Chapter Questions
How does Thomson's plum-pudding model fail to account for Rutherford scattering? How does the nuclear atom account for the backscattering events?
Would Rutherford scattering work if protons were used as projectiles instead of alpha particles?
When an electron in the Bohr atom jumps to a larger-radius orbit, does each of the following quantities increase, decrease, or remain the same? (a) Total energy, (b) kinetic energy; (c) potential energy, (d) electron speed.
Why doesn't Bohr's theory apply to the neutral helium atom?
Does the $n=3$ to $n=2$ transition produce a longer-wavelength photon in hydrogen or deuterium?
List successes and failures of Bohr's atomic theory.
What's the experimental evidence for each of the four quantum numbers that describes the state of a hydrogen atom?
Explain how to find the maximum number of electrons allowed on a given subshell.
Based on their electronic configurations, discuss the likely ionization states of sodium and chlorine. Explain why it's favorable for these atoms to combine in the form $\mathrm{NaCl}$.
Why are the noble gases so stable? Why are they unlikely to form chemical compounds?
What's the physical significance of the quantum numbers (a) $m_{l}$ and (b) $m_{s} ?$
When sodium is dropped into water, a violent reaction occurs. Why?
Aluminum isn't in the same group as copper, silver, and gold, yet it's an excellent electrical conductor. Why?
Why do you suppose that some science fiction stories have used the concept of silicon-based life?
How do you think Moseley measured the wavelengths of the x rays he created in the lab? For each element he studied, how did he know which $x$ rays were $\mathrm{K}_{\alpha}, \mathrm{K}_{\beta},$ and so on?
If the potential difference in an x-ray tube is increased, how does this affect the wavelengths of (a) bremsstrahlung $x$ rays and(b) characteristic x-rays?
A hydrogen atom makes a transition from the $n=4$ level to $n=2$. This results in the emission of a photon with wavelength(a) $410 \mathrm{nm}$(b) $434 \mathrm{nm}$(c) $486 \mathrm{nm}$(d) $656 \mathrm{nm}$.
A ground-state hydrogen atom absorbs a photon and ends up in the $n=3$ state. The photon's energy was (a) $2.18 \times 10^{-18} \mathrm{~J}$(b) $1.94 \times 10^{-18} \mathrm{~J}$(c) $1.24 \times 10^{-18} \mathrm{~J}$(d) $2.42 \times 10^{-19} \mathrm{~J}$
The ground-state energy of the helium ion $\mathrm{He}^{+}$ is equal to the(b) 2 ; ground-state energy of hydrogen multiplied by(a) 4(c) $1 / 2 ;$ (d) $1 / 4$.
What transition in hydrogen corresponds to the emission of an infrared photon with $\lambda=1282 \mathrm{nm} ?$ (a) $n=4$ to $n=2 ;$ (b) $n=4$ to $n=3 ;$ (c) $n=5$ to $n=3 ;$ (d) $n=6$ to $n=3$.
If a hydrogen atom makes a transition from the $n=5$ level directly to $n=1,$ the emitted photon will be in what part of thespectrum? (a) Microwave; (b) infrared; (c) visible; (d) ultraviolet.
The spectroscopic notation for a hydrogen atom with an electron having quantum numbers $n=3, l=2, m_{l}=-1, m_{s}=-1 / 2$ is(a) $3 s$ ( b) $3 p$, (c) $3 d$; (d) $3 f$
Which of the following transitions is not allowed for an electron (c) $3 d$ to $4 f$ in hydrogen? (a) $4 p$ to $1 s$; (b) $3 d$ to $2 s$;(d) $2 p$ to $4 s$
For a hydrogen atom in the $4 d$ state, the orbital angular momentum is (a) $2(h / 2 \pi) ;$ (b) $\sqrt{6}(h / 2 \pi)$(c) $\sqrt{12}(h / 2 \pi)$ (d) $\sqrt{20}(h / 2 \pi)$.
