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Modern Physics

Kenneth S. Krane

Chapter 8

Many-Electron Atoms - all with Video Answers

Educators

Chapter Questions

03:05

Problem 1

(a) List the six possible sets of quantum numbers $\left(n, l, m_{l}, m_{s}\right)$ of a $2 p$ electron. $(b)$ Suppose we have an atom such as carbon, which has two $2 p$ electrons. Ignoring the Pauli principle, how many different possible combinations of quantum numbers of the two electrons are there? ( $c$ ) How many of the possible combinations of part $(b)$ are eliminated by applying the Pauli principle? $(d)$ Suppose carbon is in an excited state with configuration $2 p^{1} 3 p^{1}$. Does the Pauli principle restrict the choice of quantum numbers for the electrons? How many different sets of quantum numbers are possible for the two electrons?

Suzanne W.
Suzanne W.
Numerade Educator
03:57

Problem 2

Nitrogen $(Z=7)$ has three electrons in the $2 p$ level (in addition to two electrons each in the $1 s$ and $2 s$ levels).
(a) Consistent with the Pauli principle, what is the maximum possible value of the total $m_{s}$ of all seven electrons?
(b) List the quantum numbers of the three $2 p$ electrons that result in the largest total $m_{s}$.
(c) If the electrons in the $2 p$ level occupy states that maximize $m_{s},$ what would be the maximum possible value for the total $m_{l} ?$ ( $d$ ) What would be the maximum possible total $m_{l}$ if the three $2 p$ electrons were in states that did not maximize $m_{\varepsilon} ?$

Suzanne W.
Suzanne W.
Numerade Educator
03:11

Problem 3

(a) How many different sets of quantum numbers $\left(n, l, m_{l}, m_{s}\right)$ are possible for an electron in the $4 f$ level?
(b) Suppose a certain atom has three electrons in the $4 f$ level. What is the maximum possible value of the total $m_{s}$ of the three electrons? (c) What is the maximum possible total $m_{l}$ of three $4 f$ electrons? $(d)$ Suppose an atom has ten electrons in the $4 f$ level. What is the maximum possible value of the total $m_{s}$ of the ten $4 f$ electrons? (e) What is the maximum possible total $m_{l}$ of ten $4 f$ electrons?

Suzanne W.
Suzanne W.
Numerade Educator
01:51

Problem 4

( $a$ ) Suppose a beryllium atom $(Z=4$ ) absorbs energy (such as from a beam of photons) that pushes one of the electrons to an excited state. If the photon energy is set at the minimum necessary for this to occur, from which subshell does the electron make the transition and to which subshell does it jump? ( $b$ ) Suppose the same experiment is done with neon $(Z=10)$. At the minimum energy for absorption, from which subshell does the electron make the transition and to which subshell does it jump? ( $c$ ) Would you expect the minimum absorption energy for beryllium to be larger or smaller than the minimum energy for neon? Explain.

Suzanne W.
Suzanne W.
Numerade Educator
01:15

Problem 5

(a) List all elements with a $p^{3}$ configuration. (b) List all elements with a $d^{7}$ configuration.

Suzanne W.
Suzanne W.
Numerade Educator
View

Problem 6

Give the electronic configuration of $(a) \mathrm{P} ;(b) \mathrm{V} ;(c) \mathrm{Sb} ;$ (d) $\mathrm{Pb}$.

Sarah Jones
Sarah Jones
Numerade Educator
02:42

Problem 7

(a) What is the electronic configuration of Fe? (b) In its ground state, what is the maximum possible total $m_{s}$ of its electrons? (c) When the electrons have their maximum possible total $m_{s},$ what is the maximum total $m_{l} ?$ ( $d$ ) Suppose one of the $d$ electrons is excited to the next highest level. What is the maximum possible total $m_{s},$ and when $m_{s}$ has its maximum total what is the maximum total $m_{l} ?$

Suzanne W.
Suzanne W.
Numerade Educator
01:21

Problem 8

The ground state of singly ionized lithium $(Z=3)$ is $1 s^{2}$. Use the electron screening model to predict the energies of the $1 s^{1} 2 p^{1}$ and $1 s^{1} 3 d^{1}$ excited states in singly ionized lithium. Compare your predictions with the measured energies (respectively $-13.4 \mathrm{eV}$ and $-6.0 \mathrm{eV})$.

