A sample of substance that has an atomic mass of and emits

Isotopes of nitrogen, iron and tin have stable isotopes and Give the symbols for two other isotopes of (a) nitrogen, (b) iron, and (c) tin.

Why is the decay chain not found in nature?

A decay by emission is often followed by decay. When this occurs, it is by and not decay. Why?

The half-life of is much less than the age of the universe, yet is found in nature. Why?

What effect would a long-term variation in cosmic-ray activity have on the accuracy of dating?

Why does an element that has not exist?

Why is a moderator needed in an ordinary nuclear fission reactor?

Explain why water is more effective than lead in slowing down fast neutrons.

The stable isotope of sodium is What kind of beta decay would you expect of (a) and (b)

What is the advantage of a breeder reactor over an ordinary reactor? What are the disadvantages of a breeder reactor?

True or false: (a) In a breeder reactor, fuel can be produced as fast as it is consumed. 22Na 24Na? (b) The atomic nucleus is composed of protons, neutrons, and electrons. (c) The mass of a nucleus is less than the mass of a nucleus plus the mass of a neutron. (d) After two half-lives, all the radioactive nuclei in a given sample have decayed.

Why is it that extreme changes in the temperature or the pressure of a radioactive sample have little or no effect on the radioactivity?

We found in Chapter 25 that the ratio of the resistivity of the most insulating material to the resistivity of the least resistive material (excluding superconductors) is approximately Few properties of materials show such a wide range of values. Using information in the textbook or other resources, find the ratio of largest to smallest for some nuclear properties of matter. Some examples might be the range of mass densities found in an atom, the half-life of radioactive nuclei, or the range of nuclear masses.

According to the United States Department of Energy, the U.S. population consumes approximately joules of energy each year. Estimate the mass (in kilograms) of (a) uranium that would be needed to produce this much energy using nuclear fission and (b) deuterium and tritium that would be needed to produce this much energy using nuclear fusion.

Calculate the binding energy and the binding energy per nucleon from the masses given in Table 40-1 for (a) (b) and (c)

Calculate the binding energy and the binding energy per nucleon from the masses given in Table 40-1 for (a) (b) and (c)

Use the radius formula (Equation 40-1), where to compute the radii of the following nuclei: (a) (b) and (c)

During a fission process, a nucleus splits into two nuclei whose mass number ratio is 3 to 1. Calculate the radii of the nuclei formed during the process.

The neutron, when isolated from an atomic nucleus, decays into a proton, an electron, and an antineutrino as follows: The thermal energy of a neutron is of the order of where is the Boltzmann constant. (a) In both joules and electron volts, calculate the energy of a thermal neutron at (b) What is the speed of that thermal neutron? (c) A beam of monoenergetic thermal neutrons is produced at and has an intensity After traveling the beam has an intensity of Using this information, estimate the half-life of the neutron. Express your answer in minutes.

Use (Equation 40-1), where for the radius of a spherical nucleus to calculate the density of nuclear matter. Express your answer in grams per cubic centimeter.

In 1920, 12 years before the discovery of the neutron, Ernest Rutherford argued that protonelectron pairs might exist in the confines of the nucleus in order to explain the mass number, being greater than the nuclear charge, He also used this argument to account for the source of beta particles in radioactive decay. Rutherfords scattering experiments in 1910 showed that the nucleus had a diameter of approximately Using this nuclear diameter, the uncertainty principle, and that beta particles have an energy range of to show why the hypothetical electrons cannot be confined to a region occupied by the nucleus.

Consider the following fission process: Determine the electrostatic potential energy, in of the reaction products when the surfaces of the nucleus and the nucleus are just touching immediately after being formed during the fission process.

Homer enters the visitors chambers, and his Geiger beeper sounds. He shuts off the beeper, removes the device from his shoulder patch, and holds it near the only new object in the rooman orb that is to be presented as a gift from the visiting Cartesians. Pushing a button marked monitor, Homer reads that the device is reading a counting rate of above the background counting rate. After the counting rate has dropped to above the background rate. (a) What is the half-life of the source? (b) How high will the counting rate be (above the background counting rate) after the monitoring device was switched on?

A certain source gives at time Its half-life is How many counts per second will it give after (a) 4.0 min, (b) 6.0 min, and (c) 8.0 min?

