Concepts & Calculations

Note: Each of these problems consists of Concept Questions followed by a related quantitative Problem. The Concept Questions involve little or no mathematics. They focus on the concepts with which the problems deal. Recognizing the concepts is the essential initial step in any problem-solving technique.

| 57 | Concept Questions (a) In a nucleus, each proton experirences a repulsive electrostatic force from each of the other protons. Write an expression for the magnitude of the force that one proton (charge ) applies to another proton that is located a distance r away. (b) The force that acts on either of two particular protons in the nucleus has the smallest possible magnitude. Relative to one another, where in the nucleus must these two protons be located? Explain.
Problem In a nucleus of gold , what is the magnitude of the least possible electrostatic force of repulsion that one proton can exert on another?

| 58 | Concept Questions(a) How is the total released energy related to the decrease in mass that accompanies decay? (b) When a parent nucleus undergoes decay, a daughter nucleus, a particle, and an antineutrino are produced. The released energy is shared among these three particles. Assume that the kinetic energy of the particle is known and that the antineutrino carries away the maximum possible energy. What then can you say about the kinetic energy of the recoiling daughter nucleus?
Problem The decay of phosphorus (atomic mass = 31.973 907 u) produces a daughter nucleus that is sulfur (atomic mass = 31.972070 u), a particle, and an antineutrino. The kinetic energy of the particle is 0.90 MeV. Find the maximum possible energy that the antineutrino could carry away.

| 59 | Concept Questions (a) Two radioactive nuclei A and B have half-lives of and , where is greater than . During the same time period, is the fraction of nuclei A that decay greater than, smaller than, or the same as the fraction of nuclei B that decay? (b) The numbers of these nuclei present initially are and , the ratio of the two being . Is the ratio of the number of nuclei present at a later time greater than, smaller than, or the same as ? Justify your answers.
Problem Two waste products from nuclear reactors are strontium () and cesium (). These two species are present initially in a ratio of . What is the ratio fifteen years later? Verify that your answer is consistent with your answers to the Concept Questions.

| 60 | Concept Questions(a) Write an expression for the heat Q needed to melt a mass m of water. (b) When a nucleus disintegrates via decay, how is the released energy related to the decrease in mass that accompanies the decay? (c) During decay, the energy released by each disintegration is E. What is the total energy ETotal released by a number n of disintegrations? (d) If the number of radioactive nuclei present initially is N0, what is the number of disintegrations that occur in one half-life?
Problem A one-gram sample of radium (atomic mass = 224.020186 u, ) contains nuclei and undergoes decay to produce radon (atomic mass = 220.011 368 u). The atomic mass of an particle is 4.002 603 u. With the energy released in 3.66 days, how many kilograms of ice could be melted at 0°C?

| 61 | Concept Questions Outside the nucleus, the neutron itself is radioactive and decays into a proton, an electron, and an antineutrino. (a) Suppose that, originally, a number N0 of neutrons are outside the nucleus. What is meant by the statement “The half-life of the neutron is X minutes”? (b) How is the number of neutrons remaining at any time t related to the original number and the half-life of the neutron? (c) Suppose each of the neutrons is moving with the same kinetic energy. How is the speed of each neutron related to its kinetic energy? Ignore relativistic effects.
Problem The half-life of a neutron () outside the nucleus is 10.4 min. On average, over what distance (in meters) would a beam of 5.00-eV neutrons travel before the number of neutrons decreases to 75.0% of its initial value? Ignore relativistic effects.

| 62 | Concept Questions (a) Physically, what does the binding energy of a nucleus represent? (b) What is the relationship between the binding energy of a nucleus and its mass defect? (c) How is the mass defect of a nucleus related to the mass of the intact nucleus and the masses of the individual nucleons? (d) A piece of metal, such as a coin, contains a certain number of atoms, and, hence, nuclei. How much energy would be required to break all the nuclei into their constituent protons and neutrons? Give your answer in terms of the number of atoms in the coin and the binding energy of each nucleus. (e) How is the number of atoms in a piece of metal related to its mass? Express your answer in terms of the atomic mass of the atoms and Avogadro’s number.
Problem A copper penny has a mass of 3.0 g. Determine the energy (in MeV) that would be required to break all the copper nuclei into their constituent protons and neutrons. Ignore the energy that binds the electrons to the nucleus and the energy that binds one atom to another in the structure of the metal. For simplicity, assume that all the copper nuclei are (atomic mass = 62.939 598 u).

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