Principles of Physics

David Halliday , Robert Resnick , Jearl Walker

Chapter 40

All About Atoms - all with Video Answers

Educators

Chapter Questions

Problem 1

Seven electrons are trapped in a one-dimensional infinite potential well of width $L$. What multiple of $h^{2} / 8 m L^{2}$ gives the energy of the ground state of this system? Assume that the electrons do not interact with one another, and do not neglect spin.

Alia Hamdan

Alia Hamdan

Numerade Educator

Problem 2

Show that the number of states with the same quantum number $n$ is $2 n^{2}$.

Mayank Tripathi

Mayank Tripathi

Numerade Educator

Problem 3

For the situation of Problem 1, what multiple of $h^{2} / 8 m L^{2}$ gives the energy of (a) the first excited state, (b) the second excited state, and (c) the third excited state of the system of seven electrons?
(d) Construct an energy-level diagram for the lowest four energy levels of the system.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 4

Here are the $K_{\alpha}$ wavelengths of a few elements:
\begin{tabular}{lccc}
\hline Element & $\lambda(\mathrm{pm})$ & Element & $\lambda(\mathrm{pm})$ \\
\hline $\mathrm{Ti}$ & 275 & $\mathrm{Co}$ & 179 \\
$\mathrm{~V}$ & 250 & $\mathrm{Ni}$ & 166 \\
$\mathrm{Cr}$ & 229 & $\mathrm{Cu}$ & 154 \\
$\mathrm{Mn}$ & 210 & $\mathrm{Zn}$ & 143 \\
$\mathrm{Fe}$ & 193 & $\mathrm{Ga}$ & 134 \\
\hline
\end{tabular}
Make a Moseley plot (like that in Fig. 40-16) from these data and verify that its slope agrees with the value given for $C$ in Module $40-6$.

Salamat Ali

Numerade Educator

Problem 5

The binding energies of $K$-shell and $L$-shell electrons in copper are $8.979$ and $0.951 \mathrm{keV}$, respectively. If a $K_{\alpha}$ x ray from copper is incident on a sodium chloride crystal and gives a first-order Bragg reflection at an angle of $74.1^{\circ}$ measured relative to parallel planes of sodium atoms, what is the spacing between these parallel planes?

Alia Hamdan

Numerade Educator

Problem 6

A hypothetical atom has only two atomic energy levels,
separated by 3.2 eV. Suppose that at a certain altitude in the atmosphere of a star there are $6.1 \times 10^{13} / \mathrm{cm}^{3}$ of these atoms in the higher-energy state and $2.2 \times 10^{15} / \mathrm{cm}^{3}$ in the lower-energy state. What is the temperature of the star's atmosphere at that altitude?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 7

In an NMR experiment, the RF source oscillates at $34 \mathrm{MHz}$ and magnetic resonance of the hydrogen atoms in the sample being investigated occurs when the external field $\vec{B}_{\text {ext }}$ has magnitude $0.78 \mathrm{~T}$. Assume that $\vec{B}_{\text {int }}$ and $\vec{B}_{\text {ext }}$ are in the same direction and take the proton magnetic moment component $\mu_{z}$ to be $1.41 \times 10^{-26} \mathrm{~J} / \mathrm{T}$. What is the magnitude of $\vec{B}_{\mathrm{int}}$ ?

Alia Hamdan

Numerade Educator

Problem 8

Suppose two electrons in an atom have quantum numbers $n=2$ and $\ell=1$. (a) How many states are possible for those two electrons? (Keep in mind that the electrons are indistinguishable.) (b) If the Pauli exclusion principle did not apply to the electrons, how many states would be possible?