The counting rate from a radioactive source is at time and later the rate is (a) What is the half-life? (b) What is the decay constant? (c) What is the counting rate after

The half-life of radium is Calculate the number of disintegrations per second of of radium and show that the disintegration rate is approximately

A radioactive piece of silver foil is placed near a Geiger counter and are observed at time (a) What is the counting rate at and at (b) If the counting efficiency is 20 percent, how many radioactive silver nuclei are there at time At time (c) At what time will the counting rate be about

Use Table 40-1 to calculate the energy release, in for the decay of (a) and (b)

Plutonium is very toxic to the human body. Once it enters the body it collects primarily in the bones, although it also can be found in other organs. Red blood cells are synthesized within the marrow of the bones. The isotope is an alpha emitter that has a half-life of Because alpha particles are an ionizing radiation, the blood-making ability of the marrow is, in time, destroyed by the presence of In addition, many kinds of cancers will also develop in the surrounding tissues because of the ionizing effects of the alpha particles. (a) If a person accidentally ingested of and all of it is absorbed by the bones of the person, how many alpha particles are produced per second within the body of the person? (b) When, in years, will the activity be 1000 alpha particles per second?

Consider an alpha-emitting parent nucleus initially at rest. The nucleus decays into a daughter nucleus Y and an alpha particle as follows: (a) Show that the alpha particle has a kinetic energy of (b) Show that the kinetic energy of the recoiling daughter nucleus is given by

The fissile material is an alpha emitter. Write the reaction that describes undergoing alpha decay. Given that and an alpha particle have respective masses of and use the relations appearing in Problem 30 to calculate the kinetic energies of the alpha particle and the recoiling daughter nucleus.

Through a friend in the security department at the museum, Angela obtains a sample of a wooden tool handle that contains of carbon. The decay rate of the in the sample is How long ago was the wood in the handle last alive?

A sample of a radioactive isotope is found to have an activity of immediately after it is pulled from the reactor that formed the isotope. Its activity later is measured to be (a) Calculate the decay constant and the half-life of the sample. (b) How many radioactive nuclei were there in the sample initially?

A sample of substance that has an atomic mass of and emits particles has an activity of Find the decay constant for the substance in reciprocal seconds and find the half-life in years.

Radiation has been used for a long time in medical therapy to control the development and growth of cancer cells. Cobalt-60, a gamma emitter that emits photons that have energies of and is used to irradiate and destroy deep-rooted cancers. Small needles made of of a specified activity are encased in gold and used as body implants in tumors for time periods that are related to tumor size, tumor cell reproductive rate, and the activity of the needle. (a) A 1.00-mg sample of 60Co, that has a half-life of and that is used to irradiate a small internal tumor with gamma rays, is prepared in the cyclotron of a medical center. Determine the activity of the sample in curies. (b) What will the activity of the sample be from now?

(a) Show that if the decay rate is at time and at some later time the decay constant is given by and the half-life is given by (b) Use these results to find the decay constant and the half-life if the decay rate is at and at

A wooden casket is thought to be 18 000 years old. How much carbon would have to be recovered from the object to yield a counting rate of no less than with a detection efficiency of 20 percent?

A sample of radioactive material is initially found to have an activity of After its activity is measured to be (a) Calculate the half-life of the material. (b) How long (from the initial time) will it take for the sample to reach an activity of

The rubidium isotope is a emitter that has a halflife of It decays into This nuclear decay is used to determine the age of rocks and fossils. Rocks containing the fossils of early animals have a ratio of to of 0.0100. Assuming that there was no present when the rocks were formed, calculate the age of the fossils.

Consider a single nucleus of a radioactive isotope that has a decay rate equal to The nucleus has not decayed at The probability that the nucleus will decay between time and time is equal to (a) Show that this statement is consistent with the fact that the probability is 1 that the nucleus will decay between and (b) Show that the expected lifetime of the nucleus is equal to Hint: The expected lifetime is equal to divided by (c) A sample of material contains a number of these radioactive nuclei at time What is the mean lifetime of the radioactive nuclei in the sample?

Using Table 40-1, find the values for the following reactions: (a) and (b)

Using Table 40-1, find the values for the following reactions: (a) (b) and (c)

(a) Use the values and for the atomic masses of \({ }^{14} \mathrm{C}\) and \({ }^{14} \mathrm{N}\) respectively, to calculate the Q value (in MeV) for the \(\beta \text {-decay }\) reaction \({ }_{6}^{14} \mathrm{C} \rightarrow{ }_{7}^{14} \mathrm{N}+\mathrm{e}^{-}+\bar{v}_{\mathrm{e}}\) (b) Explain why you should not add the mass of the electron to that of atomic \({ }^{14} \mathrm{N}\) for the calculation in Part (a)

(a) Use the values and for the atomic masses of and respectively, to calculate the value (in MeV) for the -decay reaction (b) Explain why you need to add twice the mass of an electron to the mass of during the calculation of the value for the reaction in Part (a).

