Chapter 8

The magnesium spectrum has a line at \(266.8 \mathrm{nm}\). Which of these statements about this radiation is (are) correct? Explain. (a) It has a higher frequency than radiation with wavelength \(402 \mathrm{nm}\) (b) It is visible to the eye. (c) It has a greater speed in a vacuum than does red light of wavelength \(652 \mathrm{nm}\) (d) Its wavelength is longer than that of X-rays.

The most intense line in the cerium spectrum is at 418.7 nm. (a) Determine the frequency of the radiation producing this line. (b) In what part of the electromagnetic spectrum does this line occur? (c) Is it visible to the eye? If so, what color is it? If not, is this line at higher or lower energy than visible light?

Without doing detailed calculations, determine which of the following wavelengths represents light of the highest frequency: (a) \(6.7 \times 10^{-4} \mathrm{cm} ;\) (b) \(1.23 \mathrm{mm}\) (c) \(80 \mathrm{nm} ;\) (d) \(6.72 \mu \mathrm{m}\)

Without doing detailed calculations, arrange the following electromagnetic radiation sources in order of increasing frequency: (a) a red traffic light, (b) a \(91.9 \mathrm{MHz}\) radio transmitter, (c) light with a frequency of \(3.0 \times 10^{14} \mathrm{s}^{-1}\) (d) light with a wavelength of \(49 \mathrm{nm}\).

How long does it take light from the sun, 93 million miles away, to reach Earth?

In astronomy, distances are measured in light-years, the distance that light travels in one year. What is the distance of one light-year expressed in kilometers?

Use the Balmer equation (8.2) to determine (a) the frequency, in \(s^{-1}\), of the radiation corresponding to \(n=5\) (b) the wavelength, in nanometers, of the line in the Balmer series corresponding to \(n=7\) (c) the value of \(n\) corresponding to the Balmer series line at \(380 \mathrm{nm}\)

Use Planck's equation (8.3) to determine (a) the energy, in joules per photon, of radiation of frequency \(7.39 \times 10^{15} \mathrm{s}^{-1}\) (b) the energy, in kilojoules per mole, of radiation of frequency \(1.97 \times 10^{14} \mathrm{s}^{-1}\)

What is \(\Delta E\) for the transition of an electron from \(n=6\) to \(n=3\) in a Bohr hydrogen atom? What is the frequency of the spectral line produced?

What is \(\Delta E\) for the transition of an electron from \(n=5\) to \(n=2\) in a Bohr hydrogen atom? What is the frequency of the spectral line produced?

The Lyman series of the hydrogen spectrum can be represented by the equation $$\nu=3.2881 \times 10^{15} \mathrm{s}^{-1}\left(\frac{1}{1^{2}}-\frac{1}{n^{2}}\right)(\text { where } n=2,3, \ldots)$$ (a) Calculate the maximum and minimum wavelength lines, in nanometers, in this series. (b) What value of \(n\) corresponds to a spectral line at 95.0 nm? (c) Is there a line at \(108.5 \mathrm{nm} ?\) Explain.

Calculate the wavelengths, in nanometers, of the first four lines of the Balmer series of the hydrogen spectrum, starting with the longest wavelength component.

A line is detected in the hydrogen spectrum at \(1880 \mathrm{nm}\). Is this line in the Balmer series? Explain.

A certain radiation has a wavelength of \(574 \mathrm{nm}\). What is the energy, in joules, of (a) one photon; (b) a mole of photons of this radiation?

Without doing detailed calculations, indicate which of the following electromagnetic radiations has the greatest energy per photon and which has the least: (a) \(662 \mathrm{nm}\) (b) \(2.1 \times 10^{-5} \mathrm{cm} ;\) (c) \(3.58 \mu \mathrm{m} ;\) (d) \(4.1 \times 10^{-6} \mathrm{m}\).

In what region of the electromagnetic spectrum would you expect to find radiation having an energy per photon 100 times that associated with 988 nm radiation?