Salamat Ali

Numerade Educator

Problem 9

A tungsten $(Z=74)$ target is bombarded by electrons in an x-ray tube. The $K, L$, and $M$ energy levels for tungsten (compare Fig. 40-15) have the energies $69.5,11.3$, and $2.30 \mathrm{keV}$, respectively. (a) What is the minimum value of the accelerating potential that will permit the production of the characteristic $K_{\alpha}$ and $K_{\beta}$ lines of tungsten? (b) For this same accelerating potential, what is $\lambda_{\min }$ ? What are the (c) $K_{\alpha}$ and (d) $K_{\beta}$ wavelengths?

Eduard Sanchez

Numerade Educator

Problem 10

A high-powered laser beam $(\lambda=600 \mathrm{~nm})$ with a beam diameter of $10 \mathrm{~cm}$ is aimed at the Moon, $3.8 \times 10^{5} \mathrm{~km}$ distant. The beam spreads only because of diffraction. The angular location of the edge of the central diffraction disk (see Eq. 36-12) is given by
$$
\sin \theta=\frac{1.22 \lambda}{d}
$$
where $d$ is the diameter of the beam aperture. What is the diameter of the central diffraction disk on the Moon's surface?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 11

Consider the elements selenium $(Z=34)$, bromine $(Z=35)$, and krypton $(Z=36) .$ In their part of the periodic table, the subshells of the electronic states are filled in the sequence
$$
1 s 2 s 2 p 3 s 3 p 3 d 4 s 4 p \ldots .
$$
What are (a) the highest occupied subshell for selenium and (b) the number of electrons in it, (c) the highest occupied subshell for bromine and (d) the number of electrons in it, and (e) the highest occupied subshell for krypton and (f) the number of electrons in it?

Alia Hamdan

Numerade Educator

Problem 12

Determine the constant $C$ in Eq. 40-27 to five significant figures by finding $C$ in terms of the fundamental constants in Eq. 40-24 and then using data from Appendix B to evaluate those constants. Using this value of $C$ in Eq. $40-27$, determine the theoretical energy $E_{\text {theory }}$ of the $K_{\alpha}$ photon for the low-mass elements listed in the following table. The table includes the value (eV) of the measured energy $E_{\exp }$ of the $K_{\alpha}$ photon for each listed element. The percentage deviation between $E_{\text {theory }}$ and $E_{\text {exp }}$ can be calculated as
percentage deviation $=\frac{E_{\text {theory }}-E_{\exp }}{E_{\exp }} 100 .$
What is the percentage deviation for (a) Li, (b) Be, (c) B, (d) C, (e) $\mathrm{N}$, (f) $\mathrm{O}$, (g) $\mathrm{F}$, (h) $\mathrm{Ne}$, (i) $\mathrm{Na}$, and (j) $\mathrm{Mg}$ ?
\begin{tabular}{lrlc}
\hline $\mathrm{Li}$ & $54.3$ & $\mathrm{O}$ & $524.9$ \\
$\mathrm{Be}$ & $108.5$ & $\mathrm{~F}$ & $676.8$ \\
$\mathrm{~B}$ & $183.3$ & $\mathrm{Ne}$ & $848.6$ \\
$\mathrm{C}$ & 277 & $\mathrm{Na}$ & 1041 \\
$\mathrm{~N}$ & $392.4$ & $\mathrm{Mg}$ & 1254 \\
\hline
\end{tabular}
(There is actually more than one $K_{\alpha}$ ray because of the splitting of the $L$ energy level, but that effect is negligible for the elements listed here.)

Keshav Singh

Numerade Educator

Problem 13

In Fig. $40-13$, the $x$ rays shown are produced when $35.0 \mathrm{keV}$ electrons strike a molybdenum $(Z=42)$ target. If the accelerating potential is maintained at this value but a silver $(Z=47)$ target is used instead, what values of (a) $\lambda_{\min }$, (b) the wavelength of the $K_{\alpha}$ line, and (c) the wavelength of the $K_{\beta}$ line result? The $K, L$, and $M$ atomic x-ray levels for silver (compare Fig. 40-15) are $25.51,3.56$, and $0.53 \mathrm{keV}$.

