A ssion reactor is hit by a nuclear weapon, causing 5.0

Use Equation 28.1 to calculate the wavelength of the rst three lines in the Balmer series for hydrogen.

What is the order of magnitude of the number of protons in your body? Of the number of neutrons? Of the number of electrons?

Using the result of Example 29.1, nd the radius of a sphere of nuclear matter that would have a mass equal to that of the Earth. The Earth has a mass of 5.98  1024 kg and average radius of 6.37  106 m.

Consider the hydrogen atom to be a sphere of radius equal to the Bohr radius, 0.53  1010 m, and calculate the approximate value of the ratio of the nuclear density to the atomic density.

An alpha particle (Z  2, mass 6.64  1027 kg) approaches to within 1.00  1014 m of a carbon nucleus (Z  6). What are (a) the maximum Coulomb force on the alpha particle, (b) the acceleration of the alpha particle at this time, and (c) the potential energy of the alpha particle at the same time?

Singly ionized carbon atoms are accelerated through 1 000 V and passed into a mass spectrometer to determine the isotopes present. (See Chapter 19.) The magnetic eld strength in the spectrometer is 0.200 T. (a) Determine the orbital radii for the 12C and the 13C isotopes as they pass through the eld. (b) Show that the ratio of the radii may be written in the form and verify that your radii in part (a) satisfy this formula.

(a) Find the speed an alpha particle requires to come within 3.2  1014 m of a gold nucleus. (b) Find the energy of the alpha particle in MeV.

Find the radius of a nucleus of (a) and (b) .

Calculate the average binding energy per nucleon of and .

Calculate the binding energy per nucleon for (a) , (b) , (c) , and (d) .

A pair of nuclei for which Z1  N2 and Z2  N1 are called mirror isobars. (The atomic and neutron numbers are interchangeable.) Binding-energy measurements on such pairs can be used to obtain evidence of the charge independence of nuclear forces. Charge independence means that the proton proton, protonneutron, and neutronneutron forces are approximately equal. Calculate the difference in binding energy for the two mirror nuclei and .

The peak of the stability curve occurs at . This is why iron is prominent in the spectrum of the Sun and stars. Show that has a higher binding energy per nucleon than its neighbors and . Compare your results with Figure 29.4.

Two nuclei having the same mass number are known as isobars. Calculate the difference in binding energy per nucleon for the isobars and . How do you account for this difference?

Calculate the binding energy of the last neutron in the nucleus. [Hint: You should compare the mass of with the mass of plus the mass of a neutron. The mass of  41.958 622 u.]

The half-life of an isotope of phosphorus is 14 days. If a sample contains 3.0  1016 such nuclei, determine its activity. Express your answer in curies.

A drug tagged with (half-life  6.05h) is prepared for a patient. If the original activity of the sample was 1.1  104 Bq, what is its activity after it has sat on the shelf for 2.0 h?

The half-life of is 8.04 days. (a) Calculate the decay constant for this isotope. (b) Find the number of nuclei necessary to produce a sample with an activity of 0.50

After 2.00 days, the activity of a sample of an unknown type of radioactive material has decreased to 84.2% of the initial activity. (a) What is the half-life of this material? (b) Can you identify it by using the table of isotopes in Appendix B?

Suppose that you start with 1.00  103 g of a pure radioactive substance and 2.0 h later determine that only 0.25  103 g of the substance remains. What is the half-life of this substance?

Radon gas has a half-life of 3.83 days. If 3.00 g of radon gas is present at time t  0, what mass of radon will remain after 1.50 days have passed?

Many smoke detectors use small quantities of the isotope in their operation. The half-life of is 432 yr. How long will it take for the activity of this material to decrease to 1.00  103 of the original activity?

After a plant or animal dies, its 14C content decreases with a half-life of 5 730 yr. If an archaeologist nds an ancient repit containing partially consumed rewood, and the 14C content of the wood is only 12.5% that of an equal carbon sample from a present-day tree, what is the age of the ancient site?