Narayan Hari
Narayan Hari
Numerade Educator
01:31

Problem 9

The ground state of neutral beryllium $(Z=4)$ is $1 s^{2} 2 s^{2} .$ Use the electron screening model to predict the energies of the following excited states: $1 s^{2} 2 s^{1} 3 p^{1}$ (measured $-2.02 \mathrm{eV}$ ) and $1 s^{2} 2 s^{1} 4 d^{1}(-0.90 \mathrm{eV})$.

Suzanne W.
Suzanne W.
Numerade Educator
03:24

Problem 10

Using the wavelengths given in Figure $8.4,$ compute the energy difference between the $3 d$ and $4 d$ states in lithium; do the same for sodium. Compare those values with the corresponding $n=4$ to $n=3$ energy difference in hydrogen. Why is the agreement so good, considering the different values of $Z$ ?

Suzanne W.
Suzanne W.
Numerade Educator
05:57

Problem 11

(a) Using the information for lithium given in Figure 8.4 compute the energy difference of the $3 p$ and $3 d$ states. (b) Compute the energy of the $3 s, 4 s,$ and $5 s$ states above the ground state. (c) The ionization energy of lithium in its ground state is $5.39 \mathrm{eV}$. What is the ionization energy of the $2 p$ state? Of the $3 s$ state?

Suzanne W.
Suzanne W.
Numerade Educator
01:13

Problem 12

A certain element emits a $K_{\alpha}$ X ray of wavelength $0.1940 \mathrm{nm}$. Identify the element.

Suzanne W.
Suzanne W.
Numerade Educator
02:10

Problem 13

Compute the $K_{\alpha}$ X ray energies of calcium $(Z=20)$, zirconium $(Z=40)$, and mercury $(Z=80)$. Compare with the measured values of $3.69 \mathrm{keV}, 15.8 \mathrm{keV},$ and $70.8 \mathrm{keV}$. (See Question 16).

Narayan Hari
Narayan Hari
Numerade Educator
02:20

Problem 14

Chromium has the electron configuration $4 s^{1} 3 d^{5}$ beyond the inert argon core. What are the ground-state $L$ and $S$ values?

Suzanne W.
Suzanne W.
Numerade Educator
04:39

Problem 15

Use Hund's rules to find the ground-state $L$ and $S$ of
(a) Ce, configuration [Xe] $6 s^{2} 4 f^{1} 5 d^{1} ;$ (b) Gd, configuration $[\mathrm{Xe}] 6 s^{2} 4 f^{7} 5 d^{1} ;(c) \mathrm{Pt},$ configuration $[\mathrm{Xe}] 6 s^{1} 4 f^{14} 5 d^{9}$.

Suzanne W.
Suzanne W.
Numerade Educator
03:28

Problem 16

Using Hund's rules, find the ground-state $L$ and $S$ of (a) fluorine $(Z=9) ;$ (b) magnesium $(Z=12) ;$ (c) titanium $(Z=22) ;(d)$ iron $(Z=26)$.

Suzanne W.
Suzanne W.
Numerade Educator
01:03

Problem 17

A certain excited state of an atom has the configuration $4 d^{1} 5 d^{1}$. What are the possible $L$ and $S$ values?

Suzanne W.
Suzanne W.
Numerade Educator
01:33

Problem 18

Use the degeneracies of the states with all possible total $L$ and $S$ to find how many different levels the $2 p^{1} 3 p^{1}$ excited state of carbon includes. (See Figure $8.18 .$ ) Compare this result with the result of counting the individual $m_{l}$ and $m_{s}$ values from Problem $1(d)$. (See also Question 11.)

Suzanne W.
Suzanne W.
Numerade Educator
02:05

Problem 19

A small helium-neon laser produces a light beam with an average power of $3.5 \mathrm{~mW}$ and a diameter of $2.4 \mathrm{~mm}$.
(a) How many photons per second are emitted by the laser?
(b) What is the amplitude of the electric field of the light wave? Compare this result with the electric field at a distance of $1 \mathrm{~m}$ from an incandescent light bulb that emits $100 \mathrm{~W}$ of visible light.

