Solved: How do the allowed orbits for electrons in atoms

Problem 2CQ Explain why patterns observed in the periodic table of the elements are evidence for the existence of atoms, and why Brownian motion is a more direct type of evidence for their existence.

Problem 1CQ Name three different types of evidence for the existence of atoms.

Problem 1PE Using the given charge-to-mass ratios for electrons and protons, and knowing the magnitudes of their charges are equal, what is the ratio of the proton’s mass to the electron’s? (Note that since the charge-to-mass ratios are given to only three-digit accuracy, your answer may differ from the accepted ratio in the fourth digit.)

Problem 3CQ If atoms exist, why can’t we see them with visible light?

Construct Your Own Problem Consider the Doppler-shifted hydrogen spectrum received from a rapidly receding galaxy. Construct a problem in which you calculate the energies of selected spectral lines in the Balmer series and examine whether they can be described with a formula like that in the equation \(\frac{1}{\lambda}=R\left(\frac{1}{n_{\mathrm{f}}^{2}}-\frac{1}{n_{\mathrm{i}}^{2}}\right)\), but with a different constant \(R\). Equation Transcription: Text Transcription: 1/lambda = R(1/n_f ^2 - 1/n_i ^2) R

Problem 2PE (a) Calculate the mass of a proton using the charge-tomass ratio given for it in this chapter and its known charge. (b) How does your result compare with the proton mass given in this chapter?

Problem 3PE If someone wanted to build a scale model of the atom with a nucleus 1.00 m in diameter, how far away would the nearest electron need to be?

Problem 4CQ What two pieces of evidence allowed the first calculation of me , the mass of the electron? (a) The ratios qe / me and q p / mp . (b) The values of qe and EB . (c) The ratio qe / me and qe . Justify your response.

Problem 5CQ How do the allowed orbits for electrons in atoms differ from the allowed orbits for planets around the sun? Explain how the correspondence principle applies here.

Problem 6PE (a) An aspiring physicist wants to build a scale model of a hydrogen atom for her science fair project. If the atom is 1.00 m in diameter, how big should she try to make the nucleus? (b) How easy will this be to do?

Problem 7CQ Explain how Bohr’s rule for the quantization of electron orbital angular momentum differs from the actual rule.

Problem 6CQ How do the allowed orbits for electrons in atoms differ from the allowed orbits for planets around the sun? Explain how the correspondence principle applies here.

Problem 7PE By calculating its wavelength, show that the first line in the Lyman series is UV radiation.

Problem 5PE In Millikan’s oil-drop experiment, one looks at a small oil drop held motionless between two plates. Take the voltage between the plates to be 2033 V, and the plate separation to be 2.00 cm. The oil drop (of density 0.81 g/cm3 ) has a diameter of 4.0×10?6 m . Find the charge on the drop, in terms of electron units.

Problem 4PE Rutherford found the size of the nucleus to be about 10?15 m . This implied a huge density. What would this density be for gold?

Problem 8CQ What is a hydrogen-like atom, and how are the energies and radii of its electron orbits related to those in hydrogen?

Problem 9CQ Explain why characteristic x rays are the most energetic in the EM emission spectrum of a given element.

Look up the values of the quantities in \(a_{\mathrm{B}}=\frac{h^{2}}{4 \pi^{2} m_{e} k q_{e}^{2}}\), and verify that the Bohr radius \(a_{\mathrm{B}}\) is \(0.529\times10^{-10}\mathrm{\ m}\). Equation Transcription: Text Transcription: a_B = h^2 /4pi^2 m_e kq_e ^2 a_B 0.529×10^-10 m

Problem 8PE Find the wavelength of the third line in the Lyman series, and identify the type of EM radiation.

Problem 11CQ Observers at a safe distance from an atmospheric test of a nuclear bomb feel its heat but receive none of its copious x rays. Why is air opaque to x rays but transparent to infrared?

Problem 11PE If a hydrogen atom has its electron in the n = 4 state, how much energy in eV is needed to ionize it?

Problem 12CQ Lasers are used to burn and read CDs. Explain why a laser that emits blue light would be capable of burning and reading more information than one that emits infrared.

Problem 10CQ Why does the energy of characteristic x rays become increasingly greater for heavier atoms?

Problem 13CQ Crystal lattices can be examined with x rays but not UV. Why?

Problem 13PE Find the radius of a hydrogen atom in the n = 2 state according to Bohr’s theory.