High-pressure sodium vapor lamps are used in street lighting. The two brightest lines in the sodium spectrum are at 589.00 and \(589.59 \mathrm{nm}\). What is the difference in energy per photon of the radiations corresponding to these two lines?

The lowest-frequency light that will produce the photoelectric effect is called the threshold frequency. (a) The threshold frequency for indium is \(9.96 \times\) \(10^{14} \mathrm{s}^{-1} .\) What is the energy, in joules, of a photon of this radiation? (b) Will indium display the photoelectric effect with UV light? With infrared light? Explain.

Calculate the increase in (a) distance from the nucleus and (b) energy when an electron is excited from the first to the third Bohr orbit.

What are the (a) frequency, in \(s^{-1}\), and (b) wavelength, in nanometers, of the light emitted when the electron in a hydrogen atom drops from the energy level \(n=7\) to \(n=4 ?\) (c) In what portion of the electromagnetic spectrum is this light?

Without doing detailed calculations, indicate which of the following electron transitions requires the greatest amount of energy to be absorbed by a hydrogen atom: from (a) \(n=1\) to \(n=2 ;\) (b) \(n=2\) to \(n=4 ;\) (c) \(n=3\) to \(n=9 ;\) (d) \(n=10\) to \(n=1\)

For the Bohr hydrogen atom determine (a) the radius of the orbit \(n=4\) (b) whether there is an orbit having a radius of \(4.00 \AA\) (c) the energy level corresponding to \(n=8\) (d) whether there is an energy level at \(-2.5 \times 10^{-17} \mathrm{J}\)

Without doing detailed calculations, indicate which of the following electron transitions in the hydrogen atom results in the emission of light of the longest wavelength. (a) \(n=4\) to \(n=3 ;\) (b) \(n=1\) to \(n=2\) (c) \(n=1\) to \(n=6 ;\) (d) \(n=3\) to \(n=2\).

What electron transition in a hydrogen atom, starting from the orbit \(n=7,\) will produce light of wavelength \(410 \mathrm{nm} ?\)

What electron transition in a hydrogen atom, ending in the orbit \(n=3,\) will produce light of wavelength \(1090 \mathrm{nm} ?\)

Which must possess a greater velocity to produce matter waves of the same wavelength (such as \(1 \mathrm{nm}\) ), protons or electrons? Explain your reasoning.

What must be the velocity, in meters per second, of a beam of electrons if they are to display a de Broglie wavelength of \(1 \mu \mathrm{m} ?\)

Calculate the de Broglie wavelength, in nanometers, associated with a \(145 \mathrm{g}\) baseball traveling at a speed of \(168 \mathrm{km} / \mathrm{h} .\) How does this wavelength compare with typical nuclear or atomic dimensions?

What is the wavelength, in nanometers, associated with a \(1000 \mathrm{kg}\) automobile traveling at a speed of \(25 \mathrm{m} \mathrm{s}^{-1},\) that is, considering the automobile to be a matter wave? Comment on the feasibility of an experimental measurement of this wavelength.

Describe how the Bohr model of the hydrogen atom appears to violate the Heisenberg uncertainty principle.

Although Einstein made some early contributions to quantum theory, he was never able to accept the Heisenberg uncertainty principle. He stated, "God does not play dice with the Universe." What do you suppose Einstein meant by this remark? In reply to Einstein's remark, Niels Bohr is supposed to have said, "Albert, stop telling God what to do." What do you suppose Bohr meant by this remark?

Show that the uncertainty principle is not significant when applied to large objects such as automobiles. Assume that \(m\) is precisely known; assign a reasonable value to either the uncertainty in position or the uncertainty in velocity, and estimate a value of the other.

What must be the velocity of electrons if their associated wavelength is to equal the radius of the first Bohr orbit of the hydrogen atom?

A standing wave in a string \(42 \mathrm{cm}\) long has a total of six nodes (including those at the ends). What is the wavelength, in centimeters, of this standing wave?

