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College Physics for AP® Courses

Problems & Exercises

College Physics for AP® CoursesProblems & Exercises

30.1 Discovery of the Atom


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.)


(a) Calculate the mass of a proton using the charge-to-mass 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?


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?

30.2 Discovery of the Parts of the Atom: Electrons and Nuclei


Rutherford found the size of the nucleus to be about 1015 m1015 m size 12{"10" rSup { size 8{ - "15"} } " m"} {}. This implied a huge density. What would this density be for gold?


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/cm30.81 g/cm3 size 12{0 "." "81 g/cm" rSup { size 8{3} } } {}) has a diameter of 4.0×106 m4.0×106 m size 12{4 "." 0 times "10" rSup { size 8{ - 6} } " m"} {}. Find the charge on the drop, in terms of electron units.


(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?

30.3 Bohr’s Theory of the Hydrogen Atom


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


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


Look up the values of the quantities in aB=h24π2mekqe2aB=h24π2mekqe2, and verify that the Bohr radius aBaB is 0.529×1010 m0.529×1010 m size 12{0 "." "529" times "10" rSup { size 8{ - "10"} } " m"} {}.


Verify that the ground state energy E0E0 size 12{E rSub { size 8{0} } } {} is 13.6 eV by using E0=2π2qe4mek2h2.E0=2π2qe4mek2h2. size 12{E rSub { size 8{o} } = { {2π rSup { size 8{2} } q rSub { size 8{e} } rSup { size 8{4} } m rSub { size 8{e} } k rSup { size 8{2} } } over {h rSup { size 8{2} } } } "." } {}


If a hydrogen atom has its electron in the n=4n=4 size 12{n=4} {} state, how much energy in eV is needed to ionize it?


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


Find the radius of a hydrogen atom in the n=2n=2 size 12{n=2} {} state according to Bohr’s theory.


Show that 13.6 eV/hc=1.097×107 m=R13.6 eV/hc=1.097×107 m=R (Rydberg’s constant), as discussed in the text.


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


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.


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


(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?


A wavelength of 4.653 μm4.653 μm size 12{4 "." "653 μm"} {} is observed in a hydrogen spectrum for a transition that ends in the nf=5nf=5 size 12{n rSub { size 8{f} } =5} {} level. What was nini size 12{n rSub { size 8{i} } } {} for the initial level of the electron?


A singly ionized helium ion has only one electron and is denoted He+He+ size 12{"He" rSup { size 8{+{}} } } {}. What is the ion’s radius in the ground state compared to the Bohr radius of hydrogen atom?


A beryllium ion with a single electron (denoted Be3+Be3+ size 12{"Be" rSup { size 8{3+{}} } } {}) is in an excited state with radius the same as that of the ground state of hydrogen.

(a) What is nn size 12{n} {} for the Be3+Be3+ size 12{"Be" rSup { size 8{3+{}} } } {} ion?

(b) How much energy in eV is needed to ionize the ion from this excited state?


Atoms can be ionized by thermal collisions, such as at the high temperatures found in the solar corona. One such ion is C+5C+5 size 12{C rSup { size 8{+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?


Verify Equations rn=n2ZaBrn=n2ZaB and aB=h24π2mekqe2=0.529×1010 maB=h24π2mekqe2=0.529×1010 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.


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 1λ=R1nf21ni21λ=R1nf21ni2 size 12{ { {1} over {λ} } =R left ( { {1} over {n rSub { size 8{f} } rSup { size 8{2} } } } - { {1} over {n rSub { size 8{i} } rSup { size 8{2} } } } right )} {}? It is amazing how well a simple formula (disconnected originally from theory) could duplicate this phenomenon.

30.4 X Rays: Atomic Origins and Applications


(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.


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.)


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.


The maximum characteristic x-ray photon energy comes from the capture of a free electron into a KK size 12{K} {} shell vacancy. What is this photon energy in keV for tungsten, assuming the free electron has no initial kinetic energy?


What are the approximate energies of the KαKα size 12{K rSub { size 8{α} } } {} and KβKβ size 12{K rSub { size 8{β} } } {} x rays for copper?

30.5 Applications of Atomic Excitations and De-Excitations


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?


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.


Ruby lasers have chromium atoms doped in an aluminum oxide crystal. The energy level diagram for chromium in a ruby is shown in Figure 30.63. What wavelength is emitted by a ruby laser?

The figure shows energy levels of chromium atoms in an aluminum oxide crystal. Ground state is at zero point zero electron volts, first metastable state is at one point seventy nine electron volts, second state is at two point three electron volts, and the third state is at three point zero electron volts.
Figure 30.63 Chromium atoms in an aluminum oxide crystal have these energy levels, one of which is metastable. This is the basis of a ruby laser. Visible light can pump the atom into an excited state above the metastable state to achieve a population inversion.

(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 30.64. (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 μm1.06 μm size 12{1 "." "06-μm"} {} radiation.

The figure shows different energy levels of neodymium atoms in glass. The ground state is at zero electron volts, first state is at zero point five zero electron volts, the metastable second state is at one point sixty seven electron volts, and the group state levels above metastable second are at two point one electron volts. The photons release one point seventeen electron volts at wavelength of one point zero six micro meters while coming from the metastable second state to first state.
Figure 30.64 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.

30.8 Quantum Numbers and Rules


If an atom has an electron in the n=5n=5 size 12{n=5} {} state with ml=3ml=3 size 12{m rSub { size 8{l} } =3} {}, what are the possible values of ll size 12{l} {}?