Problem 14CQ CT scanners do not detect details smaller than about 0.5 mm. Is this limitation due to the wavelength of x rays? Explain.

Problem 15CQ How do the allowed orbits for electrons in atoms differ from the allowed orbits for planets around the sun? Explain how the correspondence principle applies here.

Problem 15PE What is the smallest-wavelength line in the Balmer series? Is it in the visible part of the spectrum?

Problem 16CQ Atomic and molecular spectra are discrete. What does discrete mean, and how are discrete spectra related to the quantization of energy and electron orbits in atoms and molecules?

Problem 14PE Show that (13.6 eV) / hc = 1.097×107 m = R (Rydberg’s constant), as discussed in the text.

Verify that the ground state energy \(E_{0}\) is \(13.6\mathrm{\ eV}\) by using \(E_{0}=\frac{2 \pi^{2} q_{e}^{4} m_{e} k^{2}}{h^{2}}\). Equation Transcription: Text Transcription: E_0 13.6 eV E_0 = 2pi^2 q_e ^4 m_e k^2 / h^2

Problem 16PE Show that the entire Paschen series is in the infrared part of the spectrum. To do this, you only need to calculate the shortest wavelength in the series.

Problem 17CQ Hydrogen gas can only absorb EM radiation that has an energy corresponding to a transition in the atom, just as it can only emit these discrete energies. When a spectrum is taken of the solar corona, in which a broad range of EM wavelengths are passed through very hot hydrogen gas, the absorption spectrum shows all the features of the emission spectrum. But when such EM radiation passes through room-temperature hydrogen gas, only the Lyman series is absorbed. Explain the difference.

Problem 17PE Do the Balmer and Lyman series overlap? To answer this, calculate the shortest-wavelength Balmer line and the longest-wavelength Lyman line.

Problem 12PE A hydrogen atom in an excited state can be ionized with less energy than when it is in its ground state. What is n for a hydrogen atom if 0.850 eV of energy can ionize it?

Problem 18CQ Lasers are used to burn and read CDs. Explain why a laser that emits blue light would be capable of burning and reading more information than one that emits infrared.

Problem 19CQ The coating on the inside of fluorescent light tubes absorbs ultraviolet light and subsequently emits visible light. An inventor claims that he is able to do the reverse process. Is the inventor’s claim possible?

Problem 19PE A wavelength of 4.653 ?m is observed in a hydrogen spectrum for a transition that ends in the nf = 5 level. What was ni for the initial level of the electron?

Problem 20CQ What is the difference between fluorescence and phosphorescence?

Problem 21PE A beryllium ion with a single electron (denoted Be3 + ) is in an excited state with radius the same as that of the ground state of hydrogen. (a) What is n for the Be3 + ion? (b) How much energy in eV is needed to ionize the ion from this excited state?

Problem 22CQ How is the de Broglie wavelength of electrons related to the quantization of their orbits in atoms and molecules?

Problem 22PE Atoms can be ionized by thermal collisions, such as at the high temperatures found in the solar corona. One such ion is C +5 , a carbon atom with only a single electron. (a) By what factor are the energies of its hydrogen-like levels greater than those of hydrogen? (b) What is the wavelength of the first line in this ion’s Paschen series? (c) What type of EM radiation is this?

Problem 20PE A singly ionized helium ion has only one electron and is denoted He+ . What is the ion’s radius in the ground state compared to the Bohr radius of hydrogen atom?

Problem 23CQ What is the Zeeman effect, and what type of quantization was discovered because of this effect?

Problem 21CQ How can you tell that a hologram is a true three-dimensional image and that those in 3-D movies are not?

Verify Equations \(r_{n}=\frac{n^{2}}{Z} a_{\mathrm{B}}\) and \(a_{\mathrm{B}}=\frac{h^2}{4\pi^2m_ekq_e^2}=0.529\times10^{-10}\mathrm{\ m}\) using the approach stated in the text. That is, equate the Coulomb and centripetal forces and then insert an expression for velocity from the condition for angular momentum quantization. Equation Transcription: Text Transcription: r_n = n^2/Z a_B a_B = h^2 / 4pi^2 m_e kq_e ^2 = 0.529 x 10^-10 m

Problem 24CQ Define the quantum numbers n, l, ml , s , and ms .

Problem 26PE A color television tube also generates some x rays when its electron beam strikes the screen. What is the shortest wavelength of these x rays, if a 30.0-kV potential is used to accelerate the electrons? (Note that TVs have shielding to prevent these x rays from exposing viewers.)

