Modern Physics

Paul A. Tipler, Ralph A. Llewellyn

Chapter 4

The Nuclear Atom - all with Video Answers

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Chapter Questions

Problem 1

Compute the wavelength and frequency of the series limit for the Lyman, Balmer, and Paschen spectral series of hydrogen.

Narayan Hari

Narayan Hari

Numerade Educator

Problem 2

The wavelength of a particular line in the Balmer series is measured to be $379.1 \mathrm{nm}$. What transition does it correspond to?

Narayan Hari

Numerade Educator

Problem 3

An astronomer finds a new absorption line with $\lambda=164.1 \mathrm{nm}$ in the ultraviolet region of the Sun's continuous spectrum. He attributes the line to hydrogen's Lyman series. Is he right? Justify your answer.

Narayan Hari

Numerade Educator

Problem 4

The series of hydrogen spectral lines with $m=4$ is called Brackett's series. Compute the wavelengths of the first four lines of Brackett's series.

Narayan Hari

Numerade Educator

Problem 5

In a sample that contains hydrogen, among other things, four spectral lines are found in the infrared with wavelengths $7460 \mathrm{nm}, 4654 \mathrm{nm}, 4103 \mathrm{nm}$, and $3741 \mathrm{nm}$. Which one does not belong to a hydrogen spectral series?

Narayan Hari

Numerade Educator

Problem 6

A gold foil of thickness $2.0 \mu \mathrm{m}$ is used in a Rutherford experiment to scatter $\alpha$ particles with energy $7.0 \mathrm{MeV}$. ( $a$ ) What fraction of the particles will be scattered at angles greater than $90^{\circ} ?$ ( $b$ ) What fraction will be scattered at angles between $45^{\circ}$ and $75^{\circ}$ ? (c) Use $N_{A}, \rho,$ and $M$ for gold to compute the approximate radius of a gold atom. (For gold, $\rho=19.3 \mathrm{gm} / \mathrm{cm}^{3}$ and $\left.M=197 \mathrm{gm} / \mathrm{mol} .\right)$

Ronald Prasad

Numerade Educator

Problem 7

( $a$ ) What is the ratio of the number of particles per unit area on the screen scattered at $10^{\circ}$ to those at $1^{\circ} ?(b)$ What is the ratio of those scattered at $30^{\circ}$ to those at $1^{\circ} ?$

Narayan Hari

Numerade Educator

Problem 8

For $\alpha$ particles of $7.7 \mathrm{MeV}$ (those used by Geiger and Marsden), what impact parameter will result in a deflection of $2^{\circ}$ for a thin gold foil?

Narayan Hari

Numerade Educator

Problem 9

What will be the distance of closest approach $r_{d}$ to a gold nucleus for an $\alpha$ particle of $5.0 \mathrm{MeV} ? 7.7 \mathrm{MeV} ? 12 \mathrm{MeV} ?$

Umar Sohail Qureshi

Umar Sohail Qureshi

Numerade Educator

Problem 10

What energy $\alpha$ particle would be needed to just reach the surface of an Al nucleus if its radius is $4 \mathrm{fm} ?$

Eduard Sanchez

Numerade Educator

Problem 11

If a particle is deflected by $0.01^{\circ}$ in each collision, about how many collisions would be necessary to produce an rms deflection of $10^{\circ} ?$ (Use the result from the one-dimensional random walk problem in statistics stating that the rms deflection equals the magnitude of the individual deflections times the square root of the number of deflections.) Compare this result with the number of atomic layers in a gold foil of thickness $10^{-6} \mathrm{~m}$, assuming that the thickness of each atom is $0.1 \mathrm{nm}=10^{-10} \mathrm{~m}$

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 12

Consider the foil and $\alpha$ particle energy in Problem 4 - 6 . Suppose that 1000 of those particles suffer a deflection of more than $25^{\circ} .$ ( $a$ ) How many of these are deflected by more than $45^{\circ} ?(b)$ How many are deflected between $25^{\circ}$ and $45^{\circ} ?(c)$ How many are deflected between $75^{\circ}$ and $90^{\circ} ?$

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 13

The radius of the $n=1$ orbit in the hydrogen atom is $a_{0}=0.053 \mathrm{nm} .$ (a) Compute the radius of the $n=6$ orbit. $(b)$ Compute the radius of the $n=6$ orbit in singly ionized helium $\left(\mathrm{He}^{+}\right)$, which is hydrogenlike, that is, it has only a single electron outside the nucleus.