What's the shortest-wavelength $x$ ray produced in a $150-\mathrm{kV}$ x-ray tube? (a) $8.3 \mathrm{nm} ;$ (b) $0.83 \mathrm{nm} ;$ (c) $0.083 \mathrm{nm}$; (d) $8.3 \times 10^{-3} \mathrm{nm}$
Which of the following characteristic $x$ rays has the longest wavelength?(a) $\mathrm{K}_{\alpha}$ (b) $\mathrm{K}_{\beta}$; (c) $\mathrm{L}_{\alpha}$; (d) $\mathrm{L}_{\beta}$
In some Rutherford scattering experiments, the alpha particles had kinetic energies of $7.7 \mathrm{MeV}$. What's the speed of such an alpha particle?
Suppose a proton and an alpha particle each have kinetic energy $8.0 \mathrm{MeV}$. (a) Find the speed of each particle.(b) Find each particle's distance of closest approach in a head-on collision with a gold nucleus.
Find the distance of closest approach of a $7.7-\mathrm{MeV}$ alpha particle in a head-on collision with an aluminum nucleus. Compare your answer with the closest approach to a gold nucleus, given in Example 24.1 .
Alpha particles and gold nuclei have radii $1.9 \times 10^{-15} \mathrm{~m}$ and $7.0 \times 10^{-15} \mathrm{~m},$ respectively. Find the kinetic energy of an alpha particle such that the two will just touch in a head-on collision. (At this energy Rutherford's point-particle approximation would begin to break down.)
Suppose the hydrogen atom consists of a proton with radius $1.2 \times 10^{-15} \mathrm{~m},$ with an electron orbiting the proton at a distance of $5.3 \times 10^{-11} \mathrm{~m} .$ Take the atom to be a sphere with a radius equal to the electron's orbital radius.(a) What's the atom's volume? What's the volume of the nucleus? (b) What's the atom's density? Compare with that of water, $1000 \mathrm{~kg} / \mathrm{m}^{3}$. (c) What is the density of the nucleus? Compare with the atomic density you found in part (b).
Find the wavelength of the photon emitted when hydrogen makes a transition (a) $n=8 \rightarrow 5$ (b) $n=6 \rightarrow 1 ;$ (c) $n=10 \rightarrow 2$.
What approximate value of the quantum number $n$ will make the hydrogen atom's energy (a) $-1.36 \times 10^{-20} \mathrm{~J} ;$ (b) $-2.69 \times 10^{-21} \mathrm{~J}$ (c) $-1.29 \times 10^{-21} \mathrm{~J}$
What's the series limit for (a) the Lyman series; (b) the Paschen series; (c) the Brackett series?
Find the range of wavelengths seen in the hydrogen emission spectrum in the Pfund series.
A hydrogen atom absorbs a photon with a wavelength $\lambda=486 \mathrm{nm} .$ What are the atom's initial and final states?
A hydrogen atom is in the $n=2$ state. (a) How much energy is required to remove the electron completely from the atom?(b) What's the wavelength of a photon with this energy?
I wavelength $656 \mathrm{nm}$. Find the change in (a) the atom's total energy; (b) the atom's potential energy; (c) the orbiting electron's kinetic energy.
In (a) Use Bohr's theory to find the speed of an electron orbiting on the $n=1$ level. Should relativistic calculations be used?(b) Repeat for $n=2$.
Find the wavelengths of photons emitted by $\mathrm{Li}^{2+}$ for the transitions (a) $n=3 \rightarrow 2$ and (b) $n=4 \rightarrow 2$.
A hydrogen atom at rest makes a transition from the $n=4$ state to $n=2$. Find (a) the wavelength of the emitted photon and(b) the recoil speed of the atom.
For a hydrogen atom in the ground state, find the total energy, the potential energy, and the electron's kinetic energy. Verify that $E=K+U$
An electron in a hydrogen atom orbits with radius of $4 a_{0}$. (a) What's the change in the atom's total energy if the radius increases to $9 a_{0} ?$ (b) Does this process correspond to the emission or absorption of a photon? What's the photon's wavelength?