Neelesh Sharma

Numerade Educator

Problem 14

(a) From Eq. 40-26, what is the ratio of the photon energies due to $K_{\alpha}$ transitions in two atoms whose atomic numbers are $Z$ and $Z^{\prime} ?$ (b) What is this ratio for uranium and aluminum? (c) For uranium and lithium?

Salamat Ali

Numerade Educator

Problem 15

Show that a moving electron cannot spontaneously change into an x-ray photon in free space. A third body (atom or nucleus) must be present. Why is it needed? (Hint: Examine the conservation of energy and momentum.)

Alia Hamdan

Numerade Educator

Problem 16

A population inversion for two energy levels is often described by assigning a negative Kelvin temperature to the system. What negative temperature would describe a system in which the population of the upper energy level exceeds that of the lower level by $10 \%$ and the energy difference between the two levels is $2.32 \mathrm{eV} ?$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 17

Two of the three electrons in a lithium atom have quantum numbers $\left(n, \ell, m_{\ell}, m_{s}\right)$ of $\left(1,0,0,+\frac{1}{2}\right)$ and $\left(1,0,0,-\frac{1}{2}\right) .$ What quantum numbers are possible for the third electron if the atom is (a) in the ground state and (b) in the first excited state?

Keshav Singh

Numerade Educator

Problem 18

Figure 40-22 is an energy-level dia-
gram for a fictitious infinite potential well
that contains one electron. The number of
degenerate states of the levels are indi-
cated: "non" means nondegenerate
(which includes the ground state of the
electron), "double" means 2 states, and
"triple" means 3 states. We put a total of
11 electrons in the well. If the electro-
static forces between the electrons can be
neglected, what multiple of $h^{2} / 8 m L^{2}$ gives
the energy of the first excited state of the
11-electron system?

Salamat Ali

Numerade Educator

Problem 19

A pulsed laser emits light at a wavelength of $694.4 \mathrm{~nm}$. The pulse duration is $12 \mathrm{ps}$, and the energy per pulse is $0.150 \mathrm{~J}$. (a) What is the length of the pulse? (b) How many photons are emitted in each pulse?

Alia Hamdan

Numerade Educator

Problem 20

From Fig. 40-13, calculate approximately the energy difference $E_{L}-E_{M}$ for molybdenum. Compare it with the value that may be obtained from Fig. $40-15$.

Eduard Sanchez

Numerade Educator

Problem 21

A helium-neon laser emits laser light at a wavelength of $632.8 \mathrm{~nm}$ and a power of $2.3 \mathrm{~mW}$. At what rate are photons emitted by this device?

Alia Hamdan

Numerade Educator

Problem 22

For a helium atom in its ground state, what are quantum numbers $\left(n, \ell, m_{\ell}\right.$, and $m_{s}$ ) for the (a) spin-up electron and (b) spin-down electron?

Eduard Sanchez

Numerade Educator

Problem 23

A hypothetical atom has two energy levels, with a transition wavelength between them of $580 \mathrm{~nm}$. In a particular sample at $300 \mathrm{~K}$, $4.0 \times 10^{20}$ such atoms are in the state of lower energy. (a) How many atoms are in the upper state, assuming conditions of thermal equilibrium? (b) Suppose, instead, that $3.0 \times 10^{20}$ of these atoms are "pumped" into the upper state by an external process, with $1.0 \times 10^{20}$ atoms remaining in the lower state. What is the maximum energy that could be released by the atoms in a single laser pulse if each atom jumps once between those two states (either via absorption or via stimulated emission)?

Alia Hamdan

Numerade Educator

Problem 24

A $15 \mathrm{keV}$ electron is brought to rest by colliding twice with target nuclei as in Fig. 40-14. (Assume the nuclei remain stationary.) The wavelength associated with the photon emitted in the second collision is $130 \mathrm{pm}$ greater than that associated with the photon emitted in the first collision. (a) What is the kinetic energy of the electron after the first collision? What are (b) the wavelength $\lambda_{1}$ and (c) the energy $E_{1}$ associated with the first photon? What are (d) $\lambda_{2}$ and (e) $E_{2}$ associated with the second photon?

