Chapter 6

The familiar phenomenon of a rainbow results from the diffraction of sunlight through raindrops. (a) Does the wavelength of light increase or decrease as we proceed outward from the innermost band of the rainbow? (b) Does the frequency of light increase or decrease as we proceed outward? (c) Suppose that instead of sunlight, the visible light from a hydrogen discharge tube (Figure 6.12) was used as the light source. What do you think the resulting "hydrogen discharge rainbow" would look like? [Section 6.3]

Consider a fictitious one-dimensional system with one electron. The wave function for the electron, drawn at the top of the next column, is \(\psi(x)=\sin x\) from \(x=0\) to \(x=2 \pi .\) (a) Sketch the probability density, \(\psi^{2}(x)\), from \(x=0\) to \(x=2 \pi\) (b) At what value or values of \(x\) will there be the greatest probability of finding the electron? (c) What is the probability that the electron will be found at \(x=\pi ?\) What is such a point in a wave function called? \([\) Section \(6.5]\)

What are the basic SI units for (a) the wavelength of light, (b) the frequency of light, (c) the speed of light?

(a) What is the relationship between the wavelength and the frequency of radiant energy? (b) Ozone in the upper atmosphere absorbs energy in the \(210-230-\mathrm{nm}\) range of the spectrum. In what region of the electromagnetic spectrum does this radiation occur?

Label each of the following statements as true or false. For those that are false, correct the statement. (a) Visible light is a form of electromagnetic radiation. (b) The frequency of radiation increases as the wavelength increases. (c) Ultraviolet light has longer wavelengths than visible light. (d) X-rays travel faster than microwaves. (e) Electromagnetic radiation and sound waves travel at the same speed.

Arrange the following kinds of electromagnetic radiation in order of increasing wavelength: infrared, green light, red light, radio waves, X-rays, ultraviolet light.

List the following types of electromagnetic radiation in order of increasing wavelength: (a) the gamma rays produced by a radioactive nuclide used in medical imaging; (b) radiation from an FM radio station at \(93.1 \mathrm{MHz}\) on the dial; (c) a radio signal from an AM radio station at \(680 \mathrm{kHz}\) on the dial; (d) the yellow light from sodium vapor streetlights; (e) the red light of a light-emitting diode, such as in a calculator display.

(a) What is the frequency of radiation that has a wavelength of \(10 \mu \mathrm{m}\), about the size of a bacterium? (b) What is the wavelength of radiation that has a frequency of \(5.50 \times 10^{14} \mathrm{~s}^{-1} ?(\mathrm{c})\) Would the radiations in part (a) or part (b) be visible to the human eye? (d) What distance does electromagnetic radiation travel in \(50.0 \mu \mathrm{s} ?\)

(a) What is the frequency of radiation whose wavelength is \(10.0 \mathrm{~A}\) ? (b) What is the wavelength of radiation that has a frequency of \(7.6 \times 10^{10} \mathrm{~s}^{-1} ?\) (c) Would the radiations in part (a) or part (b) be detected by an \(\mathrm{X}\) -ray detector? (d) What distance does electromagnetic radiation travel in \(25.5 \mathrm{fs}\) ?

It is possible to convert radiant energy into electrical energy using photovoltaic cells. Assuming equal efficiency of conversion, would infrared or ultraviolet radiation vield more electrical energy on a per-photon basis?

If human height were quantized in one-foot increments, what would happen to the height of a child as she grows up?

Einstein's 1905 paper on the photoelectric effect was the first important application of Planck's quantum hypothesis. Describe Planck's original hypothesis, and explain how Einstein made use of it in his theory of the photoelectric effect.

(a) A red laser pointer emits light with a wavelength of \(650 \mathrm{~nm}\). What is the frequency of this light? (b) What is the energy of 1 mole of these photons? (c) The laser pointer emits light because electrons in the material are excited (by a battery) from their ground state to an upper excited state. When the electrons return to the ground state they lose the excess energy in the form of \(650 \mathrm{~nm}\) photons. What is the energy gap between the ground state and excited state in the laser material?

If you put 120 volts of electricity through a pickle, the pickle will smoke and start glowing an orange-yellow color. The light is emitted because the sodium ions in the pickle become excited; their return to the ground state results in light emission (see Figure \(6.13 \mathrm{~b}\) and Sample Exercise 6.3). (a) The wavelength of this emitted light is \(589 \mathrm{~nm}\). Calculate its frequency. (b) What is the energy of \(0.10\) mole 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? Why or why not?

