Chapter 6

(a) What are the similarities of and differences between the \(1 \mathrm{~s}\) and \(2 s\) orbitals of the hydrogen atom? (b) In what sense does a \(2 p\) orbital have directional character? Compare the "directional" characteristics of the \(p_{x}\) and \(d_{x^{2}-y^{2}}\) orbitals. (That is, in what direction or region of space is the electron density concentrated?) (c) What can you say about the average distance from the nucleus of an electron in a \(2 s\) orbital as compared with a \(3 s\) orbital? (d) For the hydrogen atom, list the following orbitals in order of increasing energy (that is, most stable ones first): \(4 f, 6 s, 3 d, 1 s, 2 p\).

(a) The average distance from the nucleus of a \(3 s\) electron in a chlorine atom is smaller than that for a \(3 p\) electron. In light of this fact, which orbital is higher in energy? (b) Would you expect it to require more or less energy to remove a \(3 s\) electron from the chlorine atom, as compared with a \(2 p\) electron?

(a) What experimental evidence is there for the electron having a "spin"? (b) Draw an energy-level diagram that shows the relative energetic positions of a \(1 s\) orbital and a \(2 s\) orbital. Put two electrons in the \(1 s\) orbital. (c) Draw an arrow showing the excitation of an electron from the \(1 s\) to the \(2 s\) orbital.

(a) State the Pauli exclusion principle in your own words. (b) The Pauli exclusion principle is, in an important sense, the key to understanding the periodic table. (c) Explain.

What is the maximum number of electrons that can occupy each of the following subshells? (a) \(3 p\), (b) \(5 d\), (c) \(2 s\), (d) \(4 f\).

What is the maximum number of electrons in an atom that can have the following quantum numbers? (a) \(n=3, m_{l}=-2 ;\) (b) \(n=4\), \(l=3 ;(\mathrm{c}) n=5, l=3, m_{l}=2,(\mathrm{~d}) n=4, l=1, m_{l}=0\).

(a) What are "valence electrons"? (b) What are "core electrons"? (c) What does each box in an orbital diagram represent? (d) What quantity is represented by the half arrows in an orbital diagram?

For each element, indicate the number of valence electrons, core electrons, and unpaired electrons in the ground state: (a) nitrogen, (b) silicon, (c) chlorine.

Write the condensed electron configurations for the following atoms, using the appropriate noble-gas core abbreviations: (a) Cs, (b) Ni, (c) \(\mathrm{Se}\), (d) \(\mathrm{Cd}\), (e) \(\mathrm{U}\), (f) \(\mathrm{Pb}\).

Write the condensed electron configurations for the following atoms and indicate how many unpaired electrons each has: (a) Mg, (b) Ge, (c) Br, (d) V, (e) \(\mathrm{Y}\), (f) Lu.

Identify the specific element that corresponds to each of the following electron configurations and indicate the number of unpaired electrons for each: (a) \(1 s^{2} 2 s^{2}\), (b) \(1 s^{2} 2 s^{2} 2 p^{4}\), (c) \([\mathrm{Ar}] 4 s^{1} 3 d^{5}\), (d) \([\mathrm{Kr}] 5 s^{2} 4 d^{10} 5 p^{4}\).

Identify the group of elements that corresponds to each of the following generalized electron configurations and indicate the number of unpaired electrons for each: (a) [noble gas \(] n s^{2} n p^{5}\) (b) [noble gas \(] n s^{2}(n-1) d^{2}\) (c) [noble gas] \(n s^{2}(n-1) d^{10} n p^{1}\) (d) [noble gas] \(n s^{2}(n-2) f^{6}\)

What is wrong with the following electron configurations for atoms in their ground states? (a) \(1 s^{2} 2 s^{2} 3 s^{1}\), (b) \([\mathrm{Ne}] 2 s^{2} 2 p^{3}\), (c) \([\mathrm{Ne}] 3 s^{2} 3 d^{5}\).