A freshly prepared sample of a certain radioactive isotope has an activity of 10.0 mCi. After 4.00 h, the activity is 8.00 mCi. (a) Find the decay constant and half-life of the isotope. (b) How many atoms of the isotope were contained in the freshly prepared sample? (c) What is the samples activity 30 h after it is prepared?

A building has become accidentally contaminated with radioactivity. The longest-lived material in the building is strontium-90. (The atomic mass of is 89.907 7.) If the building initially contained 5.0 kg of this substance, and the safe level is less than 10.0 counts/min, how long will the building be unsafe?

Complete the following radioactive decay formulas:

Complete the following radioactive decay formulas:

Complete the following radioactive decay formulas:

Figure P29.28 shows the steps by which decays to . Enter the correct isotope symbol in each square.

The mass of 56Fe is 55.934 9 u and the mass of 56Co is 55.939 9 u. Which isotope decays into the other and by what process?

Find the energy released in the alpha decay of . The following mass value will be useful: has a mass of 234.043 583 u.

Determine which of the following suggested decays can occur spontaneously:

(mass  65.9291 u) undergoes beta decay to (mass  65.9289 u). (a) Write the complete decay formula for this process. (b) Find the maximum kinetic energy of the emerging electrons.

An 3H nucleus beta decays into 3He by creating an electron and an antineutrino according to the reaction :  e  Use Appendix B to determine the total energy released in this reaction.

A piece of charcoal used for cooking is found at the remains of an ancient campsite. A 1.00-kg sample of carbon from the wood has an activity of 2.00  103 decays per minute. Find the age of the charcoal. [Hint: Living material has an activity of 15.0 decays/minute per gram of carbon present.]

A wooden artifact is found in an ancient tomb. Its carbon-14 ( ) activity is measured to be 60.0% of that in a fresh sample of wood from the same region. Assuming the same amount of 14C was initially present in the wood from which the artifact was made, determine the age of the artifact.

A living specimen in equilibrium with the atmosphere contains one atom of 14C (half-life  5730 yr) for every 7.70  1011 stable carbon atoms. An archeological sample of wood (cellulose, C12H22O11) contains 21.0 mg of carbon. When the sample is placed inside a shielded beta counter with 88.0% counting efciency, 837 counts are accumulated in one week. Assuming that the cosmic-ray ux and the Earths atmosphere have not changed appreciably since the sample was formed, nd the age of the sample.

The rst known reaction in which the product nucleus was radioactive (achieved in 1934) was one in which was bombarded with alpha particles. Produced in the reaction were a neutron and a product nucleus. (a) What was the product nucleus? (b) Find the Q value of the reaction.

Complete the following nuclear reactions:

Identify the unknown particles X and X in the following nuclear reactions:

The rst nuclear reaction utilizing particle accelerators was performed by Cockcroft and Walton. Accelerated protons were used to bombard lithium nuclei, producing the following reaction: H  Li : He  He Since the masses of the particles involved in the reaction were well known, these results were used to obtain an early proof of the Einstein massenergy relation. Calculate the Q value of the reaction.

(a) Suppose B is struck by an alpha particle, releasing a proton and a product nucleus in the reaction. What is the product nucleus? (b) An alpha particle and a product nucleus are produced when C is struck by a proton. What is the product nucleus?

(a) Determine the product of the reaction Li  He : ?  n. (b) What is the Q value of the reaction?

Natural gold has only one isotope: . If gold is bombarded with slow neutrons, e particles are emitted. (a) Write the appropriate reaction equation. (b) Calculate the maximum energy of the emitted beta particles. The mass of is 197.966 75 u.

Find the threshold energy that an incident neutron must have to produce the reaction

When 18O is struck by a proton, 18F and another particle are produced. (a) What is the other particle? (b) The reaction has a Q value of 2.453 MeV, and the atomic mass of 18O is 17.999 160 u. What is the atomic mass of 18F?

In terms of biological damage, how many rad of heavy ions is equivalent to 100 rad of x-rays?