Salamat Ali

Numerade Educator

Problem 25

If orbital angular momentum $\vec{L}$ is measured along, say, a $z$ axis to obtain a value for $L_{z}$, show that
$$
\left(L_{x}^{2}+L_{y}^{2}\right)^{1 / 2}=\left[\ell(\ell+1)-m_{\ell}^{2}\right]^{1 / 2} \hbar
$$
is the most that can be said about the other two components of the orbital angular momentum.

Alia Hamdan

Numerade Educator

Problem 26

A laser emits at $424 \mathrm{~nm}$ in a single pulse that lasts $0.500 \mu \mathrm{s}$. The power of the pulse is $3.25 \mathrm{MW}$. If we assume that the atoms contributing to the pulse underwent stimulated emission only once during the $0.500 \mu$ s, how many atoms contributed?

Salamat Ali

Numerade Educator

Problem 27

X rays are produced in an x-ray tube by electrons accelerated through an electric potential difference of $50.0 \mathrm{kV}$. Let $K_{0}$ be the kinetic energy of an electron at the end of the acceleration. The electron collides with a target nucleus (assume the nucleus remains stationary) and then has kinetic energy $K_{1}=0.500 K_{0}$. (a) What wavelength is associated with the photon that is emitted? The electron collides with another target nucleus (assume it, too, remains stationary) and then has kinetic energy $K_{2}=0.500 K_{1} .$ (b) What wavelength is associated with the photon that is emitted?

Alia Hamdan

Numerade Educator

Problem 28

The beam from an argon laser (of wavelength $515 \mathrm{~nm}$ ) has a diameter $d$ of $3.00 \mathrm{~mm}$ and a continuous energy output rate of $4.00 \mathrm{~W}$. The beam is focused onto a diffuse surface by a lens whose focal length $f$ is $3.50 \mathrm{~cm}$. A diffraction pattern such as that of Fig. $36-10$ is formed, the radius of the central disk being given by
$$
R=\frac{1.22 f \lambda}{d}
$$
(see Eq. 36-12 and Fig. 36-14). The central disk can be shown to contain $84 \%$ of the incident power. (a) What is the radius of the central disk? (b) What is the average intensity (power per unit area) in the incident beam? (c) What is the average intensity in the central disk?

Salamat Ali

Numerade Educator

Problem 29

Assume that lasers are available whose wavelengths can be precisely "tuned" to anywhere in the visible range-that is, in the range $450 \mathrm{~nm}<\lambda<650 \mathrm{~nm}$. If every television channel occupies a bandwidth of $10 \mathrm{MHz}$, how many channels can be accommodated within this wavelength range?

Alia Hamdan

Numerade Educator

Problem 30

A certain gas laser can emit light at wavelength $550 \mathrm{~nm}$, which involves population inversion between ground state and an excited state. At room temperature, how many moles of neon are needed to put 12 atoms in that excited state by thermal agitation?

Salamat Ali

Numerade Educator

Problem 31

What is the wavelength associated with a photon that will induce a transition of an electron spin from parallel to antiparallel orientation in a magnetic field of magnitude $0.200 \mathrm{~T}$ ? Assume that $\ell=0$.

Alia Hamdan

Numerade Educator

Problem 32

How many electron states are there in the following shells:
(a) $n=4$, (b) $n=1$, (c) $n=3$, (d) $n=2 ?$

Salamat Ali

Numerade Educator

Problem 33

The active volume of a laser constructed of the semiconductor $\mathrm{GaAlAs}$ is only $200 \mu \mathrm{m}^{3}$ (smaller than a grain of sand), and yet the laser can continuously deliver $5.0 \mathrm{~mW}$ of power at a wavelength of $0.80 \mu \mathrm{m}$. At what rate does it generate photons?