Narayan Hari

Numerade Educator

Problem 14

Show that Equation $4-19$ for the radius of the first Bohr orbit and Equation $4-20$ for the magnitude of the lowest energy for the hydrogen atom can be written as
$$
a_{0}=\frac{\hbar c}{\alpha m c^{2}}=\frac{\lambda_{c}}{2 \pi \alpha} \quad E_{1}=\frac{1}{2} \alpha^{2} m c^{2}
$$
where $\lambda_{c}=h / m c$ is the Compton wavelength of the electron and $\alpha=k e^{2} / \hbar c$ is the finestructure constant. Use these expressions to check the numerical values of the constants $a_{0}$ and $E_{1}$.

Eduard Sanchez

Numerade Educator

Problem 15

Calculate the three longest wavelengths in the Lyman series $\left(n_{f}=1\right)$ in $\mathrm{nm},$ and indicate their position on a horizontal linear scale. Indicate the series limit (shortest wavelength) on this scale. Are any of these lines in the visible spectrum?

Eduard Sanchez

Numerade Educator

Problem 16

If the angular momentum of Earth in its motion around the Sun were quantized like a hydrogen electron according to Equation $4-17,$ what would Earth's quantum number be? How much energy would be released in a transition to the next lowest level? Would that energy release (presumably as a gravity wave) be detectable? What would be the radius of that orbit? (The radius of Earth's orbit is $\left.1.50 \times 10^{11} \mathrm{~m} .\right)$

Eduard Sanchez

Numerade Educator

Problem 17

On the average, a hydrogen atom will exist in an excited state for about $10^{-8} \mathrm{~s}$ before making a transition to a lower energy state. About how many revolutions does an electron in the $n=2$ state make in $10^{-8} \mathrm{~s} ?$

Narayan Hari

Numerade Educator

Problem 18

An atom in an excited state will on the average undergo a transition to a state of lower energy in about $10^{-8}$ seconds. If the electron in a doubly ionized lithium atom $\left(\mathrm{Li}^{+2},\right.$ which is hydrogenlike) is placed in the $n=4$ state, about how many revolutions around the nucleus does it make before undergoing a transition to a lower energy state?

Eduard Sanchez

Numerade Educator

Problem 19

It is possible for a muon to be captured by a proton to form a muonic atom. A muon is identical to an electron except for its mass, which is $105.7 \mathrm{MeV} / \mathrm{c}^{2}$. ( $a$ ) Calculate the radius of the first Bohr orbit of a muonic atom. (b) Calculate the magnitude of the lowest energy state. ( $c$ ) What is the shortest wavelength in the Lyman series for this atom?

Eduard Sanchez

Numerade Educator

Problem 20

In the lithium atom $(Z=3)$ two electrons are in the $n=1$ orbit and the third is in the $n=2$ orbit. (Only two are allowed in the $n=1$ orbit because of the exclusion principle, which will be discussed in Chapter $7 .$ ) The interaction of the inner electrons with the outer one can be approximated by writing the energy of the outer electron as
$$
E=-Z^{\prime 2} E_{1} / n^{2}
$$
where $E_{1}=13.6 \mathrm{eV}, n=2,$ and $Z^{\prime}$ is the effective nuclear charge, which is less than 3 because of the screening effect of the two inner electrons. Using the measured ionization energy of $5.39 \mathrm{eV},$ calculate $Z^{\prime} .$

Narayan Hari

Numerade Educator

Problem 21

Draw to careful scale an energy-level diagram for hydrogen for levels with $n=1$, $2,3,4, \infty .$ Show the following on the diagram: $(a)$ the limit of the Lyman series, $(b)$ the $\mathrm{H}_{\beta}$ line, $(c)$ the transition between the state whose binding energy $(=$ energy needed to remove the electron from the atom) is $1.51 \mathrm{eV}$ and the state whose excitation energy is $10.2 \mathrm{eV},$ and $(d)$ the longest wavelength line of the Paschen series.

