Chapter 7

Discuss the similarities and differences between a \(1 s\) and a \(2 s\) orbital.

What is the difference between a \(2 p_{x}\) and a \(2 p_{y}\) orbital?

Calculate the total number of electrons that can occupy (a) one \(s\) orbital, (b) three \(p\) orbitals, (c) five \(d\) orbitals, (d) seven \(f\) orbitals.

What is the total number of electrons that can be held in all orbitals having the same principal quantum number \(n ?\)

Determine the maximum number of electrons that can be found in each of the following subshells: \(3 s\), \(3 d, 4 p, 4 f, 5 f\)

Indicate the total number of (a) \(p\) electrons in \(\mathrm{N}\) \((Z=7),(b) s\) electrons in \(\operatorname{Si}(Z=14),\) and (c) \(3 d\) electrons in \(\mathrm{S}(Z=16)\)

Make a chart of all allowable orbitals in the first four principal energy levels of the hydrogen atom. Designate each by type (for example, \(s, p\) ) and indicate how many orbitals of each type there are.

Why do the \(3 s, 3 p,\) and \(3 d\) orbitals have the same energy in a hydrogen atom but different energies in a many-electron atom?

For each of the following pairs of hydrogen orbitals, indicate which is higher in energy: (a) \(1 s, 2 s ;\) (b) \(2 p\) \(3 p ;\) (c) \(3 d_{x y}, 3 d_{y c}\) (d) \(3 s, 3 d ;\) (e) \(4 f, 5 s\)

Which orbital in each of the following pairs is lower in energy in a many- electron atom? (a) \(2 s, 2 p ;\) (b) \(3 p, 3 d ;\) (c) \(3 s, 4 s ;\) (d) \(4 d, 5 f\)

What is electron configuration? Describe the roles that the Pauli exclusion principle and Hund's rule play in writing the electron configuration of elements.

Explain the meaning of diamagnetic and paramagnetic. Give an example of an element that is diamagnetic and one that is paramagnetic. What does it mean when we say that electrons are paired?

What is meant by the term "shielding of electrons" in an atom? Using the \(\mathrm{Li}\) atom as an example, describe the effect of shielding on the energy of electrons in an atom.

Indicate which of the following sets of quantum numbers in an atom are unacceptable and explain why: \((a)\left(1,0, \frac{1}{2}, \frac{1}{2}\right),(b)\left(3,0,0,+\frac{1}{2}\right),(c)\left(2,2,1,+\frac{1}{2}\right)\) (d) \(\left(4,3,-2,+\frac{1}{2}\right),\) (e) (3,2,1,1)

The ground-state electron configurations listed here are incorrect. Explain what mistakes have been made in each and write the correct electron configurations. Al: \(1 s^{2} 2 s^{2} 2 p^{4} 3 s^{2} 3 p^{3}\) \(\mathrm{B}: 1 s^{2} 2 s^{2} 2 p^{5}\) \(\mathrm{F}: 1 \mathrm{~s}^{2} 2 \mathrm{~s}^{2} 2 \mathrm{p}^{6}\)

The atomic number of an element is 73 . Is this element diamagnetic or paramagnetic?

Indicate the number of unpaired electrons present in each of the following atoms: \(\mathrm{B}, \mathrm{Ne}, \mathrm{P}, \mathrm{Sc}, \mathrm{Mn}, \mathrm{Se}\) \(\mathrm{Kr}, \mathrm{Fe}, \mathrm{Cd}, \mathrm{I}, \mathrm{Pb}\)

State the Aufbau principle and explain the role it plays in classifying the elements in the periodic table.

Describe the characteristics of the following groups of elements: transition metals, lanthanides, actinides.

What is the noble gas core? How does it simplify the writing of electron configurations?

What are the group and period of the element osmium?

Define the following terms and give an example of each: transition metals, lanthanides, actinides.

Explain why the ground-state electron configurations of \(\mathrm{Cr}\) and \(\mathrm{Cu}\) are different from what we might expect.

Explain what is meant by a noble gas core. Write the electron configuration of a xenon core.

Comment on the correctness of the following statement: The probability of finding two electrons with the same four quantum numbers in an atom is zero.