Alia Hamdan

Numerade Educator

Problem 34

In the subshell $\ell=4$, (a) what is the greatest (most positive) $m_{\ell}$ value, (b) how many states are available with the greatest $m_{\ell}$ value, and (c) what is the total number of states available in the subshell?

Salamat Ali

Numerade Educator

Problem 35

Calculate the (a) smaller and (b) larger value of the semiclassical angle between the electron spin angular momentum vector and the magnetic field in a Stern-Gerlach experiment. Bear in mind that the orbital angular momentum of the valence electron in the silver atom is zero.

Alia Hamdan

Numerade Educator

Problem 36

The wavelength of the $K_{\alpha}$ line from iron is $193 \mathrm{pm}$. What is the energy difference between the two states of the iron atom that give rise to this transition?

Salamat Ali

Numerade Educator

Problem 37

An electron in a multielectron atom has $m_{\ell}=+4$. For this electron, what are (a) the value of $\ell$, (b) the smallest possible value of $n$, and (c) the number of possible values of $m_{s}$ ?

Alia Hamdan

Numerade Educator

Problem 38

The mirrors in the laser of Fig. $40-20$, which are separated by $6.0 \mathrm{~cm}$, form an optical cavity in which standing waves of laser light can be set up. Each standing wave has an integral number $n$ of half wavelengths in the $6.0 \mathrm{~cm}$ length, where $n$ is large and the waves differ slightly in wavelength. Near $\lambda=533 \mathrm{~nm}$, how far apart in wavelength are the standing waves?

Salamat Ali

Numerade Educator

Problem 39

A hypothetical atom has energy levels uniformly separated by $1.2 \mathrm{eV}$. At a temperature of $2000 \mathrm{~K}$, what is the ratio of the number of atoms in the 13 th excited state to the number in the 11 th excited state?

Alia Hamdan

Numerade Educator

Problem 40

Ruby lases at a wavelength of $694 \mathrm{~nm}$. A certain ruby crystal has $4.00 \times 10^{19} \mathrm{Cr}$ ions (which are the atoms that lase). The lasing transition is between the first excited state and the ground state, and the output is a light pulse lasting $1.50 \mu \mathrm{s}$. As the pulse begins, $60.0 \%$ of the Cr ions are in the first excited state and the rest are in the ground state. What is the average power emitted during the pulse?

Eduard Sanchez

Numerade Educator

Problem 41

Figure $40-23$ shows the energy levels of two types of atoms. Atoms $A$ are in one tube, and atoms $B$ are in another tube. The energies (relative to a ground-state energy of zero) are indicated; the average lifetime of atoms in each level is also indicated. All the atoms are initially pumped to levels higher than the levels shown in the figure. The atoms then drop down through the levels, and many become "stuck" on certain levels, leading to population inversion and lasing. The light emitted by $A$ illuminates $B$ and can cause stimulated emission of $B$. What is the energy per photon of that stimulated emission of $B ?$

Alia Hamdan

Numerade Educator

Problem 42

How many electron states are there in a shell defined by the quantum number $n=6$ ?

Salamat Ali

Numerade Educator

Problem 43

What is the acceleration of a silver atom as it passes through the deflecting magnet in the Stern-Gerlach experiment of Fig. $40-8$ if the magnetic field gradient is $1.4 \mathrm{~T} / \mathrm{mm}$ ?

Eduard Sanchez

Numerade Educator

Problem 44

A rectangular corral of widths $L_{x}=L$ and $L_{y}=2 L$ contains seven electrons. What multiple of $h^{2} / 8 m L^{2}$ gives the energy of the ground state of this system? Assume that the electrons do not interact with one another, and do not neglect spin.