A person whose mass is 75.0 kg is exposed to a whole-body dose of 25.0 rads. How many joules of energy are deposited in the persons body?

A 200-rad dose of radiation is administered to a patient in an effort to combat a cancerous growth. Assuming all of the energy deposited is absorbed by the growth, (a) calculate the amount of energy delivered per unit mass. (b) Assuming the growth has a mass of 0.25 kg and a specic heat equal to that of water, calculate its temperature rise.

A clever technician decides to heat some water for his coffee with an x-ray machine. If the machine produces 10 rad/s, how long will it take to raise the temperature of a cup of water by 50C. Ignore heat losses during this time.

An x-ray technician works 5 days per week, 50 weeks per year. Assume that the technician takes an average of eight x-rays per day and receives a dose of 5.0 rem/yr as a result. (a) Estimate the dose in rem per x-ray taken. (b) How does this result compare with the amount of low-level background radiation the technician is exposed to?

A patient swallows a radiopharmaceutical tagged with phosphorus-32 ( P), a emitter with a half-life of 14.3 days. The average kinetic energy of the emitted electrons is 700 keV. If the initial activity of the sample is 1.31 MBq, determine (a) the number of electrons emitted in a 10-day period, (b) the total energy deposited in the body during the 10 days, and (c) the absorbed dose if the electrons are completely absorbed in 100 g of tissue.

A particular radioactive source produces 100 mrad of 2-MeV gamma rays per hour at a distance of 1.0 m. (a) How long could a person stand at this distance before accumulating an intolerable dose of 1 rem? (b) Assuming the gamma radiation is emitted uniformly in all directions, at what distance would a person receive a dose of 10mrad/h from this source?

A 200.0-mCi sample of a radioactive isotope is purchased by a medical supply house. If the sample has a half-life of 14.0 days, how long will it keep before its activity is reduced to 20.0 mCi?

A sample of organic material is found to contain 18g of carbon. The investigators believe the material to be 20000 years old, based on samples of pottery found at the site. If so, what is the expected activity of the organic material? Take data from Example 29.7.

Deuterons that have been accelerated are used to bombard other deuterium nuclei, resulting in the reaction H  H : He  n Does this reaction require a threshold energy? If so, what is its value?

Free neutrons have a characteristic half-life of 12 min. What fraction of a group of free neutrons at a thermal energy of 0.040 eV will decay before traveling a distance of 10.0 km?

A by-product of some ssion reactors is the isotope Pu, an alpha emitter having a half-life of 24 120 yr. The reaction involved is Pu : U   Consider a sample of 1.00 kg of pure Pu at t  0. Calculate (a) the number of Pu nuclei present at t  0 and (b) the initial activity in the sample. (c) How long does the sample have to be stored if a safe activity level is 0.100 Bq?

(a) Find the radius of the C nucleus. (b) Find the force of repulsion between a proton at the surface of a C nucleus and the remaining ve protons. (c) How much work (in MeV) has to be done to overcome this electrostatic repulsion in order to put the last proton into the nucleus? (d) Repeat (a), (b), and (c) for U.

With your partner not looking, use modeling clay to build one or more mounds on top of a table. Place a piece of cardboard over your mound(s), and assign your partner the task of determining the size, shape, and number of mounds without looking. He is to do this by rolling marbles at the unseen mounds and observing how they emerge. This experiment models the Rutherford scattering experiment.

Your instructor can probably lend you a small plastic diffraction grating to enable you to examine the spectrum of different light sources. You can use these gratings to examine a source by holding the grating very close to your eye and noting the spectrum produced by glancing out of the corner of your eye while looking at a light source. You should look at light sources such as sodium vapor lights and mercury vapor lights used in many parking lots, neon lights used in many signs, black lights, ordinary incandescent light bulbs, and so forth.

In a piece of rock from the Moon, the 87Rb content is assayed to be 1.82  1010 atoms per gram of material and the 87Sr content is found to be 1.07  109 atoms per gram. (The relevant decay is 87Rb : 87Sr  e. The half-life of the decay is 4.8  1010 yr.) (a) Determine the age of the rock. (b) Could the material in the rock actually be much older? What assumption is implicit in using the radioactive-dating method?