Problem 27CQ List all the possible values of s and ms for an electron. Are there particles for which these values are different? The same?

Problem 25PE (a) What is the shortest-wavelength x-ray radiation that can be generated in an x-ray tube with an applied voltage of 50.0 kV? (b) Calculate the photon energy in eV. (c) Explain the relationship of the photon energy to the applied voltage.

The wavelength of the four Balmer series lines for hydrogen are found to be \(410.3\), \(434.2\), \(486.3\), and \(656.5 \ nm\). What average percentage difference is found between these wavelength numbers and those predicted by \(\frac{1}{\lambda}=R\left(\frac{1}{n_{\mathrm{f}}^{2}}-\frac{1}{n_{\mathrm{i}}^{2}}\right)\)? It is amazing how well a simple formula (disconnected originally from theory) could duplicate this phenomenon. Equation Transcription: Text Transcription: 410.3 434.2 486.3 656.5 nm 1/lambda = R(1/n_f ^2 - 1/n_i ^2)

Problem 26CQ For a given value of l, what are the allowed values of ml ? What are the allowed values of ml for a given value of n ? Give an example in each case.

Problem 25CQ For a given value of n , what are the allowed values of l ?

Problem 29CQ Which of the following are not allowed? State which rule is violated for any that are not allowed. (a) 1p 3 (b) 2p 8 (c) 3g 11 (d) 4 f 2

Problem 28PE The maximum characteristic x-ray photon energy comes from the capture of a free electron into a K shell vacancy. What is this photon energy in keV for tungsten, assuming the free electron has no initial kinetic energy?

A helium-neon laser is pumped by electric discharge. What wavelength electromagnetic radiation would be needed to pump it? See Figure 30.39 for energy-level information.

Problem 29PE What are the approximate energies of the K? and K? x rays for copper?

Figure 30.39 shows the energy-level diagram for neon. (a) Verify that the energy of the photon emitted when neon goes from its metastable state to the one immediately below is equal to \(1.96 \ eV\). (b) Show that the wavelength of this radiation is \(633 \ nm\). (c) What wavelength is emitted when the neon makes a direct transition to its ground state? Equation Transcription: Text Transcription: 1.96 eV 633 nm

Problem 28CQ Identify the shell, subshell, and number of electrons for the following: (a) 2p 3 . (b) 4d 9 . (c) 3s 1 . (d) 5g 16.

Ruby lasers have chromium atoms doped in an aluminum oxide crystal. The energy level diagram for chromium in a ruby is shown in Figure 30.64. What wavelength is emitted by a ruby laser? Ground

Problem 33PE (a) What energy photons can pump chromium atoms in a ruby laser from the ground state to its second and third excited states? (b) What are the wavelengths of these photons? Verify that they are in the visible part of the spectrum.

Some of the most powerful lasers are based on the energy levels of neodymium in solids, such as glass, as shown in Figure 30.65. (a) What average wavelength light can pump the neodymium into the levels above its metastable state? (b) Verify that the \(1.17 \ eV\) transition produces \(1.06\ \mu\mathrm{m}\) radiation. Figure 30.65 Neodymium atoms in glass have these energy levels, one of which is metastable. The group of levels above the metastable state is convenient for achieving a population inversion, since photons of many different energies can be absorbed by atoms in the ground state. Equation Transcription: Text Transcription: 1.17 eV 1.06 mu m

Problem 35PE If an atom has an electron in the n = 5 state with ml = 3 , what are the possible values of l ?

Problem 36PE An atom has an electron with ml = 2 . What is the smallest value of n for this electron?

Problem 40PE (a) What is the magnitude of the angular momentum for an l = 1 electron? (b) Calculate the magnitude of the electron’s spin angular momentum. (c) What is the ratio of these angular momenta?

Problem 39PE (a) Calculate the magnitude of the angular momentum for an l = 1 electron. (b) Compare your answer to the value Bohr proposed for the n = 1 state.

Problem 41PE Repeat Exercise 30.40 for l = 3 . Reference Exercise 30.40: (a) What is the magnitude of the angular momentum for an l = 1 electron? (b) Calculate the magnitude of the electron’s spin angular momentum. (c) What is the ratio of these angular momenta?

Problem 42PE (a) How many angles can L make with the z -axis for an l = 2 electron? (b) Calculate the value of the smallest angle.

Problem 37PE What are the possible values of ml for an electron in the n = 4 state?