Use the Aufbau principle to obtain the ground-state electron configuration of selenium.

Use the Aufbau principle to obtain the ground-state electron configuration of technetium.

Write the ground-state electron configurations for the following elements: \(\mathrm{B}, \mathrm{V}, \mathrm{Ni}, \mathrm{As}, \mathrm{I},\) Au.

Write the ground-state electron configurations for the following elements: Ge, Fe, Zn, Ni, W, TI.

Which of the following species has the most unpaired electrons: \(\mathrm{S}^{+}, \mathrm{S},\) or \(\mathrm{S}^{-} ?\) Explain how you arrive at your answer.

A laser produces a beam of light with a wavelength of \(532 \mathrm{nm}\). If the power output is \(25.0 \mathrm{~mW}\), how many photons does the laser emit per second? \((1 \mathrm{~W}=1 \mathrm{~J} / \mathrm{s})\)

Discuss the current view of the correctness of the following statements. (a) The electron in the hydrogen atom is in an orbit that never brings it closer than \(100 \mathrm{pm}\) to the nucleus. (b) Atomic absorption spectra result from transitions of electrons from lower to higher energy levels. (c) A many- electron atom behaves somewhat like a solar system that has a number of planets.

What is the basis for thinking that atoms are spherical in shape even though the atomic orbitals \(p, d, \ldots\) have distinctly nonspherical shapes?

What is the maximum number of electrons in an atom that can have the following quantum numbers? Specify the orbitals in which the electrons would be found. (a) \(n=2, m_{\mathrm{s}}=+\frac{1}{2}\) (b) \(n=4, m_{e}=+1\) (c) \(n=3, \ell=2 ;\) (d) \(n=2, \ell=0, m_{\mathrm{s}}=-\frac{1}{2} ;\) (e) \(n=4\) \(\ell=3, m_{\ell}=-2\)

Identify the following individuals and their contributions to the development of quantum theory: Bohr, de Broglie, Einstein, Planck, Heisenberg, Schrödinger.

What properties of electrons are used in the operation of an electron microscope?

In a photoelectric experiment a student uses a light source whose frequency is greater than that needed to eject electrons from a certain metal. However, after continuously shining the light on the same area of the metal for a long period of time the student notices that the maximum kinetic energy of ejected electrons begins to decrease, even though the frequency of the light is held constant. How would you account for this behavior?

A certain pitcher's fastballs have been clocked at about \(100 \mathrm{mph}\). (a) Calculate the wavelength of a \(0.141-\mathrm{kg}\) baseball (in \(\mathrm{nm})\) at this speed. (b) What is the wavelength of a hydrogen atom at the same speed? ( 1 mile \(=1609 \mathrm{~m}\) )

$$ \begin{array}{lccccc} \lambda(\mathrm{nm}) & 405 & 435.8 & 480 & 520 & 577.7 \\ \hline \mathrm{KE}(\mathrm{J}) & 2.360 \times & 2.029 \times & 1.643 \times & 1.417 \times & 1.067 \times \\ & 10^{-19} & 10^{-19} & 10^{-19} & 10^{-19} & 10^{-19} \end{array} $$ (a) What is the lowest possible value of the principal quantum number \((n)\) when the angular momentum quantum number \((\ell)\) is \(1 ?\) (b) What are the possible values of the angular momentum quantum number ( \(\ell\) ) when the magnetic quantum number \(\left(m_{\ell}\right)\) is 0 , given than \(n \leq 4 ?\)

$$ \begin{array}{lccccc} \lambda(\mathrm{nm}) & 405 & 435.8 & 480 & 520 & 577.7 \\ \hline \mathrm{KE}(\mathrm{J}) & 2.360 \times & 2.029 \times & 1.643 \times & 1.417 \times & 1.067 \times \\ & 10^{-19} & 10^{-19} & 10^{-19} & 10^{-19} & 10^{-19} \end{array} $$ A ruby laser produces radiation of wavelength \(633 \mathrm{nm}\) in pulses whose duration is \(1.00 \times 10^{-9} \mathrm{~s}\). (a) If the laser produces \(0.376 \mathrm{~J}\) of energy per pulse, how many photons are produced in each pulse? (b) Calculate the power (in watts) delivered by the laser per pulse. \((1 \mathrm{~W}=1 \mathrm{~J} / \mathrm{s})\)