Salamat Ali

Numerade Educator

Problem 45

Through what minimum potential difference must an electron in an $\mathrm{x}$-ray tube be accelerated so that it can produce $\mathrm{x}$ rays with a wavelength of $0.100 \mathrm{~nm}$ ?

Alia Hamdan

Numerade Educator

Problem 46

For Problem 44 , what multiple of $h^{2} / 8 m L^{2}$ gives the energy of (a) the first excited state, (b) the second excited state, and (c) the third excited state of the system of seven electrons?
(d) Construct an energy-level diagram for the lowest four energy levels.

Salamat Ali

Numerade Educator

Problem 47

Calculate the ratio of the wavelength of the $K_{\alpha}$ line for niobium (Nb) to that for gallium (Ga). Take needed data from the periodic table of Appendix G.

Alia Hamdan

Numerade Educator

Problem 48

A hydrogen atom in its ground state actually has two possible, closely spaced energy levels because the electron is in the magnetic field $\vec{B}$ of the proton (the nucleus). Accordingly, an energy is associated with the orientation of the electron's magnetic moment $\vec{\mu}$ relative to $\vec{B}$, and the electron is said to be either spin up (higher energy) or spin down (lower energy) in that field. If the electron is excited to the higher-energy level, it can de-excite by spin-flipping and emitting a photon. The wavelength associated with that photon is $21 \mathrm{~cm}$. (Such a process occurs extensively in the Milky Way galaxy, and reception of the $21 \mathrm{~cm}$ radiation by radio telescopes reveals where hydrogen gas lies between stars.) What is the effective magnitude of $\vec{B}$ as experienced by the electron in the ground-state hydrogen atom?

Salamat Ali

Numerade Educator

Problem 49

An electron is in a state with $\ell=3 .$ (a) What multiple of $\hbar$ gives the magnitude of $\vec{L}$ ? (b) What multiple of $\mu_{\mathrm{B}}$ gives the magnitude of $\vec{\mu} ?$ (c) What is the largest possible value of $m_{\ell}$, (d) what multiple of $\hbar$ gives the corresponding value of $L_{z}$, and (e) what multiple of $\mu_{\mathrm{B}}$ gives the corresponding value of $\mu_{\mathrm{orb}, z}$ ? (f) What is the value of the semiclassical angle $\theta$ between the directions of $L_{z}$ and $\vec{L}$ ? What is the value of angle $\theta$ for $(\mathrm{g})$ the second largest possible value of $m_{\ell}$ and $(\mathrm{h})$ the smallest (that is, most negative) possible value of $m_{\ell}$ ?

Keshav Singh

Numerade Educator

Problem 50

When electrons bombard a molybdenum target, they produce both continuous and characteristic x rays as shown in Fig. 40-13. In that figure the kinetic energy of the incident electrons is $35.0 \mathrm{keV}$. If the accelerating potential is increased to $55.0 \mathrm{keV}$, (a) what is the value of $\lambda_{\min }$, and (b) do the wavelengths of the $K_{\alpha}$ and $K_{\beta}$ lines increase, decrease, or remain the same?

Salamat Ali

Numerade Educator

Problem 51

An electron in a hydrogen atom is in a state with $\ell=5$. What is the minimum possible value of the semiclassical angle between $\vec{L}$ and $L_{z}$ ?

Alia Hamdan

Numerade Educator

Problem 52

An electron is in a state with $n=4$. What are (a) the number of possible values of $\ell$, (b) the number of possible values of $m_{\ell}$, (c) the number of possible values of $m_{s}$, (d) the number of states in the $n=4$ shell, and (e) the number of subshells in the $n=4$ shell?

Salamat Ali

Numerade Educator

Problem 53

A cubical box of widths $L_{x}=L_{y}=L_{z}=L$ contains eight electrons. What multiple of $h^{2} / 8 m L^{2}$ gives the energy of the ground state of this system? Assume that the electrons do not interact with one another, and do not neglect spin.