The following electron configurations represent excited states. Identify the element and write its ground-state condensed electron configuration. (a) \(1 s^{2} 2 s^{2} 2 p^{4} 3 s^{1}\), (b) \([\) Ar \(] 4 s^{1} 3 d^{10} 4 p^{2} 5 p^{1}\), (c) \([\mathrm{Kr}] 5 s^{2} 4 d^{2} 5 p^{1}\)

If you put 120 volts of electricity through a pickle, the pickle will smoke and start glowing orange-yellow. The light is emitted because sodium ions in the pickle become excited; their return to the ground state results in light emission. (a) The wavelength of this emitted light is \(589 \mathrm{~nm}\). Calculate its frequency. (b) What is the energy of \(0.10 \mathrm{~mol}\) of these photons? (c) Calculate the energy gap between the excited and ground states for the sodium ion. (d) If you soaked the pickle for a long time in a different salt solution, such as strontium chloride, would you still observe \(589-\mathrm{nm}\) light emission?

The rays of the Sun that cause tanning and burning are in the ultraviolet portion of the electromagnetic spectrum. These rays are categorized by wavelength. So-called UV-A radiation has wavelengths in the range of \(320-380 \mathrm{~nm}\), whereas UV-B radiation has wavelengths in the range of 290-320 \(\mathrm{nm}\). (a) Calculate the frequency of light that has a wavelength of \(320 \mathrm{~nm}\). (b) Calculate the energy of a mole of \(320-\mathrm{nm}\) photons. (c) Which are more energetic, photons of UV-A radiation or photons of UV-B radiation? (d) The UV-B radiation from the Sun is considered a greater cause of sunburn in humans than is UV-A radiation. Is this observation consistent with your answer to part (c)?

The rays of the Sun that cause tanning and burning are in the ultraviolet portion of the electromagnetic spectrum. These rays are categorized by wavelength. So-called UV-A radiation has wavelengths in the range of \(320-380 \mathrm{~nm}\), whereas UV-B radiation has wavelengths in the range of 290-320 \(\mathrm{nm}\). (a) Calculate the frequency of light that has a wavelength of \(320 \mathrm{~nm}\). (b) Calculate the energy of a mole of \(320-\mathrm{nm}\) photons. (c) Which are more energetic, photons of UV-A radiation or photons of UV-B radiation? (d) The UV-B radiation from the Sun is considered a greater cause of sunburn in humans than is UV-A radiation. Is this observation consistent with your answer to part (c)?

The watt is the derived SI unit of power, the measure of energy per unit time: \(1 \mathrm{~W}=1 \mathrm{~J} / \mathrm{s}\). A semiconductor laser in a CD player has an output wavelength of \(780 \mathrm{~nm}\) and a power level of \(0.10 \mathrm{~mW}\). How many photons strike the CD surface during the playing of a CD 69 minutes in length?

Consider a transition in which the hydrogen atom is excited from \(n=1\) to \(n=\infty\). (a) What is the end result of this transition? (b) What is the wavelength of light that must be absorbed to accomplish this process? (c) What will occur if light with a shorter wavelength than that in part (b) is used to excite the hydrogen atom? (d) How the results of parts (b) and (c) related to the plot shown in Exercise \(6.88\) ?

The series of emission lines of the hydrogen atom for which \(n_{\mathrm{f}}=3\) is called the Paschen series. (a) Determine the region of the electromagnetic spectrum in which the lines of the Paschen series are observed. (b) Calculate the wavelengths of the first three lines in the Paschen series-those for which \(n_{\mathrm{i}}=4,5\), and 6 .