Narayan Hari
Narayan Hari
Numerade Educator
03:11

Problem 20

(a) How many different possible ways are there to assign the sets of quantum numbers to the four $2 p$ electrons in oxygen $(Z=8) ?(b)$ List all possible values of the total $m_{s}$ for the four electrons. (c) List all possible values of the total $m_{l}$ of the four electrons. $(d)$ If the total $m_{s}$ has its largest possible value, what are the possible values of the total $m_{l} ?(e)$ If the total $m_{l}$ has its largest possible value, what are the possible values of the total $m_{s} ?$

Suzanne W.
Suzanne W.
Numerade Educator
02:16

Problem 21

(a) The ionization energy of sodium is $5.14 \mathrm{eV}$. What is the effective charge seen by the outer electron? $(b)$ If the $3 s$ electron of a sodium atom is moved to the $4 f$ state, the measured binding energy is $0.85 \mathrm{eV}$. What is the effective charge seen by an electron in this state?

Suzanne W.
Suzanne W.
Numerade Educator
02:26

Problem 22

Draw a Moseley plot, similar to Figure $8.16,$ for the $K_{\beta} X$ rays using the following energies in $\mathrm{keV}$ :$$
\begin{array}{llllll}
\text { Ne } & 0.858 & \text { Mn } & 6.51 & \text { Zr } & 17.7 \\
\text { P } & 2.14 & \text { Zn } & 9.57 & \text { Rh } & 22.8 \\
\text { Ca } & 4.02 & \text { Br } & 13.3 & \text { Sn } & 28.4
\end{array}
$$ Determine the slope and compare with the expected value. (Equation 8.4 applies only to $K_{\alpha}$ X rays; you will need to derive a similar equation for the $K_{\beta}$ X rays.) Determine the $z$ -axis intercept and give its interpretation.

Suzanne W.
Suzanne W.
Numerade Educator
02:33

Problem 23

Draw a Moseley plot, similar to Figure $8.16,$ for the $L_{\alpha} X$ rays using the following energies in $\mathrm{keV}$ :
$$\begin{array}{llll}\mathrm{Mn} & 0.721 & \mathrm{Rh} & 2.89 \\ \mathrm{Zn} & 1.11 & \mathrm{Sn} & 3.71 \\ \mathrm{Br} & 1.60 & \mathrm{Cs} & 4.65 \\ \mathrm{Zr} & 2.06 & \mathrm{Nd} & 5.72\end{array}$$ Give interpretations of the slope and intercept.

Suzanne W.
Suzanne W.
Numerade Educator
01:14

Problem 24

Because of the fine-structure splitting of the $3 p$ state, the $3 p \rightarrow 3 s$ transition in sodium actually consists of two closely spaced lines of wavelengths $589.00 \mathrm{nm}$ and $589.59 \mathrm{nm}$. Assuming a magnetic moment of one Bohr magneton, find the effective magnetic field that produces the fine-structure splitting of the $3 p$ state of sodium.

Narayan Hari
Narayan Hari
Numerade Educator
02:16

Problem 25

(a) What is the longest wavelength of the absorption spectrum of lithium? ( $b$ ) What is the longest wavelength of the absorption spectrum of helium? In what region of the spectrum does this occur? ( $c$ ) What are the shortest wavelengths in the absorption spectra of helium and lithium? In what region of the electromagnetic spectrum are these?

Suzanne W.
Suzanne W.
Numerade Educator
02:44

Problem 26

Using the wavelengths given in Figure $8.17,$ compute the energy difference between the $1 s^{1} 4 p^{1}$ and $1 s^{1} 3 p^{1}$ singlet $(S=0)$ states in helium. Compare this energy difference with the value expected using the Bohr model, assuming that the $p$ electron is screened by the $s$ electron. Repeat the calculation for the $3 d$ and $4 d$ triplet $(S=1)$ states.

Suzanne W.
Suzanne W.
Numerade Educator