Alia Hamdan

Numerade Educator

Problem 54

For the situation of Problem 53, what multiple of $h^{2} / 8 m L^{2}$ gives the energy of (a) the first excited state, (b) the second excited state, and (c) the third excited state of the system of eight electrons? (d) Construct an energy-level diagram for the lowest four energy levels of the system.

Salamat Ali

Numerade Educator

Problem 55

(a) How many $\ell$ values are associated with $n=3 ?$ (b) How many $m_{\ell}$ values are associated with $\ell=1 ?$

Mayank Tripathi

Mayank Tripathi

Numerade Educator

Problem 56

Assume that in the Stern-Gerlach experiment as described for neutral silver atoms, the magnetic field $\vec{B}$ has a magnitude of $0.60 \mathrm{~T}$.
(a) What is the energy difference between the magnetic moment orientations of the silver atoms in the two subbeams? (b) What is the frequency of the radiation that would induce a transition between these two states? (c) What is the wavelength of this radiation, and (d) to what part of the electromagnetic spectrum does it belong?

Salamat Ali

Numerade Educator

Problem 57

The active medium in a particular laser that generates laser light at a wavelength of $694 \mathrm{~nm}$ is $6.00 \mathrm{~cm}$ long and $1.00 \mathrm{~cm}$ in diameter. (a) Treat the medium as an optical resonance cavity analogous to a closed organ pipe. How many standing-wave nodes are there along the laser axis? (b) By what amount $\Delta f$ would the beam frequency have to shift to increase this number by one? (c) Show that $\Delta f$ is just the inverse of the travel time of laser light for one round trip back and forth along the laser axis. (d) What is the corresponding fractional frequency shift $\Delta f / f ?$ The appropriate index of refraction of the lasing medium (a ruby crystal) is $1.75$.

Alia Hamdan

Numerade Educator

Problem 58

A magnetic field is applied to a freely floating uniform iron sphere with radius $R=1.50 \mathrm{~mm}$. The sphere initially had no net magnetic moment, but the field aligns $12 \%$ of the magnetic moments of the atoms (that is, $12 \%$ of the magnetic moments of the loosely bound electrons in the sphere, with one such electron per atom). The magnetic moment of those aligned electrons is the sphere's intrinsic magnetic moment $\vec{\mu}_{s}$. What is the sphere's resulting angular speed $\omega$ ?

Salamat Ali

Numerade Educator

Problem 59

A recently named element is darmstadtium (Ds), which has 110 electrons. Assume that you can put the 110 electrons into the atomic shells one by one and can neglect any electronelectron interaction. With the atom in ground state, what is the spectroscopic notation for the quantum number $\ell$ for the last electron?

Alia Hamdan

Numerade Educator

Problem 60

How many electron states are in these subshells: (a) $n=4$, $\ell=3 ;(\mathrm{b}) n=3, \ell=2 ;$ (c) $n=4, \ell=1 ;$ (d) $n=2, \ell=0$ ?

Salamat Ali

Numerade Educator

Problem 61

(a) What is the magnitude of the orbital angular momentum in a state with $\ell=3$ ? (b) What is the magnitude of its largest projection on an imposed $z$ axis?

Alia Hamdan

Numerade Educator

Problem 62

Suppose that a hydrogen atom in its ground state moves $80 \mathrm{~cm}$ through and perpendicular to a vertical magnetic field that has a magnetic field gradient $d B / d z=1.6 \times 10^{2} \mathrm{~T} / \mathrm{m}$. (a) What is the magnitude of force exerted by the field gradient on the atom due to the magnetic moment of the atom's electron, which we take to be 1 Bohr magneton? (b) What is the vertical displacement of the atom in the $80 \mathrm{~cm}$ of travel if its speed is $2.5 \times 10^{5} \mathrm{~m} / \mathrm{s}$ ?

Salamat Ali

Numerade Educator