Problem 43PE What angles can the spin S of an electron make with the z -axis?

Problem 38PE What, if any, constraints does a value of ml = 1 place on the other quantum numbers for an electron in an atom?

Problem 44PE (a) How many electrons can be in the n = 4 shell? (b) What are its subshells, and how many electrons can be in each?

Problem 45PE (a) What is the minimum value of 1 for a subshell that has 11 electrons in it? (b) If this subshell is in the n = 5 shell, what is the spectroscopic notation for this atom?

Problem 46PE (a) If one subshell of an atom has 9 electrons in it, what is the minimum value of l ? (b) What is the spectroscopic notation for this atom, if this subshell is part of the n = 3 shell?

Problem 48PE Which of the following spectroscopic notations are not allowed? (a) 5s 1 (b) 1d 1 (c) 4s 3 (d) 3p 7 (e) 5g 15 . State which rule is violated for each that is not allowed.

Problem 51PE Integrated Concepts Estimate the density of a nucleus by calculating the density of a proton, taking it to be a sphere 1.2 fm in diameter. Compare your result with the value estimated in this chapter.

Problem 50PE (a) Using the Pauli exclusion principle and the rules relating the allowed values of the quantum numbers (n, l, ml , ms), prove that the maximum number of electrons in a subshell is 2n2 . (b) In a similar manner, prove that the maximum number of electrons in a shell is 2n2 .

Problem 49PE Which of the following spectroscopic notations are allowed (that is, which violate none of the rules regarding values of quantum numbers)? (a) 1s 1 (b) 1d 3 (c) 4s 2 (d) 3p 7 (e) 6h 20

Problem 47PE (a) List all possible sets of quantum numbers (n, l, ml , ms) for the n = 3 shell, and determine the number of electrons that can be in the shell and each of its subshells. (b) Show that the number of electrons in the shell equals 2n2 and that the number in each subshell is 2(2l + 1) .

Problem 53PE (a) What is the distance between the slits of a diffraction grating that produces a first-order maximum for the first Balmer line at an angle of 20.0º ? (b) At what angle will the fourth line of the Balmer series appear in first order? (c) At what angle will the second-order maximum be for the first line?

Integrated Concepts The electric and magnetic forces on an electron in the \(CRT\) in Figure 30.7 are supposed to be in opposite directions. Verify this by determining the direction of each force for the situation shown. Explain how you obtain the directions (that is, identify the rules used). Equation Transcription: Text Transcription: CRT

Problem 55PE Integrated Concepts Calculate the velocity of a star moving relative to the earth if you observe a wavelength of 91.0 nm for ionized hydrogen capturing an electron directly into the lowest orbital (that is, a ni = ? to nf = 1 , or a Lyman series transition).

Problem 54PE Integrated Concepts A galaxy moving away from the earth has a speed of 0.0100c . What wavelength do we observe for an ni = 7 to nf = 2 transition for hydrogen in that galaxy?

Problem 58PE Integrated Concepts In a laboratory experiment designed to duplicate Thomson’s determination of qe / me , a beam of electrons having a velocity of 6.00×107 m/s enters a 5.00×10?3 T magnetic field. The beam moves perpendicular to the field in a path having a 6.80-cm radius of curvature. Determine qe / me from these observations, and compare the result with the known value.

Problem 59PE Integrated Concepts Find the value of l, the orbital angular momentum quantum number, for the moon around the earth. The extremely large value obtained implies that it is impossible to tell the difference between adjacent quantized orbits for macroscopic objects.

Problem 57PE Integrated Concepts What double-slit separation would produce a first-order maximum at 3.00º for 25.0-keV x rays? The small answer indicates that the wave character of x rays is best determined by having them interact with very small objects such as atoms and molecules.

Particles called muons exist in cosmic rays and can be created in particle accelerators. Muons are very similar to electrons, having the same charge and spin, but they have a mass \(207\) times greater. When muons are captured by an atom, they orbit just like an electron but with a smaller radius, since the mass in \(a_{\mathrm{B}}=\frac{h^2}{4\pi^2m_ekq_e^2}=0.529\times10^{-10}\ m\) is \(207\) \(m_{e}\). (a) Calculate the radius of the \(n=1\) orbit for a muon in a uranium ion \((Z=92)\) (b) Compare this with the \(7.5-fm\) radius of a uranium nucleus. Note that since the muon orbits inside the electron, it falls into a hydrogen-like orbit. Since your answer is less than the radius of the nucleus, you can see that the photons emitted as the muon falls into its lowest orbit can give information about the nucleus. Equation Transcription: Text Transcription: 207 a_B = h^2 / 4pi^2 m_e kq_e ^2 = 0.529 x 10^-10 m 207 m_e n = 1 Z = 92 7.5-fm

Problem 61PE Integrated Concepts Calculate the minimum amount of energy in joules needed to create a population inversion in a helium-neon laser containing 1.00×10?4 moles of neon.