Eduard Sanchez

Numerade Educator

Problem 22

A hydrogen atom at rest in the laboratory emits the Lyman $\alpha$ radiation. ( $a$ ) Compute the recoil kinetic energy of the atom. (b) What fraction of the excitation energy of the $n=2$ state is carried by the recoiling atom? (Hint: Use conservation of momentum.)

Penny Riley

Numerade Educator

Problem 23

(a) Draw accurately to scale and label completely a partial energy-level diagram for $\mathrm{C}^{5+}$ including at minimum the energy levels for $n=1,2,3,4,5,$ and $\infty .(b)$ Compute the wavelength of the spectral line resulting from the $n=3$ to the $n=2$ transition, the $\mathrm{C}^{5+} H_{\alpha}$ line. $(c)$ In what part of the EM spectrum does this line lie?

Ronald Prasad

Numerade Educator

Problem 24

The electron-positron pair that was discussed in Chapter 2 can form a hydrogenlike system called positronium. Calculate $(a)$ the energies of the three lowest states and $(b)$ the wavelength of the Lyman $\alpha$ and $\beta$ lines. (Detection of those lines is a "signature" of positronium formation.)

Eduard Sanchez

Numerade Educator

Problem 25

With the aid of tunable lasers, Rydberg atoms of sodium have been produced with $n \approx 100$. The resulting atomic diameter would correspond in hydrogen to $n \approx 600$. (a) What would be the diameter of a hydrogen atom whose electron is in the $n \approx 600$ orbit? $(b)$ What would be the speed of the electron in that orbit? $(c)$ How does the result in $(b)$ compare with the speed in the $n \approx 1$ orbit?

Christopher Provencher

Christopher Provencher

Numerade Educator

Problem 26

(a) Calculate the next two longest wavelengths in the $K$ series (after the $K_{\alpha}$ line) of molybdenum. (b) What is the wavelength of the shortest wavelength in this series?

Narayan Hari

Numerade Educator

Problem 27

The wavelength of the $K_{\alpha}$ x-ray line for an element is measured to be $0.0794 \mathrm{nm}$. What is the element?

Narayan Hari

Numerade Educator

Problem 28

Moseley pointed out that elements with atomic numbers $43,61,$ and 75 should exist and (at that time) had not been found. ( $a$ ) Using Figure $4-19,$ what frequencies would Moseley's graphical data have predicted for the $K_{\alpha}$ x ray for each of these elements? (b) Compute the wavelengths for these lines predicted by Equation $4-37$.

Narayan Hari

Numerade Educator

Problem 29

What is the approximate radius of the $n=1$ orbit of gold $(Z=79)$ ? Compare this with the radius of the gold nucleus, about $7.1 \mathrm{fm} .$

Urvashi Arora

Numerade Educator

Problem 30

An electron in the $K$ shell of Fe is ejected by a high-energy electron in the target of an x-ray tube. The resulting hole in the $n=1$ shell could be filled by an electron from the $n=2$ shell, the $L$ shell; however, instead of emitting the characteristic Fe $K_{\alpha}$ x ray, the atom ejects an Auger electron from the $n=2$ shell. Using Bohr theory, compute the energy of the Auger electron.