Assuming an average energy of per fission, calculate the number of fissions per second needed for a reactor. SSM

If the reproduction factor in a reactor is 1.1, find the number of generations needed for the power level to (a) double, (b) increase by a factor of 10, and (c) increase by a factor of 100. Find the time needed in each case if (d) there are no delayed neutrons, so that the time between generations is and (e) there are delayed neutrons that make the average time between generations

Consider the following fission reaction: The masses of the neutron, and are and respectively. Calculate the value, in for the fission reaction.

In 1989, researchers claimed to have achieved fusion in an electrochemical cell at room temperature. Their now thoroughly discredited claim was that a power output of was produced by deuterium fusion reactions in the palladium electrode of their apparatus. The two most likely reactions are and Of the deuterium nuclei that participated in these reactions, assume half of the deuterium nuclei participated in the first reaction and the other half participated in the second reaction. How many neutrons per second would we expect to be emitted in the generation of of power?

A fusion reactor that uses only deuterium for fuel would have the two reactions in Problem 48 taking place in the reactor. The produced in the second reaction reacts immediately with another in the reaction The ratio of to atoms in naturally occurring hydrogen is How much energy would be produced from of water if all of the nuclei undergo fusion?

The fusion reaction between and is Using the conservation of momentum and the given value, find the final energies of both the nucleus and the neutron, assuming the initial kinetic energy of the system is and the initial momentum of the system is zero.

Energy is generated in the Sun and other stars by fusion. One of the fusion cycles, the protonproton cycle, consists of the following reactions: followed by (a) Show that the net effect of these reactions is (b) Show that is released during this cycle (not counting the additional energy of that is released when each positron meets an electron and the two annihilate). (c) The Sun radiates energy at the rate of approximately Assuming that this is due to the conversion of four protons into helium, rays, and neutrinos, which releases what is the rate of proton consumption in the Sun? How long will the Sun last if it continues to radiate at its present level? (Assume that protons constitute about half of the total mass, 2.0 _ 1030 kg, of the Sun.)

(a) Show that where is the Coulomb constant and is the magnitude of the electron charge. (b) Show that

The counting rate from a radioactive source is The half-life of the source is Make a plot of the counting rate as a function of time for times up to What is the decay constant for the source?

Find the energy needed to remove a neutron from (a) and (b)

The isotope decays according to The atomic mass of is Determine the maximum kinetic energy of the electron. (Neglect recoil of the nitrogen atom.)

The density of a neutron star is the same as the density of a nucleus. If our Sun were to collapse to a neutron star, what would be the radius of that object?

Show that the nucleus is stable and does not undergo alpha decay, The mass of the nucleus is and the products of the decay are and respectively.

Gamma rays can be used to induce photofission (fission triggered by the absorption of a photon) in nuclei. Calculate the threshold photon wavelength for the following nuclear reaction: Use Table 40-1 for the masses of the interacting particles.

The relative abundance of (potassium 40) is The isotope has a molar mass of is radioactive, and has a half-life of Potassium is an essential element of every living cell. In the human body the mass of potassium constitutes approximately 0.36 percent of the total mass. Determine the activity of this radioactive source in a student whose mass is

When a positron makes contact with an electron, the electronpositron pair annihilate by way of the reaction Calculate the minimum total energy, in of the two photons created when a positronelectron pair annihilate.

The isotope is a emitter and has a half-life of A saline solution containing the radioactive isotope has an activity of and is injected into the bloodstream of a patient. Ten hours later, the activity of of blood from the individual yields a counting rate of at a counting efficiency of 20 percent. Determine the volume of blood in the patient.

(a) Determine the distance of closest approach of an particle in a head-on collision with a stationary nucleus of and with a stationary nucleus of neglecting the recoil of the struck nuclei. (b) Repeat the calculation taking into account the recoil of the struck nuclei.

Twelve nucleons are in a one-dimensional infinite square well of length (a) Using the approximation that the mass of a nucleon is find the lowest energy of a nucleon in the well. Express your answer in MeV. What is the ground-state energy of the system of 12 nucleons in the well if (b) all the nucleons are neutrons so that there can be no more than 2 in each spatial state and (c) 6 of the nucleons are neutrons and 6 are protons so that there can be as many as 4 nucleons in each spatial state? (Neglect the energy of Coulomb repulsion of the protons.)

