# Problems & Exercises

### 13.1 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?

### 13.2 Discovery of the Parts of the Atom: Electrons and Nuclei

Rutherford found the size of the nucleus to be about ${\text{10}}^{-\text{15}}\phantom{\rule{0.25em}{0ex}}\text{m}\text{.}$ 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 2,033 V, and the plate separation to be 2.00 cm. The oil drop, (of density $0\text{.}{\text{81 g/cm}}^{3}\text{,)}$ has a diameter of $4\text{.}0\phantom{\rule{0.25em}{0ex}}\times \phantom{\rule{0.25em}{0ex}}{\text{10}}^{-6}\phantom{\rule{0.25em}{0ex}}\text{m}\text{.}$ 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?

### 13.3 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 ${a}_{\text{B}}=\frac{{h}^{2}}{{4\pi}^{2}{m}_{e}{\text{kq}}_{e}^{2}}{}^{}\text{,}$ and verify that the Bohr radius ${a}_{\text{B}}$ is $\text{0.529}\phantom{\rule{0.25em}{0ex}}\times \phantom{\rule{0.25em}{0ex}}{\text{10}}^{-\text{10}}\phantom{\rule{0.25em}{0ex}}\text{m.}$

Verify that the ground state energy ${E}_{0}$ is 13.6 eV by using ${E}_{0}=\frac{{2\pi}^{2}{q}_{e}^{4}{m}_{e}{k}^{2}}{{h}^{2}}\text{.}$

If a hydrogen atom has its electron in the $n=4$ 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 *$n$* for a hydrogen atom if 0.850 eV of energy can ionize it?

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

Show that $\left(\mathrm{13.6\; eV}\right)/\text{hc}=\text{1.097}\phantom{\rule{0.25em}{0ex}}\times \phantom{\rule{0.25em}{0ex}}{\text{10}}^{7}\phantom{\rule{0.25em}{0ex}}\text{m}=R$—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 $4\text{.}\text{653 \mu m}$ is observed in a hydrogen spectrum for a transition that ends in the ${n}_{\text{f}}=5$ level. What was ${n}_{\text{i}}$ for the initial level of the electron?

A singly ionized helium ion has only one electron and is denoted ${\text{He}}^{+}\text{.}$ 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 ${\text{Be}}^{3+}\text{)}$ is in an excited state with radius the same as that of the ground state of hydrogen.

(a) What is *$n$* for the ${\text{Be}}^{3+}$ 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 ${\mathrm{C}}^{+5}\text{,}$ 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 ${r}_{n}=\frac{{n}^{2}}{Z}{a}_{\text{B}}$ and ${a}_{\mathrm{B}}=\frac{{h}^{2}}{{4\pi}^{2}{m}_{e}{\text{kq}}_{e}^{2}}=\text{0.529}\phantom{\rule{0.25em}{0ex}}\times \phantom{\rule{0.25em}{0ex}}{\text{10}}^{-\text{10}}\phantom{\rule{0.25em}{0ex}}\text{m}$ 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 $\frac{1}{\lambda}=R\left(\frac{1}{{n}_{\text{f}}^{2}}-\frac{1}{{n}_{\text{i}}^{2}}\right)\text{?}$ It is amazing how well a simple formula, disconnected originally from theory, could duplicate this phenomenon.

### 13.4 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 $K$ 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 ${K}_{\alpha}$ and ${K}_{\beta}$ X-rays for copper?

### 13.5 Applications of Atomic Excitations and De-Excitations

How do variations in transition probabilities contribute to population inversion and lasing?

Figure 13.39 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 13.39 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 13.64. What wavelength is emitted by a ruby laser?

(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 13.65. (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 $1\text{.}\text{06 \mu m}$ radiation.

### 13.8 Quantum Numbers and Rules

If an atom has an electron in the $n=5$ state with ${m}_{l}=3\text{,}$ what are the possible values of $l\text{?}$

An atom has an electron with ${m}_{l}=2\text{.}$ What is the smallest value of $n$ for this electron?

What are the possible values of ${m}_{l}$ for an electron in the $n=4$ state?

What, if any, constraints does a value of ${m}_{l}=1$ place on the other quantum numbers for an electron in an atom?

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

(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?

Repeat Exercise 13.41 for $l=3\text{.}$

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

What angles can the spin $S$ of an electron make with the $z\text{-}$axis?

### 13.9 The Pauli Exclusion Principle

(a) How many electrons can be in the $n=4$ shell?