Many radioisotopes have important industrial, medical, and research applications. One such radioisotope is 60Co, which has a half-life of 5.2 yr and decays by the emission of a beta particle (energy 0.31 MeV) and two gamma photons (energies 1.17 MeV and 1.33 MeV). A scientist wishes to prepare a 60Co sealed source that will have an activity of at least 10 Ci after 30 months of use. What is the minimum initial mass of 60Co required?

A medical laboratory stock solution is prepared with an initial activity due to 24Na of 2.5 mCi/ml, and 10.0 ml of the stock solution is diluted at t0  0 to a working solution whose total volume is 250 ml. After 48 h, a 5.0-ml sample of the working solution is monitored with a counter. What is the measured activity? (Note that 1 ml  1 milliliter.)

A theory of nuclear astrophysics is that all the heavy elements such as uranium are formed in supernova explosions of massive stars, which immediately release the elements into space. If we assume that at the time of an explosion there were equal amounts of 235U and 238U, how long ago were the elements that formed our Earth released, given that the present 235U/238U ratio is 0.007? (The half-lives of 235U and 238U are 0.70  109 yr and 4.47  109 yr, respectively.)

A ssion reactor is hit by a nuclear weapon, causing 5.0  106 Ci of 90Sr (T1/2  28.7 yr) to evaporate into the air. The 90Sr falls out over an area of 104 km2. How long will it take the activity of the 90Sr to reach the agriculturally safe level of 2.0  Ci/m2?

After the sudden release of radioactivity from the Chernobyl nuclear reactor accident in 1986, the radioactivity of milk in Poland rose to 2 000 Bq/L dueto iodine-131, with a half-life of 8.04 days. Radioactive iodine is particularly hazardous, because the thyroid gland concentrates iodine. The Chernobyl accident caused a measurable increase in thyroid cancers among children in Belarus. (a) For comparison, nd the activity of milk due to potassium. Assume that 1 liter of milk contains 2.00 g of potassium, of which 0.011 7% is the isotope 40K, which has a half-life of 1.28  109 yr. (b) After what length of time would the activity due to iodine fall below that due to potassium?

During the manufacture of a steel engine component, radioactive iron (59Fe) is included in the total mass of 0.20 kg. The component is placed in a test engine when the activity due to the isotope is 20.0  Ci. After a 1 000-h test period, oil is removed from the engine and is found to contain enough 59Fe to produce 800 disintegrations/min per liter of oil. The total volume of oil in the engine is 6.5 L. Calculate the total mass worn from the engine component per hour of operation. (The half-life of 59Fe is 45.1 days.)

After determining that the Sun has existed for hundreds of millions of years, but before the discovery of nuclear physics, scientists could not explain why the Sun has continued to burn for such a long time. For example, if it were a coal re, the Sun would have burned up in about 3 000 yr. Assume that the Sun, whose mass is 1.99  1030 kg, originally consisted entirely of hydrogen and that its total power output is 3.76  1026 W. (a) If the energy-generating mechanism of the Sun is the transforming of hydrogen into helium via the net reaction 4H 2e : He  2    calculate the energy (in joules) given off by this reaction. (b) Determine how many hydrogen atoms constitute the Sun. Take the mass of one hydrogen atom to be 1.67  1027 kg. (c) Assuming that the total power output remains constant, after what time will all the hydrogen be converted into helium, making the Sun die? The actual projected lifetime of the Sun is about 10 billion years, because only the hydrogen in a relatively small core is available as a fuel. (Only in the Suns core are temperatures and densities high enough for the fusion reaction to be self-sustaining).