An atom has an electron with ml=2ml=2 size 12{m rSub { size 8{l} } =2} {}. What is the smallest value of nn size 12{n} {} for this electron?


What are the possible values of mlml size 12{m rSub { size 8{l} } } {} for an electron in the n=4n=4 size 12{n=4} {} state?


What, if any, constraints does a value of ml=1ml=1 size 12{m rSub { size 8{l} } =1} {} place on the other quantum numbers for an electron in an atom?


(a) Calculate the magnitude of the angular momentum for an l=1l=1 size 12{l=1} {} electron. (b) Compare your answer to the value Bohr proposed for the n=1n=1 size 12{n=1} {} state.


(a) What is the magnitude of the angular momentum for an l=1l=1 size 12{l=1} {} electron? (b) Calculate the magnitude of the electron’s spin angular momentum. (c) What is the ratio of these angular momenta?


Repeat Exercise 30.40 for l=3l=3 size 12{l=3} {}.


(a) How many angles can LL size 12{L} {} make with the zz size 12{z} {}-axis for an l=2l=2 size 12{l=2} {} electron? (b) Calculate the value of the smallest angle.


What angles can the spin SS size 12{S} {} of an electron make with the zz size 12{z} {}-axis?

30.9 The Pauli Exclusion Principle


(a) How many electrons can be in the n=4n=4 size 12{n=4} {} shell?

(b) What are its subshells, and how many electrons can be in each?


(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 n = 5 shell, what is the spectroscopic notation for this atom?


(a) If one subshell of an atom has 9 electrons in it, what is the minimum value of ll size 12{l} {}? (b) What is the spectroscopic notation for this atom, if this subshell is part of the n=3n=3 size 12{n=3} {} shell?


(a) List all possible sets of quantum numbers n,l,ml,msn,l,ml,ms for the n=3n=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 2n22n2 size 12{2n rSup { size 8{2} } } {} and that the number in each subshell is 22l+122l+1 size 12{2 left (2l+1 right )} {}.


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


Which of the following spectroscopic notations are allowed (that is, which violate none of the rules regarding values of quantum numbers)? (a) 1s11s1 size 12{1s rSup { size 8{1} } } {}(b) 1d31d3 size 12{1d rSup { size 8{3} } } {}(c) 4s24s2 size 12{4s rSup { size 8{2} } } {} (d) 3p73p7 size 12{3p rSup { size 8{7} } } {}(e) 6h206h20 size 12{6h rSup { size 8{"20"} } } {}


(a) Using the Pauli exclusion principle and the rules relating the allowed values of the quantum numbers n,l,ml,msn,l,ml,ms size 12{ left (n,`l,`m rSub { size 8{l} } ,`m rSub { size 8{s} } right )} {}, prove that the maximum number of electrons in a subshell is 2n22n2 size 12{2n rSup { size 8{2} } } {}.

(b) In a similar manner, prove that the maximum number of electrons in a shell is 2n2.


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.


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).


(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 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

A galaxy moving away from the earth has a speed of 0.0100c 0.0100c . What wavelength do we observe for an ni=7ni=7 size 12{n rSub { size 8{i} } =7} {} to nf=2nf=2 size 12{n rSub { size 8{f} } =2} {} transition for hydrogen in that galaxy?


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=ni= size 12{n rSub { size 8{i} } = infinity } {} to nf=1nf=1 size 12{n rSub { size 8{f} } =1} {}, or a Lyman series transition).


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?


Integrated Concepts

What double-slit separation would produce a first-order maximum at 3.00º3.00º size 12{3 "." "00"°} {} 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.


Integrated Concepts

In a laboratory experiment designed to duplicate Thomson’s determination of qe/meqe/me size 12{q rSub { size 8{e} } /m rSub { size 8{e} } } {}, a beam of electrons having a velocity of 6.00×107m/s6.00×107m/s size 12{6 "." "00" times "10" rSup { size 8{7} } `"m/s"} {} enters a 5.00×103T5.00×103T magnetic field. The beam moves perpendicular to the field in a path having a 6.80-cm radius of curvature. Determine qe/meqe/me from these observations, and compare the result with the known value.


Integrated Concepts

Find the value of l 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.


Integrated Concepts

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 aB=h22mekqe2=0.529×1010 maB=h22mekqe2=0.529×1010 m is 207 meme.

(a) Calculate the radius of the n=1n=1 size 12{n=1} {} orbit for a muon in a uranium ion (Z=92Z=92 size 12{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.


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 1 . 00 × 10 4 size 12{1 "." "00" times "10" rSup { size 8{ - 4} } } {} moles of neon.


Integrated Concepts

A carbon dioxide laser used in surgery emits infrared radiation with a wavelength of 10.6 μm 10.6 μm . In 1.00 ms, this laser raised the temperature of 1.00 cm31.00 cm3 of flesh to 100ºC100ºC 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?


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?


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

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.)


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±50.0 km/s size 12{ +- "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.)


Integrated Concepts

Prove that the velocity of charged particles moving along a straight path through perpendicular electric and magnetic fields is v=E/Bv=E/B size 12{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.


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?


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 nini size 12{n rSub { size 8{i} } } {}, the principal quantum number of the initial state. (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?


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.


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 1λ=R1nf21ni21λ=R1nf21ni2 size 12{ { {1} over {λ} } =R left ( { {1} over {n rSub { size 8{f} } rSup { size 8{2} } } } - { {1} over {n rSub { size 8{i} } rSup { size 8{2} } } } right )} {}, but with a different constant R R .

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