Problem 62PE Integrated Concepts A carbon dioxide laser used in surgery emits infrared radiation with a wavelength of 10.6 ?m . In 1.00 ms, this laser raised the temperature of 1.00 cm3 of flesh to 100ºC and evaporated it. (a) How many photons were required? You may assume flesh has the same heat of vaporization as water. (b) What was the minimum power output during the flash?

Problem 63PE Integrated Concepts Suppose an MRI scanner uses 100-MHz radio waves. (a) Calculate the photon energy. (b) How does this compare to typical molecular binding energies?

Problem 64PE Integrated Concepts (a) An excimer laser used for vision correction emits 193-nm UV. Calculate the photon energy in eV. (b) These photons are used to evaporate corneal tissue, which is very similar to water in its properties. Calculate the amount of energy needed per molecule of water to make the phase change from liquid to gas. That is, divide the heat of vaporization in kJ/kg by the number of water molecules in a kilogram. (c) Convert this to eV and compare to the photon energy. Discuss the implications.

Integrated Concepts In a Millikan oil-drop experiment using a setup like that in Figure 30.9, a \(500-V\) potential difference is applied to plates separated by \(2.50 \ cm\). (a) What is the mass of an oil drop having two extra electrons that is suspended motionless by the field between the plates? (b) What is the diameter of the drop, assuming it is a sphere with the density of olive oil? Equation Transcription: Text Transcription: 500-V 2.50 cm

Problem 65PE Integrated Concepts A neighboring galaxy rotates on its axis so that stars on one side move toward us as fast as 200 km/s, while those on the other side move away as fast as 200 km/s. This causes the EM radiation we receive to be Doppler shifted by velocities over the entire range of ±200 km/s. What range of wavelengths will we observe for the 656.0-nm line in the Balmer series of hydrogen emitted by stars in this galaxy. (This is called line broadening.)

Problem 66PE Integrated Concepts A pulsar is a rapidly spinning remnant of a supernova. It rotates on its axis, sweeping hydrogen along with it so that hydrogen on one side moves toward us as fast as 50.0 km/s, while that on the other side moves away as fast as 50.0 km/s. This means that the EM radiation we receive will be Doppler shifted over a range of ±50.0 km/s . What range of wavelengths will we observe for the 91.20-nm line in the Lyman series of hydrogen? (Such line broadening is observed and actually provides part of the evidence for rapid rotation.)

Problem 67PE Integrated Concepts Prove that the velocity of charged particles moving along a straight path through perpendicular electric and magnetic fields is v = E / B . Thus crossed electric and magnetic fields can be used as a velocity selector independent of the charge and mass of the particle involved.

Problem 68PE Unreasonable Results (a) What voltage must be applied to an X-ray tube to obtain 0.0100-fm-wavelength X-rays for use in exploring the details of nuclei? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

Problem 70PE Construct Your Own Problem The solar corona is so hot that most atoms in it are ionized. Consider a hydrogen-like atom in the corona that has only a single electron. Construct a problem in which you calculate selected spectral energies and wavelengths of the Lyman, Balmer, or other series of this atom that could be used to identify its presence in a very hot gas. You will need to choose the atomic number of the atom, identify the element, and choose which spectral lines to consider.

Problem 69PE Unreasonable Results A student in a physics laboratory observes a hydrogen spectrum with a diffraction grating for the purpose of measuring the wavelengths of the emitted radiation. In the spectrum, she observes a yellow line and finds its wavelength to be 589 nm. (a) Assuming this is part of the Balmer series, determine ni , the principal quantum number of the initial state. (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

Problem 18PE (a) Which line in the Balmer series is the first one in the UV part of the spectrum? (b) How many Balmer series lines are in the visible part of the spectrum? (c) How many are in the UV?

Problem 27PE An x ray tube has an applied voltage of 100 kV. (a) What is the most energetic x-ray photon it can produce? Express your answer in electron volts and joules. (b) Find the wavelength of such an X–ray.