This experiment will take a little longer to do than most that we have suggested, but the time spent is worthwhile to help you understand the concept of half-life. Obtain a box of sugar cubes and with a pencil make a mark on one side of each of about 200 cubes. Each of these cubes will represent the nucleus of a radioactive substance. Thus, at t  0, you have 200 undecayed nuclei. Now, put the 200 marked cubes in a box and roll them out on a table, just as you would roll dice. Next, count and remove any cubes that have landed marked-side up. These cubes represent nuclei that emitted radiation during the roll. They are no longer radioactive and thus do not participate in the rest of the action. Record the number of undecayed cubes remaining as the number of undecayed nuclei at t  1 roll. Continue rolling, counting, and removing until you have completed 12 to 15 rolls. By then, you should have only a few cubes remaining. Plot a graph of undecayed cubes versus the roll number and from this determine the half-roll of the cubes.

Use a nail to punch a hole in the bottom of a large tin can. Hold the can beneath a faucet and adjust the water ow from the faucet to a ne constant stream. Although water ows from the hole at the bottom, you will note that the level of the water in the can rises. As it does so, however, the ow of water leaving the can increases due to increased water pressure caused by the greater depth of water. Unless the ow of water is too great, an equilibrium point will be reached at which the amount of water owing out of the can each second exactly equals the amount owing in each second. When this happens, the level of water in the can is constant. As noted in the text, carbon-14 is continually being produced in the atmosphere and is also continually disappearing as it decays into nitrogen. What is the analogy between water entering the can, remaining in the can, and owing out of the can and the behavior of carbon-14 in the atmosphere?

Show that the wavelengths for the Balmer series satisfy the equation

The size of the atom in Rutherfords model is about 1.0  1010 m. (a) Determine the attractive electrostatic force between an electron and a proton separated by this distance. (b) Determine (in eV) the electrostatic potential energy of the atom.

The size of the nucleus in Rutherfords model of the atom is about 1.0 fm  1.0  1015 m. (a) Determine the repulsive electrostatic force between two protons separated by this distance. (b) Determine (in MeV) the electrostatic potential energy of the pair of protons.

The size of the atom in Rutherfords model is about 1.0  1010 m. (a) Determine the speed of an electron moving about the proton using the attractive electrostatic force between an electron and a proton separated by this distance. (b) Does this speed suggest that Einsteinian relativity must be considered in studying the atom? (c) Compute the de Broglie wavelength of the electron as it moves about the proton. (d) Does this wavelength suggest that wave effects, such as diffraction and interference, must be considered in studying the atom?

In a Rutherford scattering experiment, an  -particle (charge 
2e) heads directly toward a gold nucleus (charge 
79e). The  -particle had a kinetic energy of 5.0 MeV when very far ( ) from the nucleus. Assuming the gold nucleus to be fixed in space, determine the distance of closest approach. [Hint: Use conservation of energy with PE  keq1q2/r.]

A hydrogen atom is in its rst excited state (n  2). Using the Bohr theory of the atom, calculate (a) the radius of the orbit, (b) the linear momentum of the electron, (c) the angular momentum of the electron, (d) the kinetic energy, (e) the potential energy, and (f) the total energy.

For a hydrogen atom in its ground state, use the Bohr model to compute (a) the orbital speed of the electron, (b) the kinetic energy of the electron, and (c) the electrical potential energy of the atom.

Show that the speed of the electron in the nth Bohr orbit in hydrogen is given by

A photon is emitted as a hydrogen atom undergoes a transition from the n  6 state to the n  2 state. Calculate (a) the energy, (b) the wavelength, and (c) the frequency of the emitted photon.

A hydrogen atom emits a photon of wavelength 656 nm. From what energy orbit to what lower energy orbit did the electron jump?

Following are four possible transitions for a hydrogen atom I. ni  2; nf  5 II. ni  5; nf  3 III. ni  7; nf  4 IV. ni  4; nf  7 (a) Which transition will emit the shortest-wavelength photon? (b) For which transition will the atom gain the most energy? (c) For which transition(s) does the atom lose energy?

What is the energy of a photon that, when absorbed by a hydrogen atom, could cause (a) an electronic transition from the n  3 state to the n  5 state and (b) an electronic transition from the n  5 state to the n  7 state?

