Chapter 6

The speed of sound in dry air at \(20^{\circ} \mathrm{C}\) is 343 \(\mathrm{m} / \mathrm{s}\) and the lowest frequency sound wave that the human ear can detect is approximately 20 \(\mathrm{Hz}\) (a) What is the wavelength of such a sound wave? (b) What would be the frequency of electromagnetic radiation with the same wavelength? (c) What type of electromagnetic radiation would that correspond to? [Section 6.1]

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 out-wavelength of light increase or decrease as we proceed out- frequency of light increase or decrease as we proceed out-ward? [Section 6.3]

Consider a fictitious one-dimensional system with one electron. The wave function for the electron, drawn below, is \(\psi(x)=\sin x\) from \(x=0\) to \(x=2 \pi\) . (a) Sketch the probability density, \(\psi^{2}(x),\) from \(x=0\) to \(x=2 \pi .(\mathbf{b})\) At 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\(]\)

State where in the periodic table these elements appear: $$ \begin{array}{l}{\text { (a) elements with the valence-shell electron configuration }} \\ {n s^{2} n p^{5}} \\ {\text { (b) elements that have three unpaired p electrons }} \\ {\text { (c) an element whose valence electrons are } 4 s^{2} 4 p^{1}} \\ {\text { (d) the } d \text { -block elements [ Section } 6.9 ]}\end{array} $$

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) Ultraviolet light has longer wavelengths than visible light. (c) \(X\) rays travel faster than microwaves. (d) Electromagnetic radiation and sound waves travel at the same speed.

Determine which of the following statements are false and correct them. (a) The frequency of radiation increases as the wavelength increases. (b) Electromagnetic radiation travels through a vacuum at a constant speed, regardless of wavelength. (c) Infrared light has higher frequencies than visible light. (d) The glow from a fireplace, the energy within a microwave oven, and a foghorn blast are all forms of electromagnetic radiation.

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.

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

If human height were quantized in 1 -foot increments, what would happen to the height of a child as she grows up: (i) The child's height would never change, (ii) the child's height would continuously get greater, (iii) the child's height would increase in "jumps" of 1 foot at a time, or (iv) the child's height would increase in jumps of 6 inches?

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) Calculate the energy of a photon of electromagnetic radiation whose frequency is \(2.94 \times 10^{14} \mathrm{s}^{-1} .\) (b) Calculate the energy of a photon of radiation whose wavelength is 413 nm. (c) What wavelength of radiation has photons of energy \(6.06 \times 10^{-19} \mathrm{J} ?\)

(a) A green laser pointer emits light with a wavelength of 532 nm. What is the frequency of this light? (b) What is the energy of one 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 532 -nm photons. What is the energy gap between the ground state and excited state in the laser material?

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 kJ/mol.

The energy from radiation can be used to cause the rupture of chemical bonds. A minimum energy of 242 \(\mathrm{kJ} / \mathrm{mol}\) is required to break the chlorine-chlorine bond in \(\mathrm{Cl}_{2} .\) 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 \(\mathrm{J}\) over a period of 32 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 a detector is capturing \(3.2 \times 10^{8}\) photons per second at this wavelength, what is the total energy of the photons detected in 1.0 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?

Does the hydrogen atom "expand" or "contract" when an electron is excited from the \(n=1\) state to the \(n=3\) state?

Classify each of the following statements as either true or false: (a) A hydrogen atom in the \(n=3\) state can emit light at only two specific wavelengths, (b) a hydrogen atom in the \(n=2\) state is at a lower energy than one in the \(n=1\) state, and (c) the energy of an emitted photon equals the energy difference of the two states involved in the emission.

Is energy emitted or absorbed when the following electronic transitions occur in hydrogen? (a) from \(n=4\) to \(n=2\) , (b) from an orbit of radius 2.12 A to one of radius \(8.46 \hat{A},(\mathbf{c})\) 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 \(n=6,(\mathbf{b})\) from an orbit of radius 4.76\(\hat{\mathrm{A}}\) to one of radius \(0.529 \mathrm{A},(\mathbf{c})\) 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 \(n=2 .\) (b) Is this line in the visible region of the electromagnetic spectrum? If so, what color is it?