Narayan Hari

Numerade Educator

Problem 31

In a particular x-ray tube, an electron approaches the target moving at $2.25 \times 10^{8} \mathrm{~m} / \mathrm{s}$. It slows down on being deflected by a nucleus of the target, emitting a photon of energy $32.5 \mathrm{keV}$. Ignoring the nuclear recoil, but not relativity, compute the final speed of the electron

Eduard Sanchez

Numerade Educator

Problem 32

( $a$ ) Compute the energy of an electron in the $n=1$ ( $K$ shell) of tungsten, using $Z-1$ for the effective nuclear charge. ( $b$ ) The experimental result for this energy is $69.5 \mathrm{keV}$. Assume that the effective nuclear charge is $Z-\sigma,$ where $\sigma$ is called the screening constant, and calculate $\sigma$ from the experimental result.

Narayan Hari

Numerade Educator

Problem 33

Construct a Moseley plot similar to Figure $4-19$ for the $K_{\beta}$ x rays of the elements listed below (the x-ray energies are given in $\mathrm{keV}$ ):
$$
\begin{array}{|c|c|c|c|}
\hline \mathrm{Al} & \mathrm{Ar} & \mathrm{Sc} & \mathrm{Fe} \\
1.56 & 3.19 & 4.46 & 7.06 \\
\hline \mathrm{Ge} & \mathrm{Kr} & \mathrm{Zr} & \mathrm{Ba} \\
10.98 & 14.10 & 17.66 & 36.35 \\
\hline
\end{array}
$$
Determine the slope of your plot, and compare it with the $K_{\beta}$ line in Figure $4-19$.

Suzanne W.

Numerade Educator

Problem 34

Suppose that, in a Franck-Hertz experiment, electrons of energy up to $13.0 \mathrm{eV}$ can be produced in the tube. If the tube contained atomic hydrogen, $(a)$ what is the shortestwavelength spectral line that could be emitted from the tube? (b) List all of the hydrogen lines that can be emitted by this tube.

Eduard Sanchez

Numerade Educator

Problem 35

Using the data in Figure $4-24 b$ and a good ruler, draw a carefully scaled energylevel diagram covering the range from $0 \mathrm{eV}$ to $60 \mathrm{eV}$ for the vibrational states of this solid. What approximate energy is typical of the transitions between adjacent levels corresponding to the larger of each pair of peaks?

Carlos Henrique De Lima

Carlos Henrique De Lima

Numerade Educator

Problem 36

The transition from the first excited state to the ground state in potassium results in the emission of a photon with $\lambda=770 \mathrm{nm}$. If potassium vapor is used in a Franck-Hertz experiment, at what voltage would you expect to see the first decrease in current?

Narayan Hari

Numerade Educator

Problem 37

If we could somehow fill a Franck-Hertz tube with positronium, what cathode-grid voltage would be needed to reach the second current decrease in the positronium equivalent of Figure $4-23 ?$ (See Problem $4-24 .)$

Narayan Hari

Numerade Educator

Problem 38

Electrons in the Franck-Hertz tube can also have elastic collisions with the Hg atoms. If such a collision is a head-on, what fraction of its initial kinetic energy will an electron lose, assuming the Hg atom to be at rest? If the collision is not head-on, will the fractional loss be greater or less than this?

Eduard Sanchez

Numerade Educator

Problem 39

A Rydberg hydrogen atom is in the $n=45$ energy state. ( $a$ ) What is the energy difference (in eV) between this state and the $n=46$ level? (b) What is the ionization energy of the atom in the $n=45$ level? $(c)$ What are the frequency and wavelength of a photon emitted in the $n=46 \rightarrow n=45$ transition? $(d)$ What is the radius of the atom in the $n=45$ level? How does this compare with the Bohr radius?

Eduard Sanchez

Numerade Educator

Problem 40

Three isotopes of hydrogen occur in nature; ordinary hydrogen, deuterium, and tritium. Their nuclei consist of, respectively, 1 proton, 1 proton and 1 neutron (deuteron), and 1 proton and 2 neutrons (triton). The masses of the three nuclei are given in Table $11-1$. (a) Use Equation $4-26$ to determine Rydberg constants for deuterium and tritium. (b) Determine the wavelength difference between the Balmer $\alpha$ lines of deuterium and tritium. (c) Determine the wavelength difference between the Balmer $\alpha$ lines of hydrogen and tritium.