A hydrogen atom initially in its ground state (n  1) absorbs a photon and ends up in the state for which n  3. (a) What is the energy of the absorbed photon? (b) If the atom eventually returns to the ground state, what photon energies could the atom emit?

Determine both the longest and the shortest wavelengths in (a) the Lyman series (nf  1) and (b) the Paschen series (nf  3) of hydrogen.

Show that the speed of the electron in the rst (ground-state) Bohr orbit of the hydrogen atom may be expressed as

A monochromatic beam of light is absorbed by a collection of ground-state hydrogen atoms in such a way that six different wavelengths are observed when the hydrogen relaxes back to the ground state. What is the wavelength of the incident beam?

A particle of charge q and mass m, moving with a constant speed v, perpendicular to a constant magnetic eld, B, follows a circular path. If in this case the angular momentum about the center of this circle is quantized so that , show that the allowed radii for the particle are where n  1, 2, 3, . . .

(a) If an electron makes a transition from the n  4 Bohr orbit to the n  2 orbit, determine the wavelength of the photon created in the process. (b) Assuming that the atom was initially at rest, determine the recoil speed of the hydrogen atom when this photon is emitted.

Consider a large number of hydrogen atoms, with electrons all initially in the n  4 state. (a) How many different wavelengths would be observed in the emission spectrum of these atoms? (b) What is the longest wavelength that could be observed? To which series does it belong?

Analyze the EarthSun system by following the Bohr model, where the gravitational force between Earth (mass m) and Sun (mass M) replaces the Coulomb force between the electron and proton (so that F  GMm/r2 and PE GMm/r). Show that (a) the total energy of the Earth in an orbit of radius r is given by (a) E GMm/2r, (b) the radius of the nth orbit is given by rn  r0n2, where r0  2/GMm2  2.32  10138 m, and (c) the energy of the nth orbit is given by En E0/n2, where E0  G2M2m3/22  1.71  10182 J. (d) Using the EarthSun orbit radius of r  1.49  1011 m, determine the value of the quantum number n. (e) Should you expect to observe quantum effects in the EarthSun system?

An electron is in the nth Bohr orbit of the hydrogen atom. (a) Show that the period of the electron is T  ton3, and determine the numerical value of to. (b) On the average, an electron remains in the n  2 orbit for about 10  s before it jumps down to the n  1 (ground-state) orbit. How many revolutions does the electron make before it jumps to the ground state? (c) If one revolution of the electron is dened as an electron year (analogous to an Earth year being one revolution of the Earth around the Sun), does the electron in the n  2 orbit live very long? Explain. (d) How does the above calculation support the electron cloud concept?

Consider a hydrogen atom. (a) Calculate the frequency f of the n  2 : n  1 transition, and compare it with the frequency forb of the electron orbital motion in the n  2 state. (b) Make the same calculation for the n  10000 : n  9 999 transition. Comment on the results.

Two hydrogen atoms collide head-on and end up with zero kinetic energy. Each then emits a 121.6-nm photon (n  2 to n  1 transition). At what speed were the atoms moving before the collision?

Two hydrogen atoms, both initially in the ground state, undergo a head-on collision. If both atoms are to be excited to the n  2 level in this collision, what is the minimum speed each atom can have before the collision?

(a) Calculate the angular momentum of the Moon due to its orbital motion about the Earth. In your calculation, use 3.84  108 m as the average EarthMoon distance and 2.36  106 s as the period of the Moon in its orbit. (b) If the angular momentum of the moon obeys Bohrs quantization rule ( ), determine the value of the quantum number n. (c) By what fraction would the EarthMoon radius have to be increased to increase the quantum number by 1?

(a) Find the energy of the electron in the ground state of doubly ionized lithium, which has an atomic number Z  3. (b) Find the radius of its ground-state orbit.

(a) Construct an energy level diagram for the He
ion, for which Z  2. (b) What is the ionization energy for He
?

The orbital radii of a hydrogen-like atom is given by the equation . What is the radius of the rst Bohr orbit in (a) He
, (b) Li2
, and (c) Be3
?