One type of sunburn occurs on exposure to UV light of wavelength in the vicinity of \(325 \mathrm{~nm}\). (a) What is the energy of a photon of this wavelength? (b) What is the energy of a mole of these photons? (c) How many photons are in a \(1.00 \mathrm{~mJ}\) burst of this radiation? (d) These UV photons can break chemical bonds in your skin to cause sunburn-a form of radiation damage. If the 325-nm radiation provides exactly the energy to break an average chemical bond in the skin, estimate the average energy of these bonds in \(\mathrm{kJ} / \mathrm{mol}\).

The energy from radiation can be used to cause the rupture of chemical bonds. A minimum energy of \(941 \mathrm{~kJ} / \mathrm{mol}\) is required to break the nitrogen-nitrogen bond in \(\mathrm{N}_{2}\). What is the longest wavelength of radiation that possesses the necessary energy to break the bond? What type of electromagnetic radiation is this?

A diode laser emits at a wavelength of \(987 \mathrm{~nm}\). (a) In what portion of the electromagnetic spectrum is this radiation found? (b) All of its output energy is absorbed in a detector that measures a total energy of \(0.52\) J over a period of \(32 \mathrm{~s}\). How many photons per second are being emitted by the laser?

A stellar object is emitting radiation at \(3.55 \mathrm{~mm}\). (a) What type of electromagnetic spectrum is this radiation? (b) If the detector is capturing \(3.2 \times 10^{8}\) photons per second at this wavelength, what is the total energy of the photons detected in one hour?

Molybdenum metal must absorb radiation with a minimum frequency of \(1.09 \times 10^{15} \mathrm{~s}^{-1}\) before it can eject an electron from its surface via the photoelectric effect. (a) What is the minimum energy needed to eject an electron? (b) What wavelength of radiation will provide a photon of this energy? (c) If molybdenum is irradiated with light of wavelength of \(120 \mathrm{~nm}\), what is the maximum possible kinetic energy of the emitted electrons?

Sodium metal requires a photon with a minimum energy of \(4.41 \times 10^{-19} \mathrm{~J}\) to emit electrons. (a) What is the minimum frequency of light necessary to emitelectrons from sodium via the photoelectric effect? (b) What is the wavelength of this light? (c) If sodium is irradiated with light of \(439 \mathrm{~nm}\), what is the maximum possible kinetic energy of the emitted electrons? (d) What is the maximum number of electrons that can be freed by a burst of light whose total energy is \(1.00 \mu] ?\)

Explain how the existence of line spectra is consistent with Bohr's theory of quantized energies for the electron in the hydrogen atom.

(a) In terms of the Bohr theory of the hydrogen atom, what process is occurring when excited hydrogen atoms emit radiant energy of certain wavelengths and only those wavelengths? (b) Does a hydrogen atom "expand" or "contract" as it moves from its ground state to an excited state?

Is energy emitted or absorbed when the following electronic transitions occur in hydrogen: (a) from \(n=4\) to \(n=2,(\mathrm{~b})\) from an orbit of radius \(2.12 \AA\) to one of radius \(8.46 \AA\), (c) an electron adds to the \(\mathrm{H}^{+}\) ion and ends up in the \(n=3\) shell?

Indicate whether energy is emitted or absorbed when the following electronic transitions occur in hydrogen: (a) from \(n=2\) to \(n=6,(b)\) from an orbit of radius \(4.76 \AA\) to one of radius \(0.529 \AA,(c)\) from the \(n=6\) to the \(n=9\) state.

(a) Using Equation \(6.5\), calculate the energy of an electron in the hydrogen atom when \(n=2\) and when \(n=6\). Calculate the wavelength of the radiation released when an electron moves from \(n=6\) to \(n=2\). Is this line in the visible region of the electromagnetic spectrum? If so, what color is it? (b) Calculate the energies of an electron in the hydrogen atom for \(n=1\) and for \(n=(\infty)\). How much energy does it require to move the electron out of the atom completely (from \(n=1\) to \(n=\infty\) ), according to Bohr? Put your answer in \(\mathrm{kJ} / \mathrm{mol}\). (c) The energy for the process \(\mathrm{H}+\) energy \(\rightarrow \mathrm{H}^{+}+\mathrm{e}^{-}\) is called the ionization energy of hydrogen. The experimentally determined value for the ionization energy of hydrogen is \(1310 \mathrm{~kJ} / \mathrm{mol}\). How does this compare to your calculation?

For each of the following electronic transitions in the hydrogen atom, calculate the energy, frequency, and wavelength of the associated radiation, and determine whether the radiation is emitted or absorbed during the transition: (a) from \(n=4\) to \(n=1,(b)\) from \(n=5\) to \(n=2,(\mathrm{c})\) from \(n=3\) to \(n=6\). Does any of these transitions emit or absorb visible light?