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

(a) What is the minimum value of one for a subshell that has 11 electrons in it?

(b) If this subshell is in the $n=\text{5}$ shell, what is the spectroscopic notation for this atom?

(a) If one subshell of an atom has nine 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?

(a) List all possible sets of quantum numbers $\left(n,\phantom{\rule{0.25em}{0ex}}l,\phantom{\rule{0.25em}{0ex}}{m}_{l},\phantom{\rule{0.25em}{0ex}}{m}_{s}\right)$ 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 $2{n}^{2}$ and that the number in each subshell is $2\left(2l+1\right)\text{.}$

Which of the following spectroscopic notations are not allowed? (a) $5{s}^{1}\text{;}$ (b) $1{d}^{1}\text{;}$ (c) $4{s}^{3}\text{;}$ (d) $3{p}^{7}\text{;}$ (e) $5{g}^{15}\text{.}$ 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) $1{s}^{\mathrm{}}\text{;}$(b) $1{d}^{3}\text{;}$(c) $4{s}^{2}\text{;}$ (d) $3{p}^{\mathrm{}}\text{;}$(e) $6{h}^{\text{20}}\text{.}$

(a) Using the Pauli exclusion principle and the rules relating the allowed values of the quantum numbers $\left(n,\phantom{\rule{0.25em}{0ex}}l,\phantom{\rule{0.25em}{0ex}}{m}_{l},\phantom{\rule{0.25em}{0ex}}{m}_{s}\right)\text{,}$ prove that the maximum number of electrons in a subshell is $2{n}^{2}\text{.}$

(b) In a similar manner, prove that the maximum number of electrons in a shell is 2*n*^{2}.

**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 13.7 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 $\text{20.0\xba?}$

(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 $\text{0.0100}c\text{.}$ What wavelength do we observe for an ${n}_{\text{i}}=7$ to ${n}_{\text{f}}=2$ 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 ${n}_{\text{i}}=\infty $ to ${n}_{\text{f}}=1\text{,}$ or a Lyman series transition.

**Integrated Concepts**

In a Millikan oil-drop experiment using a setup like that in Figure 13.9, 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 $3\text{.}\text{00\xba}$ 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 ${q}_{e}/{m}_{e}\text{,}$ a beam of electrons having a velocity of $6\text{.}\text{00}\phantom{\rule{0.25em}{0ex}}\times \phantom{\rule{0.25em}{0ex}}{\text{10}}^{7}\phantom{\rule{0.25em}{0ex}}\text{m/s}$ enters a $\text{5.00}\phantom{\rule{0.25em}{0ex}}\times \phantom{\rule{0.25em}{0ex}}{\text{10}}^{-3}\phantom{\rule{0.25em}{0ex}}\text{T}$ magnetic field. The beam moves perpendicular to the field in a path having a 6.80-cm radius of curvature. Determine ${q}_{e}/{m}_{e}$ from these observations, and compare the result with the known value.

**Integrated Concepts**

Find the value of $l$ the orbital angular momentum quantum number, for the moon around 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 ${a}_{\text{B}}=\frac{{h}^{2}}{{\mathrm{4\pi}}^{2}{m}_{e}{\text{kq}}_{e}^{2}}=\text{0.529}\phantom{\rule{0.25em}{0ex}}\times \phantom{\rule{0.25em}{0ex}}{\text{10}}^{-\text{10}}\phantom{\rule{0.25em}{0ex}}\text{m}$ is 207 ${m}_{e}\text{.}$

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

(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 $1\text{.}\text{00}\phantom{\rule{0.25em}{0ex}}\times \phantom{\rule{0.25em}{0ex}}{\text{10}}^{-4}$ moles of neon.

**Integrated Concepts**

A carbon dioxide laser used in surgery emits infrared radiation with a wavelength of $\text{10.6 \mu m.}$ In 1.00 ms, this laser raised the temperature of ${\text{1.00 cm}}^{3}$ of flesh to $\text{100 \xbaC}$ 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 $\pm \text{50}\mathrm{.0\; km/s}\text{.}$ 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 $v=E/B\text{.}$ 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 ${n}_{\text{i}}\text{,}$ 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 $\frac{1}{\lambda}=R\left(\frac{1}{{n}_{\text{f}}^{2}}-\frac{1}{{n}_{\text{i}}^{2}}\right)\text{,}$ but with a different constant $R\text{.}$