Anand Jangid

Numerade Educator

Problem 41

Derive Equation $4-8$ along the lines indicated in the paragraph that immediately precedes it.

Eduard Sanchez

Numerade Educator

Problem 42

Geiger and Marsden used $\alpha$ particles with $7.7 \mathrm{MeV}$ kinetic energy and found that when they were scattered from thin gold foil, the number observed to be scattered at all angles agreed with Rutherford's formula. Use this fact to compute an upper limit on the radius of the gold nucleus.

Narayan Hari

Numerade Educator

Problem 43

( $a$ ) The current $i$ due to a charge $q$ moving in a circle with frequency $f_{\text {rev }}$ is $q f_{\text {rev }}$. Find the current due to the electron in the first Bohr orbit. ( $b$ ) The magnetic moment of a current loop is $i A,$ where $A$ is the area of the loop. Find the magnetic moment of the electron in the first Bohr orbit in units $\mathrm{A}-\mathrm{m}^{2}$. This magnetic moment is called a $\mathrm{Bohr}$ magneton.

Narayan Hari

Numerade Educator

Problem 44

Use a spreadsheet to calculate the wavelengths (in $\mathrm{nm}$ ) of the first five spectral lines of the Lyman, Balmer, Paschen, and Brackett series of hydrogen. Show the positions of these lines on a linear scale and indicate which ones lie in the visible.

Mayank Tripathi

Mayank Tripathi

Numerade Educator

Problem 45

Show that a small change in the reduced mass of the electron produces a small change in a spectral line given by $\Delta \lambda / \lambda=\Delta \mu / \mu .$ Use this to calculate the difference $\Delta \lambda$ in the Balmer red line $\lambda=656.3 \mathrm{nm}$ between hydrogen and deuterium, which has a nucleus with twice the mass of hydrogen.

Manish Kumar

Numerade Educator

Problem 46

Consider the Franck-Hertz experiment with Hg vapor in the tube and the voltage between the cathode and the grid equal to $4.0 \mathrm{~V},$ that is, not enough to for the electrons to excite the Hg atom's first excited state. Therefore, the electron-Hg atom collisions are elastic. ( $a$ ) If the kinetic energy of the electrons is $E_{k}$, show that the maximum kinetic energy that a recoiling Hg atom can have is approximately $4 m E_{k} / M,$ where $M$ is the $\mathrm{Hg}$ atom mass. (b) What is the approximate maximum kinetic energy that can be lost by an electron with $E_{k}=2.5 \mathrm{eV} ?$

Chai Santi

Numerade Educator

Problem 47

The $\mathrm{Li}^{2+}$ ion is essentially identical to the $\mathrm{H}$ atom in Bohr's theory, aside from the effect of the different nuclear charges and masses. ( $a$ ) What transitions in $\mathrm{Li}^{2+}$ will yield emission lines whose wavelengths are very nearly equal to the first two lines of the Lyman series in hydrogen? (b) Calculate the difference between the wavelength of the Lyman $\alpha$ line of hydrogen and the emission line from $\mathrm{Li}^{2+}$ that has very nearly the same wavelength.

Eduard Sanchez

Numerade Educator

Problem 48

In an $\alpha$ scattering experiment, the area of the $\alpha$ particle detector is $0.50 \mathrm{~cm}^{2}$. The detector is located $10 \mathrm{~cm}$ from a 1.0 - $\mu$ m-thick silver foil. The incident beam carries a current of $1.0 \mathrm{nA}$, and the energy of each $\alpha$ particle is $6.0 \mathrm{MeV}$. How many $\alpha$ particles will be counted per second by the detector at $(a) \theta=60^{\circ} ?(b) \theta=120^{\circ} ?$