(a) Substitute numerical values into Equation 28.19 to nd a value for the Rydberg constant for singly ionized helium, He
. (b) Use the result of part (a) to nd the wavelength associated with a transition from the n  2 state to the n  1 state of He
. (c) Identify the region of the electromagnetic spectrum associated with this transition.

Determine the wavelength of an electron in the third excited orbit of the hydrogen atom, with n  4.

Using the concept of standing waves, de Broglie was able to derive Bohrs stationary orbit postulate. He assumed that a confined electron could exist only in states where its de Broglie waves form standing-wave patterns, as in Figure 28.10a. Consider a particle confined in a box of length L to be equivalent to a string of length L and fixed at both ends. Apply de Broglies concept to show that (a) the linear momentum of this particle is quantized with p  mv  nh/2L and (b) the allowed states correspond to particle energies of En  n2E0, where .

List the possible sets of quantum numbers for electrons in the 3p subshell.

When the principal quantum number is n  4, how many different values of (a)
and (b) m
are possible?

The  -meson has a charge of e, a spin quantum number of 1, and a mass 1 507 times that of the electron. If the electrons in atoms were replaced by  -mesons, list the possible sets of quantum numbers for  -mesons in the 3d subshell.

(a) Write out the electronic conguration of the ground state for oxygen (Z  8). (b) Write out the values for the set of quantum numbers n,
, m
, and ms for each of the electrons in oxygen.

Two electrons in the same atom have n  3 and
 1. (a) List the quantum numbers for the possible states of the atom. (b) How many states would be possible if the exclusion principle did not apply to the atom?

How many different sets of quantum numbers are possible for an electron for which (a) n  1, (b) n  2, (c) n  3, (d) n  4, and (e) n  5? Check your results to show that they agree with the general rule that the number of different sets of quantum numbers is equal to 2n2.

Zirconium (Z  40) has two electrons in an incomplete d subshell. (a) What are the values of n and
for each electron? (b) What are all possible values of m
and ms? (c) What is the electron conguration in the ground state of zirconium?

The K-shell ionization energy of copper is 8 979 eV. The L-shell ionization energy is 951 eV. Determine the wavelength of the K  emission line of copper. What must the minimum voltage be on an x-ray tube with a copper target in order to see the K  line?

The K  x-ray is emitted when an electron undergoes a transition from the L shell (n  2) to the K shell (n  1). Use the method illustrated in Example 28.5 to calculate the wavelength of the K  x-ray from a nickel target (Z  28).

When an electron drops from the M shell (n  3) to a vacancy in the K shell (n  1), the measured wavelength of the emitted x-ray is found to be 0.101 nm. Identify the element.

The K series of the discrete spectrum of tungsten contains wavelengths of 0.018 5 nm, 0.020 9 nm, and 0.021 5 nm. The K-shell ionization energy is 69.5 keV. Determine the ionization energies of the L, M, and N shells. Sketch the transitions that produce the above wavelengths.

In a hydrogen atom, what is the principle quantum number of the electron orbit with a radius closest to 1.0  m?

(a) How much energy is required to cause an electron in hydrogen to move from the n  1 state to the n  2 state? (b) If the electrons gain this energy by collision between hydrogen atoms in a high-temperature gas, nd the minimum temperature of the heated hydrogen gas. The thermal energy of the heated atoms is given by 3kBT/2, where kB is the Boltzmann constant.

A pulsed ruby laser emits light at 694.3 nm. For a 14.0-ps pulse containing 3.00 J of energy, nd (a) the physical length of the pulse as it travels through space and (b) the number of photons in it. (c) If the beam has a circular cross section 0.600 cm in diameter, nd the number of photons per cubic millimeter.

The Lyman series for a (new?) one-electron atom is observed in a distant galaxy. The wavelengths of the rst four lines and the short-wavelength limit of this Lyman series are given by the energy-level diagram in Figure P28.47. Based on this information, calculate (a) the energies of the ground state and rst four excited states for this oneelectron atom and (b) the longest-wavelength (alpha) lines and the short-wavelength series limit in the Balmer series for this atom.