Consider a transition of the electron in the hydrogen atom from \(n=4\) to \(n=9 .\) (a) Is \(\Delta E\) for this process positive or negative? (b) Determine the wavelength of light that is associated with this transition. Will the light be absorbed or emitted? (c) In which portion of the electromagnetic spectrum is the light in part (b)?

The Lyman series of emission lines of the hydrogen atom are those for which \(n_{1}=1 .\) (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 \(n_{1}=2,3,\) and \(4 .\)

One of the emission lines of the hydrogen atom has a wavelength of 93.07 nm. (a) In what region of the elertromagnetic spectrum is this emission found? (b) Determine the initial and final values of \(n\) associated with this emission.

Order the following transitions in the hydrogen atom from smallest to largest frequency of light absorbed: \(n=3\) to \(n=6, n=4\) to \(n=9, n=2\) to \(n=3,\) and \(n=1\) to \(n=2 .\)

Place the following transitions of the hydrogen atom in order from shortest to longest wavelength of the photon emitted: \(n=5\) to \(n=3, n=4\) to \(n=2, n=7\) to \(n=4,\) and \(n=3\) to \(n=2\) .

Use the de Broglie relationship to determine the wavelengths of the following objects: (a) an 85-kg person skiing at \(50 \mathrm{km} / \mathrm{hr},\) (b) a 10.0 -g bullet fired at \(250 \mathrm{m} / \mathrm{s},\) (c) a lithium atom moving at \(2.5 \times 10^{5} \mathrm{m} / \mathrm{s},(\mathbf{d})\) an ozone \(\left(\mathrm{O}_{3}\right)\) molecule in the upper atmosphere moving at 550 \(\mathrm{m} / \mathrm{s}\) .

Among the elementary subatomic particles of physics is the muon, which decays within a few microseconds 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 \(8.85 \times 10^{5} \mathrm{cm} / \mathrm{s}\) .

Neutron diffraction is an important technique for determining the structures of molecules. Calculate the velocity of a neutron needed to achieve a wavelength of 1.25 A. (Refer to the inside cover for the mass of the neutron.)

Using Heisenberg's uncertainty principle, calculate the uncertainty in the position of (a) a 1.50-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.)

The radial probability function for a 2 s orbital is shown here. Classify the following statements as either true or false: (a) There are two maxima in this function because one electron spends most of its time at an approximate distance of 0.5 A from the nucleus and the other electron spends most of its time at an approximate distance of 3 A from the nucleus. (b) The radial probability function shown here and the probability density \([\psi(r)]^{2}\) both go to zero at the same distance from the nucleus, approximately 1 A. (c) For an \(s\) orbital, the number of radial nodes is equal to the principal quantum number, \(n .\)

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

How many unique combinations of the quantum numbers \(l\) and \(m_{l}\) are there when (a) \(n=3,\) (b) \(n=4 ?\)

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

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.

A certain orbital of the hydrogen atom has \(n=4\) and \(l=2 .\) (a) What are the possible values of \(m_{l}\) for this orbital? (b) What are the possible values of \(m_{s}\) for the orbital?

A hydrogen atom orbital has \(n=5\) and \(m_{l}=-2 .\) (a) What are the possible values of \(l\) for this orbital? (b) What are the possible values of \(m_{s}\) for the orbital?

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

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

(a) What are the similarities of and differences between the 1s 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 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) For an He^ + ion, do the 2 s and 2\(p\) orbitals have the same energy? If not, which orbital has a lower energy? (b) If we add one electron to form the He atom, would your answer to part (a) change?

(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?

An experiment called the Stern-Gerlach experiment helped establish the existence of electron spin. In this experiment, a beam of silver atoms is passed through a magnetic field, which deflects half of the silver atoms in one direction and half in the opposite direction. The separation between the two beams increases as the strength of the magnetic field increases. (a) What is the electron configuration for a silver atom? (b) Would this experiment work for a beam of cadmium (Cd) atoms? (c) Would this experiment work for a beam of fluorine (F) atoms?

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

What is the maximum number of electrons in an atom that can have the following quantum numbers? (a) \(n=3, m_{l}=-2 ;(\mathbf{b}) n=4, l=3 ;(\mathbf{c}) n=5, l=3, m_{l}=2\) (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 object is represented by the half arrows in an orbital diagram? What does the direction of the arrow signify?