Determine whether each of the following sets of quantum numbers for the hydrogen atom are valid. If a set is not valid, indicate which of the quantum numbers has a value that is not valid: (a) \(n=4, l=1, m_{l}=2, m_{s}=-\frac{1}{2}\) (b) \(n=4, l=3, m_{l}=-3, m_{s}=+\frac{1}{2}\) (c) \(n=3, l=2, m_{l}=-1, m_{s}=+\frac{1}{2}\) (d) \(n=5, l=0, m_{l}=0, m_{s}=0\) (e) \(n=2, l=2, m_{l}=1, m_{s}=+\frac{1}{2}\)

Bohr's model can be used for hydrogen-like ions-ions that have only one electron, such as \(\mathrm{He}^{+}\)and \(\mathrm{Li}^{2+}\). (a) Why is the Bohr model applicable to \(\mathrm{He}^{+}\)ions but not to neutral He atoms? (b) The ground-state energies of \(\mathrm{H}, \mathrm{He}^{+}\), and \(\mathrm{Li}^{2+}\) are tabulated as follows:

An electron is accelerated through an electric potential to a kinetic energy of \(13.4 \mathrm{keV}\). What is its characteristic wavelength? [Hint: Recall that the kinetic energy of a moving object is \(E=\frac{1}{2} m v^{2}\), where \(m\) is the mass of the object and \(\nu\) is the speed of the object.]

In the television series Star Trek, the transporter beam is a device used to "beam down" people from the Starship Enterprise to another location, such as the surface of a planet. The writers of the show put a "Heisenberg compensator" into the transporter beam mechanism. Explain why such a compensator (which is entirely fictional) would be necessary to get around Heisenberg's uncertainty principle.

As discussed in the A Closer Look box on "Measurement and the Uncertainty Principle" the essence of the uncertainty principle is that we can't make a measurement without disturbing the system that we are measuring. (a) Why can't we measure the position of a subatomic particle without disturbing it? (b) How is this concept related to the paradox discussed in the Closer Look box on "Thought Experiments and Schrödinger's Cat"?

For orbitals that are symmetric but not spherical, the contour representations (as in Figures \(6.23\) and 6.24) suggest where nodal planes exist (that is, where the electron density is zero). For example, the \(p_{x}\) orbital has a node wherever \(x=0\). This equation is satisfied by all points on the \(y z\) plane, so this plane is called a nodal plane of the \(p_{x}\) orbital. (a) Determine the nodal plane of the \(p_{z}\) orbital. (b) What are the two nodal planes of the \(d_{x y}\) orbital? (c) What are the two nodal planes of the \(d_{x^{2}-y^{2}}\) orbital?

The Chemistry and Life box in Section 6.7 described the techniques called NMR and MRI. (a) Instruments for obtaining MRI data are typically labeled with a frequency, such as 600 MHz. In what region of the electromagnetic spectrum does a photon with this frequency belong? (b) What is the value of \(\Delta E\) in Figure 6.27 that would correspond to the absorption of a photon of radiation with frequency \(450 \mathrm{MHz} ?(\mathbf{c})\) When the 450 -MHz photon is absorbed, does it change the spin of the electron or the proton on a hydrogen atom?

Using the periodic table as a guide, write the condensed electron configuration and determine the number of unpaired electrons for the ground state of (a) \(\mathrm{Br}\), (b) Ga, (c) Hf, (d) Sb, (e) Bi, (f) Sg.

In the experiment shown schematically below, a beam of neutral atoms is passed through a magnetic field. Atoms that have unpaired electrons are deflected in different directions in the magnetic field depending on the value of the electron spin quantum number. In the experiment illustrated, we envision that a beam of hydrogen atoms splits into two beams. (a) What is the significance of the observation that the single beam splits into two beams? (b) What do you think would happen if the strength of the magnet were increased? (c) What do you think would happen if the beam of hydrogen atoms were replaced with a beam of helium atoms? Why? (d) The relevant experiment was first performed by Otto Stern and Walter Gerlach in 1921 . They used a beam of Ag atoms in the experiment. By considering the electron configuration of a silver atom, explain why the single beam splits into two beams.