Find the de Broglie wavelengths of electrons in hydrogen's(a) $n=1$ state;(b) $n=2$ state; (c) $n=10$ state.
Find the minimum quantum number needed to make a hydrogen atom at least $0.50 \mu \mathrm{m}$ in diameter.
A hydrogen atom in the $n=5$ state drops to the $n=2$ state by undergoing two downward transitions. What are all possible combinations of the resulting photon wavelengths?
A light source emits a continuous wavelength range from $400 \mathrm{nm}$ to $1000 \mathrm{nm}$. If this light is incident on a gas of atomic hydrogen, find all the transitions that can occur due to photon absorption.
A hydrogen atom is in the $n=8$ state. (a) What's its electron's orbital radius? (b) Use your answer to find the electron's de Broglie wavelength. (c) From your answer to part (b), determine the electron's speed. (d) Compute the electron's speed using Bohr theory, and compare with your answer to part (c).
Find the ground-state energies of (a) $\mathrm{He}^{+}$; (b) $\mathrm{Li}^{2+}$.
Consider the spectrum of $\mathrm{He}^{+}$ resulting from downward transitions to the $n=4$ level. (a) What wavelength results when initially $n=6 ?$ (b) To which transition in hydrogen does that wavelength correspond? (c) What transitions (initial $n$ to final $n$ ) in $\mathrm{He}^{+}$ produce the rest of the wavelengths seen in hydrogen's Balmer series?
Give spectroscopic notation for each of these electron states:(a) $n=2, l=0$(b) $n=4, l=1$(c) $n=4, l=3$(d) $n=3$, $l=2$
For each of the following electronic states in hydrogen, find the magnitudes of the orbital and spin angular momenta:(a) $3 s$(b) $3 p$(c) $3 d$.
Which of the following transitions are allowed for an electron in hydrogen? For each allowed transition, find the wave-; (b) $3 d$ to $2 s$; (c) 4 fto length of the emitted photon: (a) $3 p$ to $2 s$; $3 d ;$ (d) $3 p$ to $1 s$
For an electron at the $n=4$ level in hydrogen, find all sets of allowed quantum numbers. How many different sets are there?
For an electron in the $3 p$ state of hydrogen, calculate all allowed values of (a) orbital angular momentum; (b) z-component of orbital angular momentum; (c) z-component of spin angular momentum.
A hydrogen atom has energy $-1.36 \times 10^{-19} \mathrm{~J},$ and the electron can be in any one of 10 different quantum states. Find the spectroscopic notation for this atom.
By what factor does a hydrogen atom's angular momentum change when it makes the following transitions: (a) $3 d$ to $2 p$(b) $3 p$ to $4 d ;$ (c) $4 f$ to $3 d$ ?
In a Zeeman experiment done in a 2.0-T magnetic field, the atomic energy levels in hydrogen change by $\left(1.86 \times 10^{-23} \mathrm{~J}\right) m_{l}$ When each Balmer series line is split into three, what's the difference in wavelength between the two outer lines? Is this easy to detect?
Write the ground-state electron configuration for each of the following atoms: (a) calcium; (b) iron; (c) gold; (d) uranium.
Identify the atoms with ground-state electron configuration(a) $2 s^{2} 2 p^{4}$(b) $3 s^{2} 3 p^{6}$(c) $3 d^{10} 4 s^{2} 4 p^{1}$(d) $4 f^{14} 5 d^{4} 6 s^{2}$.
For argon in the ground state, list (a) the electronic configuration and (b) the set of four quantum numbers for each of the electrons.
Find the maximum number of electrons allowed on each of the following shells by writing out all the possible sets of quantumstates for electrons on that shell: (a) $n=2 ;$ (b) $n=3 ;$ (c) $n=4$.
What type of excess charge carriers (electrons or holes) are 3 created when (a) silicon is doped with gallium and (b) germanium is doped with antimony?
What's the minimum wavelength of x rays produced in an X-ray tube with potential difference(a) $100 \mathrm{kV}$(b) $1.0 \mathrm{MV} ?$
A dental x-ray source has minimum wavelength 0.031 nm. What's the potential difference in the x-ray tube?