Photodissociation of water \(\mathrm{H}_{2} \mathrm{O}(l)+h \nu \longrightarrow \mathrm{H}_{2}(g)+\frac{1}{2} \mathrm{O}_{2}(g)\) has been suggested as a source of hydrogen. The \(\Delta H_{\mathrm{xn}}^{\circ}\) for the reaction, calculated from thermochemical data, is \(285.8 \mathrm{~kJ}\) per mole of water decomposed. Calculate the maximum wavelength (in \(\mathrm{nm}\) ) that would provide the necessary energy. In principle, is it feasible to use sunlight as a source of energy for this process?

Spectral lines of the Lyman and Balmer series do not overlap. Verify this statement by calculating the longest wavelength associated with the Lyman series and the shortest wavelength associated with the Balmer series (in \(\mathrm{nm}\) ).

The \(\mathrm{He}^{+}\) ion contains only one electron and is therefore a hydrogenlike ion. Calculate the wavelengths, in increasing order, of the first four transitions in the Balmer series of the \(\mathrm{He}^{+}\) ion. Compare these wavelengths with the same transitions in a \(\mathrm{H}\) atom. Comment on the differences. (The Rydberg constant for \(\mathrm{He}^{+}\) is \(\left.8.72 \times 10^{-18} \mathrm{~J} .\right)\)

The retina of a human eye can detect light when radiant energy incident on it is at least \(4.0 \times 10^{-17} \mathrm{~J}\). For light of 600 -nm wavelength, how many photons does this correspond to?

A helium atom and a xenon atom have the same kinetic energy. Calculate the ratio of the de Broglie wavelength of the helium atom to that of the xenon atom.

A laser is used in treating retina detachment. The wavelength of the laser beam is \(514 \mathrm{nm}\) and the power is \(1.6 \mathrm{~W}\). If the laser is turned on for \(0.060 \mathrm{~s}\) during surgery, calculate the number of photons emitted by the laser. \((1 \mathrm{~W}=1 \mathrm{~J} / \mathrm{s})\)

An electron in an excited state in a hydrogen atom can return to the ground state in two different ways: (a) via a direct transition in which a photon of wavelength \(\lambda_{1}\) is emitted and (b) via an intermediate excited state reached by the emission of a photon of wavelength \(\lambda_{2}\). This intermediate excited state then decays to the ground state by emitting another photon of wavelength \(\lambda_{3}\). Derive an equation that relates \(\lambda_{1}\) to \(\lambda_{2}\) and \(\lambda_{3}\)

A photoelectric experiment was performed by separately shining a laser at \(450 \mathrm{nm}\) (blue light) and a laser at \(560 \mathrm{nm}\) (yellow light) on a clean metal surface and measuring the number and kinetic energy of the ejected electrons. Which light would generate more electrons? Which light would eject electrons with greater kinetic energy? Assume that the same amount of energy is delivered to the metal surface by each laser and that the frequencies of the laser lights exceed the threshold frequency.

Draw the shapes (boundary surfaces) of the following orbitals: (a) \(2 p_{y},\) (b) \(3 d_{z^{2}}\) (c) \(3 d_{x^{2}-y^{2}}\), (Show coordinate axes in your sketches.)

The electron configurations described in this chapter all refer to gaseous atoms in their ground states. An atom may absorb a quantum of energy and promote one of its electrons to a higher-energy orbital. When this happens, we say that the atom is in an excited state. The electron configurations of some excited atoms are given. Identify these atoms and write their ground-state configurations: (a) \(1 s^{1} 2 s^{1}\) (b) \(1 s^{2} 2 s^{2} 2 p^{2} 3 d^{1}\) (c) \(1 s^{2} 2 s^{2} 2 p^{6} 4 s^{1}\) (d) \([\mathrm{Ar}] 4 s^{1} 3 d^{10} 4 p^{4}\) (e) \([\mathrm{Ne}] 3 s^{2} 3 p^{4} 3 d^{1}\)