Exercises

An FM radio station found at 103.1 on the FM dial broadcasts at a frequency of 1.031 $\times$ 10⁸ s⁻¹ (103.1 MHz). What is the wavelength of these radio waves in meters?

A bright violet line occurs at 435.8 nm in the emission spectrum of mercury vapor. What amount of energy, in joules, must be released by an electron in a mercury atom to produce a photon of this light?

Heated lithium atoms emit photons of light with an energy of 2.961 $\times$ 10⁻¹⁹ J. Calculate the frequency and wavelength of one of these photons. What is the total energy in 1 mole of these photons? What is the color of the emitted light?

When rubidium ions are heated to a high temperature, two lines are observed in its line spectrum at wavelengths (a) 7.9 $\times$ 10⁻⁷ m and (b) 4.2 $\times$ 10⁻⁷ m. What are the frequencies of the two lines? What color do we see when we heat a rubidium compound?

Photons of infrared radiation are responsible for much of the warmth we feel when holding our hands before a fire. These photons will also warm other objects. How many infrared photons with a wavelength of 1.5 $\times$ 10⁻⁶ m must be absorbed by the water to warm a cup of water (175 g) from 25.0 °C to 40 °C?

The eyes of certain reptiles pass a single visual signal to the brain when the visual receptors are struck by photons of a wavelength of 850 nm. If a total energy of 3.15 $\times$ 10⁻¹⁴ J is required to trip the signal, what is the minimum number of photons that must strike the receptor?

Answer the following questions about a Blu-ray laser:

(a) The laser on a Blu-ray player has a wavelength of 405 nm. In what region of the electromagnetic spectrum is this radiation? What is its frequency?

(b) A Blu-ray laser has a power of 5 milliwatts (1 watt = 1 J s⁻¹). How many photons of light are produced by the laser in 1 hour?

(c) The ideal resolution of a player using a laser (such as a Blu-ray player), which determines how close together data can be stored on a compact disk, is determined using the following formula: Resolution = 0.60(λ/NA), where λ is the wavelength of the laser and NA is the numerical aperture. Numerical aperture is a measure of the size of the spot of light on the disk; the larger the NA, the smaller the spot. In a typical Blu-ray system, NA = 0.95. If the 405-nm laser is used in a Blu-ray player, what is the closest that information can be stored on a Blu-ray disk?

(d) The data density of a Blu-ray disk using a 405-nm laser is 1.5 $\times$ 10⁷ bits mm⁻². Disks have an outside diameter of 120 mm and a hole of 15-mm diameter. How many data bits can be contained on the disk? If a Blu-ray disk can hold 9,400,000 pages of text, how many data bits are needed for a typed page? (Hint: Determine the area of the disk that is available to hold data. The area inside a circle is given by A = πr², where the radius r is one-half of the diameter.)

Why is the electron in a Bohr hydrogen atom bound less tightly when it has a quantum number of 3 than when it has a quantum number of 1?

Using the Bohr model, determine the energy, in joules, necessary to ionize a ground-state hydrogen atom. Show your calculations.

Using the Bohr model, determine the lowest possible energy, in joules, for the electron in the Li²⁺ ion.

Using the Bohr model, determine the energy of an electron with n = 6 in a hydrogen atom.

How far from the nucleus in angstroms (1 angstrom = 1 $\times$ 10^–10 m) is the electron in a hydrogen atom if it has an energy of –8.72 $\times$ 10^–20 J?

Using the Bohr model, determine the energy in joules of the photon produced when an electron in a He⁺ ion moves from the orbit with n = 5 to the orbit with n = 2.

Consider a large number of hydrogen atoms with electrons randomly distributed in the n = 1, 2, 3, and 4 orbits.

(a) How many different wavelengths of light are emitted by these atoms as the electrons fall into lower-energy orbits?

(b) Calculate the lowest and highest energies of light produced by the transitions described in part (a).

(c) Calculate the frequencies and wavelengths of the light produced by the transitions described in part (b).

The spectra of hydrogen and of calcium are shown here.