Old-fashioned tube televisions were a weak source of $x$ rays, due to electrons stopping at the screen (leaded glass helped prevent x-ray exposure). For a TV tube operating at $30 \mathrm{kV}$, what minimum x-ray wavelength was produced?
What's the shortest-wavelength photon that can be produced by the Stanford Linear Accelerator, where electrons are accelerated to energies of $50 \mathrm{GeV} ?$
What wavelengths would Moseley have predicted for $\mathrm{K}_{\alpha} \mathrm{x}$ rays from the three missing elements, $Z=43,61,$ and $75 ?$
Find the minimum potential difference required to produce $\mathrm{K}_{\alpha} \mathrm{x}$ rays in an $\mathrm{x}$ -ray tube using a cobalt target.
A $\mathrm{K}_{\alpha}$ x ray is observed with wavelength $0.337 \mathrm{nm}$. Identify the target element.
Helium-neon lasers can be made to produce green laser light with a wavelength of $532 \mathrm{nm}$. What's the gap between energy levels in neon responsible for this emission?
(a) How many photons per second are emitted from a $1.0-\mathrm{mW}$ helium-neon laser with wavelength $632.8 \mathrm{nm} ?$(b) If a laser contains 0.025 mole of neon, what fraction of the neon atoms participate in the lasing process each second?
A laser emits $4.50 \times 10^{18}$ photons per second, using a transition from $2.33 \mathrm{eV}$ above the ground state to the ground state. Find (a) the laser light's wavelength and (b) the laser's power output.
A laser used to repair a detached retina has wavelength of $532 \mathrm{nm}$. The beam comes in short pulses, each lasting $25 \mathrm{~ms}$ and carrying $1.0 \mathrm{~J}$ of energy. (a) What's the power of this beam while it's on? (b) How many photons are in each pulse?
What transition in hydrogen corresponds to emission of an infrared photon with $\lambda=1282 \mathrm{nm} ?$
For an electron on the $n=6$ level of hydrogen, how many different sets of quantum numbers are there?
Find all the sets of allowed quantum numbers for an electron with the spectroscopic designation $5 f$
Determine the electronic configuration $(n, l)$ for a hydrogen atom with energy $-1.36 \times 10^{-19} \mathrm{~J}$ and angular momentum of magnitude $\sqrt{12} h / 2 \pi$
A hydrogen atom is in the $3 d$ state. (a) What are the possible angles its orbital angular momentum vector can make with a given axis? (b) What are the possible values for the component of angular momentum along that axis?
Bohr's corres pondence principle says that as quantized objects approach macroscopic size, quantum theory should reduce to the classical result. As one example, consider a hydrogen atom with a large value of $n$. (a) Find the orbital frequency of an electron at the $n=100$ level. (b) Find the frequency of a photon emitted when an electron makes a transition from the $n=101$ state to the $n=100$ state, and compare with the frequency you computed in part (a). Are the discrete steps in frequency as noticeable as for lower-level transitions?
A solid-state laser made from lead-tin selenide has a lasing transition at a wavelength of $30 \mu \mathrm{m}$. If its power output is $2.0 \mathrm{~mW},$ how many transitions occur each second?
Following the method outlined in the text for the $\mathrm{K}_{\alpha}$ x rays, find a formula that predicts the wavelengths of $\mathrm{K}_{\beta} \mathrm{x}$ rays. Use this formula to predict the wavelength of the $\mathrm{K}_{\beta} \mathrm{x}$ ray for copper, and compare it with the experimental result shown in the Moseley plot in the text.
For the L-series x rays, Moseley found he could predict the observed wavelengths by assuming that an electron falling to the $L$ shell of an atom feels the equivalent of 7.4 electrons inside the $L$ shell. (a) Use this information to find a formula that predicts the wavelengths of $L_{\alpha}$ x rays.(b) Use this formula to predict the wavelengths of an $L_{\alpha}$ x ray from tungsten, and compare with the experimental result shown in the Moseley plot in the text.