Microwave ovens use microwave radiation to heat food. The energy of the microwaves is absorbed by water molecules in food and then transferred to other components of the food. (a) Suppose that the microwave radiation has a wavelength of \(11.2 \mathrm{~cm}\). How many photons are required to heat \(200 \mathrm{~mL}\) of coffee from 23 to \(60^{\circ} \mathrm{C}\) ? (b) Suppose the microwave's power is \(900 \mathrm{~W}\) ( 1 watt \(=1\) joule-second). How long would you have to heat the coffee in part (a)?

The discovery of hafnium, element number 72 , provided a controversial episode in chemistry. G. Urbain, a French chemist, claimed in 1911 to have isolated an element number 72 from a sample of rare earth (elements 58-71) compounds. However, Niels Bohr believed that hafnium was more likely to be found along with zirconium than with the rare earths. D. Coster and G. von Hevesy, working in Bohr's laboratory in Copenhagen, showed in 1922 that element 72 was present in a sample of Norwegian zircon, an ore of zirconium. (The name hafnium comes from the Latin name for Copenhagen, Hafnia). (a) How would you use electron configuration arguments to justify Bohr's prediction? (b) Zirconium, hafnium's neighbor in group \(4 \mathrm{~B}\), can be produced as a metal by reduction of solid \(\mathrm{ZrCl}_{4}\) with molten sodium metal. Write a balanced chemical equation for the reaction. Is this an oxidation-reduction reaction? If yes, what is reduced and what is oxidized? (c) Solid zirconium dioxide, \(\mathrm{ZrO}_{2}\), reacts with chlorine gas in the presence of carbon. The products of the reaction are \(\mathrm{ZrCl}_{4}\) and two gases, \(\mathrm{CO}_{2}\) and \(\mathrm{CO}\) in the ratio \(1: 2\). Write a balanced chemical equation for the reaction. Starting with a 55.4-g sample of \(\mathrm{ZrO}_{2}\), calculate the mass of \(\mathrm{ZrCl}_{4}\) formed, assuming that \(\mathrm{ZrO}_{2}\) is the limiting reagent and assuming \(100 \%\) yield. (d) Using their electron configurations, account for the fact that \(\mathrm{Zr}\) and \(\mathrm{Hf}\) form chlorides \(\mathrm{MCl}_{4}\) and oxides \(\mathrm{MO}_{2}\).

(a) Account for formation of the following series of oxides in terms of the electron configurations of the elements and the discussion of ionic compounds in Section \(2.7 :\) $\mathrm{K}_{2} \mathrm{O}, \mathrm{CaO}, \mathrm{Sc}_{2} \mathrm{O}_{3}, \mathrm{Ti} \mathrm{O}_{2}, \mathrm{V}_{2} \mathrm{O}_{5}, \mathrm{CrO}_{3} .$ (b) Name these oxides. (c) Consider the metal oxides whose enthalpies of formation (in kJ mol \(^{-1}\) ) are listed here. Calculate the enthalpy changes in the following general reaction for each case: $$ \mathrm{M}_{n} \mathrm{O}_{m}(s)+\mathrm{H}_{2}(g) \longrightarrow n \mathrm{M}(s)+m \mathrm{H}_{2} \mathrm{O}(g) $$ (You will need to write the balanced equation for each case and then compute \(\Delta H^{\circ} . )(\mathbf{d})\) Based on the data given, estimate a value of \(\Delta H_{f}^{\circ}\) for \(S c_{2} \mathrm{O}_{3}(s) .\)

The first 25 years of the twentieth century were momentous for the rapid pace of change in scientists' understanding of the nature of matter. (a) How did Rutherford's experiments on the scattering of \(\alpha\) particles by a gold foil set the stage for Bohr's theory of the hydrogen atom? (b) In what ways is de Broglie's hypothesis, as it applies to electrons, consistent with J. J. Thomson's conclusion that the electron has mass? In what sense is it consistent with proposals preceding Thomson's work that the cathode rays are a wave phenomenon?