The Lyman series of emission lines of the hydrogen atom are those for which \(n_{f}=1\). (a) Determine the region of the electromagnetic spectrum in which the lines of the Lyman series are observed. (b) Calculate the wavelengths of the first three lines in the Lyman series - those for which \(n_{i}=2,3\), and 4 .

Among the elementary subatomic particles of physics is the muon, which decays within a few nanoseconds after formation. The muon has a rest mass \(206.8\) times that of an electron. Calculate the de Broglie wavelength associated with a muon traveling at a velocity of \(8.85 \times 10^{5} \mathrm{~cm} / \mathrm{s}\).

Neutron diffraction is an important technique for determining the structures of molecules. Calculate the velocity of a neutron needed to achieve a wavelength of \(0.955 \AA\). (Refer to the inside cover for the mass of the neutron).

The electron microscope has been widely used to obtain highly magnified images of biological and other types of materials. When an electron is accelerated through a particular potential field, it attains a speed of \(9.38 \times 10^{6} \mathrm{~m} / \mathrm{s}\). What is the characteristic wavelength of this electron? Is the wavelength comparable to the size of atoms?

Using Heisenberg's uncertainty principle, calculate the uncertainty in the position of (a) a \(1.50-\mathrm{mg}\) mosquito moving at a speed of \(1.40 \mathrm{~m} / \mathrm{s}\) if the speed is known to within \(\pm 0.01 \mathrm{~m} / \mathrm{s} ;\) (b) a proton moving at a speed of \((5.00 \pm 0.01) \times 10^{4} \mathrm{~m} / \mathrm{s}\). (The mass of a proton is given in the table of fundamental constants in the inside cover of the text.)

(a) Why does the Bohr model of the hydrogen atom violate the uncertainty principle? (b) In what way is the description of the electron using a wave function consistent with de Broglie's hypothesis? (c) What is meant by the term probability density? Given the wave function, how do we find the probability density at a certain point in space?

(a) According to the Bohr model, an electron in the ground state of a hydrogen atom orbits the nucleus at a specific radius of \(0.53 \AA\). In the quantum mechanical description of the hydrogen atom, the most probable distance of the electron from the nucleus is \(0.53 \AA\). Why are these two statements different? (b) Why is the use of Schrödinger's wave equation to describe the location of a particle very different from the description obtained from classical physics? (c) In the quantum mechanical description of an electron, what is the physical significance of the square of the wave function, \(\psi^{2} ?\)

(a) For \(n=4\), what are the possible values of \(l\) (b) For \(I=2\), what are the possible values of \(m_{l} ?(c)\) If \(m_{l}\) is 2 , what are the possible values for \(l ?\)

Give the numerical values of \(n\) and \(l\) corresponding to each of the following orbital designations: (a) \(3 p,(\) b) 2 s, (c) \(4 f\), (d) \(5 d\).

Give the values for \(n, l\), and \(m_{l}\) for (a) each orbital in the \(2 p\) subshell, (b) each orbital in the \(5 d\) subshell.

Sketch the shape and orientation of the following types of orbitals: (a) \(s\), (b) \(p_{z}\), (c) \(d_{x y}\).

Sketch the shape and orientation of the following types of orbitals: (a) \(p_{x}\), (b) \(d_{z^{2}}\), (c) \(d_{x^{2}-y^{2}}\).

(a) What are the similarities and differences between the \(1 s\) 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 \(2 \mathrm{~s}\) 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): \(4 f, 6 s, 3 d, 1 s, 2 p\)

For a given value of the principal quantum number, \(n\), how dothe energies of the \(s, p, d\), and \(f\) subshells vary for (a) hydrogen, (b) a many-electron atom?

(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? Explain.

(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 1s 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 1s 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. Explain why.

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 ?

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

(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 direction (up or down) of the half- arrows in an orbital diagram?

For each element, count the number of valence electrons, core electrons, and unpaired electrons in the ground state: (a) carbon, (b) phosphorus, (c) neon.

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

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

Ions also have electron configurations (Section 7.4). Cations have fewer valence electrons, and anions have more valence electrons, respectively, than their parent atoms. For example, chloride, \(\mathrm{Cl}^{-}\), has an electron configuration of \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6}\), for a total of 18 electrons, compared to 17 for neutral chlorine, the element. Na has an electron configuration of \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{1}\), but \(\mathrm{Na}^{+}\) has an electron configuration of \(1 s^{2} 2 s^{2} 2 p^{6} .\) Write out the electron configurations for (a) \(\mathrm{F}^{-}\), (b) \(\mathrm{I}^{-}\), (c) \(\mathrm{O}^{2-}\), (d) \(\mathrm{K}^{+}\), (e) \(\mathrm{Mg}^{2+}\), (f) \(\mathrm{Al}^{3+}\).