A dimensionless number that often appears in atomic physics is the ne-structure constant , where ke is the Coulomb constant. (a) Obtain a numerical value for 1/  . (b) In terms of  , what is the ratio of the Bohr radius a0 to the Compton wavelength  C  h/mec ? (d) In terms of  , what is the ratio of the reciprocal of the Rydberg constant 1/RH to the Bohr radius?

Mercurys ionization energy is 10.39 eV. The three longest wavelengths of the absorption spectrum of mercury are 253.7 nm, 185.0 nm, and 158.5 nm. (a) Construct an energy-level diagram for mercury. (b) Indicate all emission lines that can occur when an electron is raised to the third level above the ground state. (c) Disregarding recoil of the mercury atom, determine the minimum speed an electron must have in order to make an inelastic collision with a mercury atom in its ground state.

Suppose the ionization energy of an atom is 4.100 eV. In this same atom, we observe emission lines that have wavelengths of 310.0 nm, 400.0 nm, and 1 378 nm. Use this information to construct the energy-level diagram with the least number of levels. Assume the higher energy levels are closer together.

A laser used in eye surgery emits a 3.00-mJ pulse in 1.00 ns, focused to a spot 30.0  m in diameter on the retina. (a) Find (in SI units) the power per unit area at the retina. (This quantity is called the irradiance.) (b) What energy is delivered per pulse to an area of molecular sizesay, a circular area 0.600 nm in diameter.

An electron has a de Broglie wavelength equal to the diameter of a hydrogen atom in its ground state. (a) What is the kinetic energy of the electron? (b) How does this energy compare with the groundstate energy of the hydrogen atom?

Use Bohrs model of the hydrogen atom to show that, when the atom makes a transition from the state n to the state n  1, the frequency of the emitted light is given by

Calculate the classical frequency for the light emitted by an atom. To do so, note that the frequency of revolution is v/2  r, where r is the Bohr radius. Show that as n approaches innity in the equation of the preceding problem, the expression given there varies as 1/n3 and reduces to the classical frequency. (This is an example of the correspondence principle, which requires that the classical and quantum models agree for large values of n.)

A pi meson (  ) of charge e and mass 273 times greater than that of the electron is captured by a helium nucleus (Z 
2) as shown in Figure P28.55. (a) Draw an energy-level diagram (in units of eV) for this Bohr-type atom up to the rst six energy levels. (b) When the  -meson makes a transition between two orbits, a photon is emitted that Compton scatters off a free electron initially at rest, producing a scattered photon of wavelength  0.089 929 3 nm at an angle of   42.68, as shown on the right-hand side of Figure P28.55. Between which two orbits did the  -meson make a transition?

When a muon with charge e is captured by a proton, the resulting bound system forms a muonic atom, which is the same as hydrogen, except with a muon (of mass 207 times the mass of an electron) replacing the electron. For this muonic atom, determine (a) the Bohr radius and (b) the three lowest energy levels.

In this problem, you will estimate the classical lifetime of the hydrogen atom. An accelerating charge loses electromagnetic energy at a rate given by   2keq2a2/(3c3), where ke is the Coulomb constant, q is the charge of the particle, a is its acceleration, and c is the speed of light in a vacuum. Assume that the electron is one Bohr radius (0.052 9 nm) from the center of the hydrogen atom. (a) Determine its acceleration. (b) Show that  has units of energy per unit time and determine the rate of energy loss. (c) Calculate the kinetic energy of the electron and determine how long it will take for all of this energy to be converted into electromagnetic waves, assuming that the rate calculated in part (b) remains constant throughout the electrons motion.

An electron in a hydrogen atom jumps from some initial Bohr orbit ni to some nal Bohr orbit nf, as in Figure P28.58. (a) If the photon emitted in the process is capable of ejecting a photoelectron from tungsten (work function  4.58 eV), determine nf. (b) If a minimum stopping potential of V0  7.51 volts is required to prevent the photoelectron from hitting the anode, determine the value of ni.