The helium nucleus or particle is a very tightly bound system. Nuclei with where is an integer (for example, and ), can be modeled as agglomerates of particles. (a) Use this model to estimate the binding energy of a pair of particles from the atomic masses of and Assume that the four particles in form a regular tetrahedron that has one particle at each vertex. (b) From the result obtained in Part (a) determine, on the basis of the model, the binding energy of and compare your result with the result obtained from the atomic mass of

Nuclei of a radioactive isotope that has a decay constant of are produced in an accelerator at a constant rate The number of radioactive nuclei then obeys the equation (a) If is zero at sketch versus for the situation. (b) The isotope is produced at a rate of 100 per second by placing ordinary copper in a beam of high-energy photons. The reaction is The isotope decays by decay and has a half-life of After a time long enough so that how many nuclei are there?

The total energy consumed in the United States in is approximately How many kilograms of would be needed to provide this amount of energy if we assume that of energy is released by each fissioning uranium nucleus, that all of the uranium atoms undergo fission, and that all of the energy- conversion mechanisms used are 100 percent efficient?

(a) Find the wavelength of a particle in the ground state of a one-dimensional infinite square well of length (b) Find the momentum in units of for a particle that has this wavelength. (c) Show that the total energy of an electron that has this wavelength is approximately (d) What is the kinetic energy of an electron in the ground state of the well? This calculation shows that if an electron were confined in a region of space as small as a nucleus, it would have a very large kinetic energy.

If and have respective masses of and determine the minimum energy, in MeV, required to remove a proton from a nucleus.

Assume that a neutron decays into a proton and an electron without the emission of a neutrino. The kinetic energy shared by the proton and the electron is then In the rest frame of the neutron, the total momentum is zero, so the momentum of the proton must be equal and opposite the momentum of the electron. This determines the ratio of the kinetic energies of the two particles, but because the electron is relativistic, the exact calculation of these relative kinetic energies is somewhat challenging. (a) Assume that the kinetic energy of the electron is and calculate the momentum of the electron in units of Hint: Use (Equation 39-27). (b) Using your result from Part (a), calculate the kinetic energy of the proton. (c) Because the total kinetic energy of the electron and the proton is the calculation in Part (b) gives a correction to the assumption that the kinetic energy of the electron is What percentage of is this correction?

In the laboratory reference frame, a neutron of mass moving with speed makes an elastic head-on collision with a nucleus of mass that is at rest. (a) Show that the speed of the center of mass in the lab frame is (b) What is the speed of the nucleus in the center-of-mass frame before the collision and after the collision? (c) What is the speed of the nucleus in the laboratory frame after the collision? (d) Show that the energy of the nucleus after the collision in the laboratory frame is (e) Show that the fraction of the energy lost by the neutron in the elastic collision is

(a) Use the result from Part (e) of Problem 70 to show that after head-on collisions of a neutron with carbon nuclei at rest, the energy of the neutron is approximately where is its original energy. (b) How many head-on collisions are required to reduce the energy of the neutron from to assuming stationary carbon nuclei?

On the average, a neutron loses 63 percent of its energy in a collision with a hydrogen atom and 11 percent of its energy in a collision with a carbon atom. Calculate the number of collisions needed to reduce the energy of a neutron from to if the neutron collides with (a) hydrogen atoms and (b) carbon atoms.

Frequently, the daughter nucleus of a radioactive parent nucleus is itself radioactive. Suppose the parent nucleus, designated by has a decay constant while the daughter nucleus, designated by D, has a decay constant The number of daughter nuclei are then given by the solution to the differential equation where is the number of parent nuclei. (a) Justify this differential equation. (b) Show that the solution for the equation is where is the number of parent nuclei present at when there are no daughter nuclei. (c) Show that the expression for in Part (b) gives whether or (d) Make a plot of and as a function of time when where and are the mean lifetimes of the daughter and parent nuclei, respectively.

Suppose isotope A decays to isotope B and has a decay constant and isotope B in turn decays and has a decay constant Suppose a sample contains, at only isotope A nuclei. Derive an expression for the time at which the number of isotope B nuclei will be a maximum. (See Problem 73.)

An example of the situation discussed in Problem 73 is the radioactive isotope an emitter that has a half-life of Its daughter, is a emitter that has a half-life of In this instance, as in many instances, the half-life of the parent is much longer than the half-life of the daughter. Using the expression given in Problem 73, Part (b), starting with a sample of pure containing nuclei, show that the number, of nuclei will, after several years, be given by where is the number of nuclei. The number of daughter nuclei are said to be in secular equilibrium.