What is the length of a string that has a standing wave with four nodes (including those at the ends) and \(\lambda=17 \mathrm{cm} ?\)

An electron in a one-dimensional box requires a wavelength of \(618 \mathrm{nm}\) to excite an electron from the \(n=2\) level to the \(n=4\) level. Calculate the length of the box.

Calculate the wavelength of the electromagnetic radiation required to excite a proton from the ground state to the level with \(n=4\) in a one-dimensional box 50. pm long.

Describe some of the differences between the orbits of the Bohr atom and the orbitals of the wave mechanical atom. Are there any similarities?

The greatest probability of finding the electron in a small-volume element of the 1 s orbital of the hydrogen atom is at the nucleus. Yet the most probable distance of the electron from the nucleus is \(53 \mathrm{pm}\). How can you reconcile these two statements?

Select the correct answer and explain your reasoning. An electron having \(n=3\) and \(m_{\ell}=0\) (a) must have \(m_{s}=+\frac{1}{2} ;(\mathbf{b})\) must have \(\ell=1 ;(\mathbf{c})\) may have \(\ell=0,1\) or \(2 ;\) (d) must have \(\ell=2\).

Write an acceptable value for each of the missing quantum numbers. (a) \(n=3, \ell=?, m_{\ell}=2, m_{s}=+\frac{1}{2}\) (b) \(n=?, \ell=2, m_{\ell}=1, m_{s}=-\frac{1}{2}\) (c) \(n=4, \ell=2, m_{\ell}=0, m_{s}=?\) (d) \(n=?, \ell=0, m_{\ell}=?, m_{s}=?\)

What type of orbital (i.e., \(3 s, 4 p, \ldots)\) is designated by these quantum numbers? (a) \(n=5, \ell=1, m_{\ell}=0\) (b) \(n=4, \ell=2, m_{\ell}=-2\) (c) \(n=2, \ell=0, m_{\ell}=0\)

Which of the following statements is (are) correct for an electron with \(n=4\) and \(m_{\ell}=2 ?\) Explain. (a) The electron is in the fourth principal shell. (b) The electron may be in a \(d\) orbital. (c) The electron may be in a \(p\) orbital. (d) The electron must have \(m_{s}=+\frac{1}{2}\)

Concerning the electrons in the shells, subshells, and orbitals of an atom, how many can have (a) \(n=4, \ell=2, m_{\ell}=1,\) and \(m_{s}=+\frac{1}{2} ?\) (b) \(n=4, \ell=2,\) and \(m_{\ell}=1 ?\) (c) \(n=4\) and \(\ell=2 ?\) (d) \(n=4 ?\) (e) \(n=4, \ell=2,\) and \(m_{s}=+\frac{1}{2} ?\)

Concerning the concept of subshells and orbitals, (a) How many subshells are found in the \(n=3\) level? (b) What are the names of the subshells in the \(n=3\) level? (c) How many orbitals have the values \(n=4\) and \(\ell=3 ?\) (d) How many orbitals have the values \(n=3, \ell=2\) and \(m_{\ell}=-2 ?\) (e) What is the total number of orbitals in the \(n=4\) level?

Show that the probability of finding a \(2 p_{y}\) electron in the \(x z\) plane is zero.

Show that the probability of finding a \(3 d_{x z}\) electron in the \(x y\) plane is zero.

Identify the orbital that has (a) one radial node and one angular node; (b) no radial nodes and two angular nodes; (c) two radial nodes and three angular nodes.

Identify the orbital that has (a) two radial nodes and one angular node; (b) five radial nodes and zero angular nodes; (c) one radial node and four angular nodes.

On the basis of the periodic table and rules for electron configurations, indicate the number of (a) \(2 p\) electrons in \(\mathrm{N} ;\) (b) \(4 \mathrm{s}\) electrons in \(\mathrm{Rb} ;\) (c) \(4 \mathrm{d}\) electrons in As; (d) \(4 f\) electrons in \(\mathrm{Au} ;\) (e) unpaired electrons in \(\mathrm{Pb} ;\) (f) elements in group 14 of the periodic table; (g) elements in the sixth period of the periodic table.