What causes the lines in these spectra? Why are the colors of the lines different? Suggest a reason for the observation that the spectrum of calcium is more complicated than the spectrum of hydrogen.

What are the allowed values for each of the four quantum numbers: n, l, m_l, and m_s?

Answer the following questions:

(a) Without using quantum numbers, describe the differences between the shells, subshells, and orbitals of an atom.

(b) How do the quantum numbers of the shells, subshells, and orbitals of an atom differ?

Which of the subshells described in the previous question contain degenerate orbitals? How many degenerate orbitals are in each?

Which of the subshells described in the previous question contain degenerate orbitals? How many degenerate orbitals are in each?

Sketch the p_x and d_xz orbitals. Be sure to show and label the coordinates.

State the Heisenberg uncertainty principle. Describe briefly what the principle implies.

Which of the following equations describe particle-like behavior? Which describe wavelike behavior? Do any involve both types of behavior? Describe the reasons for your choices.

(a) c = λν

(b) $E=\frac{m{\nu }^2}{2}$

(c) $r=\frac{n^2{a}_0}{Z}$

(d) E = hν

(e) $\lambda =\frac{h}{m\nu }$

Read the labels of several commercial products and identify monatomic ions of at least four transition elements contained in the products. Write the complete electron configurations of these cations.

Using complete subshell notation (not abbreviations, 1s²2s²2p⁶, and so forth), predict the electron configuration of each of the following atoms:

(a) C

(b) P

(c) V

(d) Sb

(e) Sm

Is 1s²2s²2p⁶ the symbol for a macroscopic property or a microscopic property of an element? Explain your answer.

Draw the orbital diagram for the valence shell of each of the following atoms:

(a) C

(b) P

(c) V

(d) Sb

(e) Ru

Using complete subshell notation (1s²2s²2p⁶, and so forth), predict the electron configurations of the following ions.

(a) N^3–

(b) Ca²⁺

(c) S^–

(d) Cs²⁺

(e) Cr²⁺

(f) Gd³⁺

Which atom has the electron configuration 1s²2s²2p⁶3s²3p⁶3d⁷4s²?

Which of the following atoms contains only three valence electrons: Li, B, N, F, Ne?

Which atom would be expected to have a half-filled 6p subshell?

In one area of Australia, the cattle did not thrive despite the presence of suitable forage. An investigation showed the cause to be the absence of sufficient cobalt in the soil. Cobalt forms cations in two oxidation states, Co²⁺ and Co³⁺. Write the electron structure of the two cations.

Write the electron configurations for the following atoms or ions:

(a) B³⁺

(b) O^–

(c) Cl³⁺

(d) Ca²⁺

(e) Ti

Write a set of quantum numbers for each of the electrons with an n of 3 in a Sc atom.

Based on their positions in the periodic table, predict which has the largest atomic radius: Li, Rb, N, F, I.

Based on their positions in the periodic table, predict which has the smallest first ionization energy: Li, Cs, N, F, I.

Based on their positions in the periodic table, rank the following atoms in order of increasing first ionization energy: Mg, O, S, Si

Atoms of which group in the periodic table have a valence shell electron configuration of ns²?

Based on their positions in the periodic table, list the following atoms in order of increasing radius: Sr, Ca, Si, Cl.

List the following ions in order of increasing radius: Li⁺, Mg²⁺, Br^–, Te^2–.

Which of the following atoms and ions is (are) isoelectronic with S²⁺: Si⁴⁺, Cl³⁺, Ar, As³⁺, Si, Al³⁺?

Of the five elements Al, Cl, I, Na, Rb, which has the most exothermic reaction? (E represents an atom.) What name is given to the energy for the reaction? (Hint: Note the process depicted does not correspond to electron affinity.)
${\text{E}}^{\text{+}}(g)+{\text{e}}^{\text{−}}⟶\text{E}(g)$

The ionic radii of the ions S^2–, Cl^–, and K⁺ are 184, 181, 138 pm respectively. Explain why these ions have different sizes even though they contain the same number of electrons.

Explain why Al is a member of group 13 rather than group 3?