrogen atom, what is the probability of finding an electron anywhere in a sphere of radius (a) $a_{0},$ or $\left(\text { b) } 2 a_{0} ?\right.$
When atoms in excited states collide with unexcited atoms they can transfer their excitation energy to those atoms. The most efficient energy transfer occurs when the excitation energy matches the energy of an excited state in the unexcited atom. Assuming that we have a collection of excited hydrogen atoms in the $2 s^{1}$ excited state, are there any transitions of $\mathrm{He}^{+}$ that could be most efficiently excited by the hydrogen atoms?
We have noted that an emission spectrum is a kind of "atomic fingerprint." The various steels are alloys of iron and carbon, usually containing one or more other metals. Based on the principal lines of their atomic spectra, which of the metals in the table below are likely to be present in a steel sample whose hypothetical emission spectrum is pictured? Is it likely that still other metals are present in the sample? Explain.
Balmer seems to have deduced his formula for the visible spectrum of hydrogen just by manipulating numbers. A more common scientific procedure is to graph experimental data and then find a mathematical equation to describe the graph. Show that equation (8.2) describes a straight line. Indicate which variables must be plotted, and determine the numerical values of the slope and intercept of this line. Use data from Figure $8-10$ to confirm that the four lines in the visible spectrum of hydrogen fall on the straight-line graph.
Follow up on the dartboard analogy of Figure $8-34$ by plotting a graph of the scoring summary tabulated below. That is, plot the number of hits as a function of the scoring ring: $50,40, \ldots \ldots$ What illustration in the text does this plot most resemble? Explain similarities and differences between the two.
Emission and absorption spectra of the hydrogen atom exhibit line spectra characteristic of quantized systems. In an absorption experiment, a sample of hydrogen atoms is irradiated with light with wavelengths ranging from 100 to 1000 nm. In an emission spectrum experiment, the hydrogen atoms are excited through an energy source that provides a range of energies from 1230 to $1240 \mathrm{kJ} \mathrm{mol}^{-1}$ to the atoms. Assume that the absorption spectrum is obtained at room temperature, when all atoms are in the ground state.(a) Calculate the position of the lines in the absorption spectrum.(b) Calculate the position of the lines in the emission spectrum.(c) Compare the line spectra observed in the two experiments. In particular, will the number of lines
Diffraction of radiation takes place when the distance between the scattering centers is comparable to the wavelength of the radiation.(a) What velocity must helium atoms possess to be diffracted by a film of silver atoms in which the spacing is $100 \mathrm{pm}$ ?(b) Electrons accelerated through a certain potential are diffracted by a thin film of gold. Would you expect a beam of protons accelerated through the same potential to be diffracted when it strikes the film of gold? If not, what would you expect to see instead?
The emission spectrum below is for hydrogen atoms in the gas phase. The spectrum is of the first few emission lines from principal quantum number 6 down to all possible lower levels.Not all possible de-excitations are possible; the transitions are governed by what is known as a selection rule. The selection rule states that the allowed transitions correspond to $\Delta n$ being arbitrary while $\Delta \ell=\pm 1 .$ This selection rule can be understood in terms of conservation of momentum, since a photon has an angular momentum of unity. Using this selection rule, identify the transitions, in terms of the types of orbital $(s, p, d, f),$ involved, that are observed in the spectrum shown on page 358 .In the presence of a magnetic field, the lines split into more lines according to the magnetic quantum number. The selection rule for transitions between different magnetic quantum numbers is $\Delta m_{\ell}=0$ and $\pm 1 .$ Identify the line(s) in the spectrum that splits into the greatest number of lines.
In your own words, define the following terms or symbols: (a) $\lambda ;$ (b) $\nu ;$ (c) $h ;$ (d) $\psi ;$ (e) principal quantum number, $n$.
Briefly describe each of the following ideas or phenomena: (a) atomic (line) spectrum; (b) photoelectric effect; (c) matter wave; (d) Heisenberg uncertainty principle; (e) electron spin; (f) Pauli exclusion principle; (g) Hund's rule; (h) orbital diagram; (i) electron charge density; (j) radial electron density.
Explain the important distinctions between each pair of terms: (a) frequency and wavelength; (b) ultraviolet and infrared light; (c) continuous and discontinuous spectra; (d) traveling and standing waves;(e) quantum number and orbital; (f) spd f notation and orbital diagram; (g) $s$ block and $p$ block; (h) main group and transition element; (i) the ground state and excited state of a hydrogen atom.
Describe two ways in which the orbitals of multielectron atoms resemble hydrogen orbitals and two ways in which they differ from hydrogen orbitals.
Explain the phrase $e$ffective nuclear charge. How is this related to the shielding effect?
With the help of sketches, explain the difference between a $p_{x}, p_{y},$ and $p_{z}$ orbital.
With the help of sketches, explain the difference between a $2 p_{z}$ and $3 p_{z}$ orbital.
If traveling at equal speeds, which of the following matter waves has the longest wavelength? Explain.(a) electron; (b) proton; (c) neutron; (d) $\alpha$ particle $\left(\mathrm{He}^{2+}\right)$.
For electromagnetic radiation transmitted through a vacuum, state whether each of the following properties is directly proportional to, inversely proportional to, or independent of the frequency: (a) velocity;(b) wavelength; (c) energy per mole. Explain.
Sir James Jeans described the photoelectric effect in this way: "It not only prohibits killing two birds with one stone but also the killing of one bird with two stones." By referring to the margin note on page 303 comment on the appropriateness of this analogy.
Construct a concept map representing the ideas of modern quantum mechanics.
Construct a concept map representing the atomic orbitals of hydrogen and their properties.
Construct a concept map for the configurations of multielectron atoms.