Chapter 6

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) \(\mathrm{Cs},(\mathbf{b}) \mathrm{Ni},(\mathbf{c}) \mathrm{Se},(\mathbf{d}) \mathrm{Cd},(\mathbf{e}) \mathrm{U},(\mathbf{f}) \mathrm{Pb}\)

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

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

Identify the group of elements that corresponds to each of the following generalized electron configurations and indicate the number of unpaired electrons for each: $$ \begin{array}{l}{\text { (a) [noble gas ln }^{2} n p^{5}} \\ {\text { (b) }\left[\text { noble gas } \ln s^{2}(n-1) d^{2}\right.}\\\\{\text { (c) [noble gas } \operatorname{ns}^{2}(n-1) d^{10} n p^{1}} \\ {\text { (d) }[\text { noble gas }] n s^{2}(n-2) f^{6}}\end{array} $$

The following do not represent valid ground-state electron configurations for an atom either because they violate the Pauli exclusion principle or because orbitals are not filled in order of increasing energy. Indicate which of these two principles is violated in each example. (a) 1\(s^{2} 2 s^{2} 3 s^{1}\) (b) \([\mathrm{Xe}] 6 s^{2} 5 d^{4}(\mathbf{c})[\mathrm{Ne}] 3 s^{2} 3 d^{5} .\)

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 \(] 4 s^{1} 3 d^{10} 4 p^{2} 5 p^{1}\) \((\mathbf{c})[\mathrm{Kr}] 5 s^{2} 4 d^{2} 5 p^{1}\)

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 -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)?

Consider a transition in which the electron of a 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 that in part (b) is used to excite the hydrogen atom? (d) How are 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 \(n_{1}=3\) 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 \(n_{1}=4,5,\) 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: $$ \begin{array}{l}{\text { (a) } n=4, l=1, m_{l}=2, m_{s}=-\frac{1}{2}} \\\ {\text { (b) } n=4, l=3, m_{l}=-3, m_{s}=+\frac{1}{2}}\\\\{\text { (c) } n=3, l=2, m_{l}=-1, m_{s}=+\frac{1}{2}} \\ {\text { (d) } n=5, l=0, m_{l}=0, m_{s}=0} \\ {\text { (e) } n=2, l=2, m_{l}=1, m_{s}=+\frac{1}{2}}\end{array} $$

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 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: $$ \begin{array}{l}{\text { Atom or ion } \quad \quad\quad\quad\quad\quad \mathrm{H} \quad\quad\quad\quad\quad\quad \text { He }^{+} \quad\quad\quad\quad\quad\quad\quad \mathrm{Li}^{2+}} \\ {\text { Ground- state }\quad-2.18 \times 10^{-18} \mathrm{J}\quad-8.72 \times 10^{-18} \mathrm{J}\quad-1.96 \times 10^{-17} \mathrm{J}} \\ {\text { energy }}\end{array} $$ By examining these numbers, propose a relationship between the ground-state energy of hydrogen-like systems and the nuclear charge, \(Z .(\mathbf{c})\) Use the relationship you derive in part (b) to predict the ground-state energy of the \(\mathrm{C}^{5+}\) ion.

An electron is accelerated through an electric potential to a kinetic energy of \(2.15 \times 10^{-15} \mathrm{J}\) . 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 boxin 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 \(\mathrm{MHz}\) ? (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) Br, (b) Ga, (c) Hf, (d) Sb, (e) Bi, (f) Sg.

Scientists have speculated that element 126 might have a moderate stability, allowing it to be synthesized and characterized. Predict what the condensed electron configuration of this element might be.

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.

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 hafnum 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 \(Z r \mathrm{Cl}_{4}\) 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 \(55.4-\mathrm{g}\) sample of \(\mathrm{ZrO}_{2},\) calculate the mass of \(\mathrm{ZrCl}_{4}\) formed, assuming that \(Z r O_{2}\) 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 \(\mathrm{MO}_{2}\)