Narayan Hari

Numerade Educator

Problem 49

The $K_{\alpha}, L_{\alpha}$, and $M_{\alpha}$ x rays are emitted in the $n=2 \rightarrow n=1, n=3 \rightarrow n=2$, and $n=4 \rightarrow n=3$ transitions respectively. For calcium $(Z=20)$ the energies of these transitions are $3.69 \mathrm{keV}, 0.341 \mathrm{keV},$ and $0.024 \mathrm{keV},$ respectively. Suppose that energetic photons impinging on a calcium surface cause ejection of an electron from the $K$ shell of the surface atoms. Compute the energies of the Auger electrons that may be emitted from the $L, M,$ and $N$ shells $(n=2,3,$ and 4$)$ of the sample atoms, in addition to the characteristic X rays.

Pritesh Ranjan

Numerade Educator

Problem 50

Figure $3-15 b$ shows the $K_{\alpha}$ and $K_{\beta}$ characteristic $x$ rays emitted by a molybdenum
(Mo) target in an x-ray tube whose accelerating potential is $35 \mathrm{kV}$. The wavelengths are $K_{\alpha}=0.071 \mathrm{nm}$ and $K_{\beta}=0.063 \mathrm{nm} .(a)$ Compute the corresponding energies of these photons. (b) Suppose we wish to prepare a beam consisting primarily of $K_{\alpha}$ x rays by passing the molybdenum $x$ rays through a material that absorbs $K_{\beta}$ x rays more strongly than $K_{\alpha}$ x rays by photoelectric effect on $K$ -shell electrons of the material. Which of the materials listed in the accompanying table with their $K$ -shell binding energies would you choose? Explain your answer.
$$
\begin{array}{llllll}
\hline \text { Element } & \text { Zr } & \text { Nb } & \text { Mo } & \text { Tc } & \text { Ru } \\
\hline Z & 40 & 41 & 42 & 43 & 44 \\
E_{K}(\mathrm{keV}) & 18.00 & 18.99 & 20.00 & 21.04 & 22.12 \\
\hline
\end{array}
$$

Keshav Singh

Numerade Educator

Problem 51

A small shot of negligible radius hits a stationary smooth, hard sphere of radius $R,$ making an angle $\beta$ with the normal to the sphere, as shown in Figure $4-25 .$ It is reflected at an equal angle to the normal. The scattering angle is $\theta=180^{\circ}-2 \beta,$ as shown. (a) Show by the geometry of the figure that the impact parameter $b$ is related to $\theta$ by $b=R \cos \frac{1}{2} \theta .(b)$ If the incoming intensity of the shot is $I_{0}$ particles $/ \mathrm{s} \cdot$ area how many are scattered through angles greater than $\theta ?(c)$ Show that the cross section for scattering through angles greater than $0^{\circ}$ is $\pi R^{2} .(d)$ Discuss the implication of the fact that the Rutherford cross section for scattering through angles greater than $0^{\circ}$ is infinite.

Zachary Warner

Numerade Educator

Problem 52

Singly ionized helium $\mathrm{He}^{+}$ is hydrogenlike. $(a)$ Construct a carefully scaled energylevel diagram for $\mathrm{He}^{+}$ similar to that in Figure $4-16$, showing the levels for $n=1,2,3$, $4,5,$ and $\infty$. ( $b$ ) What is the ionization energy of $\mathrm{He}^{+}$ ? ( $c$ ) Compute the difference in wavelength between each of the first two lines of the Lyman series of hydrogen and the first two lines of the $\mathrm{He}^{+}$ Balmer series. Be sure to include the reduced mass correction for both atoms. ( $d$ ) Show that for every spectral line of hydrogen, $\mathrm{He}^{+}$ has a spectral line of very nearly the same wavelength. (Mass of $\mathrm{He}^{+}=6.65 \times 10^{-27} \mathrm{~kg} .$

Stanley Enemuo

Numerade Educator

Problem 53

Listed in the table are the $L_{\alpha}$ x-ray wavelengths for several elements. Construct a Moseley plot from these data. Compare the slope with the appropriate one in Figure $4-19 .$ Determine and interpret the intercept on your graph, using a suitably modified version of Equation $4-35$.
$$
\begin{array}{lcccccc}
\hline \text { Element } & \text { P } & \text { Ca } & \text { Co } & \text { Kr } & \text { Mo } & \text { I } \\
\hline Z & 15 & 20 & 27 & 36 & 42 & 53 \\
\text { Wavelength (nm) } & 10.41 & 4.05 & 1.79 & 0.73 & 0.51 & 0.33 \\
\hline
\end{array}
$$

Salamat Ali

Numerade Educator

Problem 54

In this problem you are to obtain the Bohr results for the energy levels in hydrogen without using the quantization condition of Equation $4-17$. In order to relate Equation $4-14$ to the Balmer-Ritz formula, assume that the radii of allowed orbits are given by $r_{n}=n^{2} r_{0}$, where $n$ is an integer and $r_{0}$ is a constant to be determined. ( $a$ ) Show that the frequency of radiation for a transition to $n_{f}=n-1$ is given by $f \approx k Z e^{2} / h r_{0} n^{3}$ for large $n$
(b) Show that the frequency of revolution is given by
$$
f_{\mathrm{rev}}^{2}=\frac{k Z e^{2}}{4 \pi^{2} m r_{0}^{3} n^{6}}
$$
(c) Use the correspondence principle to determine $r_{0}$ and compare with Equation $4-19 .$

Mayukh Banik

Numerade Educator

Problem 55

Calculate the energies and speeds of electrons in circular Bohr orbits in a hydrogenlike atom using the relativistic expressions for kinetic energy and momentum.

Eduard Sanchez

Numerade Educator

Problem 56

( $a$ ) Write a computer program for your personal computer or programmable calculator that will provide you with the spectral series of H-like atoms. Inputs to be included are $n_{i}, n_{f}, Z,$ and the nuclear mass $M$. Outputs are to be the wavelengths and frequencies of the first six lines and the series limit for the specified $n_{f}, Z,$ and $M .$ Include the reduced mass correction. (b) Use the program to compute the wavelengths and frequencies of the Balmer series. (c) Pick an $n_{f}>100,$ name the series the [your name] series, and use your program to compute the wavelengths and frequencies of the first three lines and the limit.

Narayan Hari

Numerade Educator

Problem 57

Figure $4-26$ shows an energy loss spectrum for He measured in an apparatus such as that shown in Figure $4-24 a$. Use the spectrum to construct and draw carefully to scale an energy-level diagram for He.

Crystal Wang

Numerade Educator

Problem 58

If electric charge did not exist and electrons were bound to protons by the gravitational force to form hydrogen, derive the corresponding expressions for $a_{0}$ and $E_{n}$ and compute the energy and frequency of the $H_{\alpha}$ line and the limit of the Balmer series. Compare these with the corresponding quantities for "real" hydrogen.

Penny Riley

Numerade Educator

Problem 59

A sample of hydrogen atoms are all in the $n=5$ state. If all the atoms return to the ground state, how many different photon energies will be emitted, assuming all possible transitions occur? If there are 500 atoms in the sample and assuming that from any state all possible downward transitions are equally probable, what is the total number of photons that will be emitted when all of the atoms have returned to the ground state?

Sheh Lit Chang

University of Washington

Problem 60

Consider muonic atoms (see Problem $4-19$ ). ( $a$ ) Draw a correctly scaled and labeled partial energy level diagram including levels with $n=1,2,3,4,5,$ and $\infty$ for muonic hydrogen. ( $b$ ) Compute the radius of the $n=1$ muon orbit in muonic $\mathrm{H}, \mathrm{He}^{1+}, \mathrm{Al}^{12+}$, and $\mathrm{Au}^{78+}$. (c) Compare the results in (b) with the radii of these nuclei. ( $d$ ) Compute the wavelength of the photon emitted in the $n=2$ to $n=1$ transition for each of these muonic atoms.

Raj Bala

Numerade Educator