1. Compute the approximate nuclear radius of carbon (A = 12), germanium (A = 70), and bismuth (A = 209). Use radius constant r0 = 1.2 fm. 2. Compute the density of a typical nucleus, and find the resultant mass if we could produce a nucleus with a radius of 1 cm. 3. Find the total binding energy B and also the average binding energy per nucleon B/A for 2 24 56 238 2 24 56 1H , 2H e , 26 Fe and 92U . The nuclear masses of 1 H , 2 H e , 26Fe and 238 92
U
are 55.934937 a.m.u., 238.050788 a.m.u., respectively.
4. Find the proton separation energy and the neutron separation energy of 125Te. The nuclear 124 125 124 51 Sb , 52Te 52Te masses of and are 123.905936 a.m.u., 124.904431 a.m.u. and 123.902818
a.m.u., respectively.
5. (a) Compute the Coulomb repulsion energy between two nuclei of 16O that just touch at their surfaces. (b) Do the same for two nuclei of 238U. 6. Find the nuclear radius of (a) 197Au; (b) 4He; (c) 20Ne. 7. Find the total binding energy, and the binding energy per nucleon, for (a) 90 Zr; (d) 59Co.
208
Pb; (b)
133
Cs; (c)
8. Find the total binding energy, and the binding energy per nucleon, for (a) 4He; (b) 20Ne; (c) 40 Ca; (d) 55Mn. 9. Calculate the total nuclear binding energy of 3He and 3H. Account for any difference by considering the Coulomb interaction of the extra proton of 3He. 10. Find the neutron separation energy of: (a) 17O; (b) 7Li; (c) 57Fe. 11. Find the proton separation energy of: (a) 4He; (b) 12C; (c) 40Ca. Note - 1. The proton separation energy for A Z
A Z
A−1
X → Z−1Y + p is Sp = [m(
A−1 Z−1
Y ) + m(p) − m(
X )]c2.
2. The proton separation energy for
A Z
X→
A−1 Z
Y +n is Sn = [m(
A−1 Z
Y ) + m(n) − m(
A Z
X )]c2
3. The masses of proton and neutron are 1.007825 a.m.u. and 1.008665 a.m.u. respectively.
Answer Keys 1. 2.7 fm, 4.9 fm, 7.1 fm 2. 2 × 1017 kg/m3, 8 × 1011 kg 3. 492.3 MeV, 8.790 MeV per nucleon, 1802 MeV, 7.570 MeV per nucleon 4. 8.691 MeV, 6.569 MeV
Radioactive Decay The half-life of 198Au is 2.70 days. (a) What is the decay constant of 198Au? (b) What is the probability that any 198Au nucleus will decay in one second? (c) Suppose we had a 1.00-μg sample of 198Au. What is its activity? (d) How many decays per second occur when the sample is one week old? 2.97 × 10−6 s−1, 2.97 × 10−6, 0.244 Ci, 1.50 × 109 Bq
The half-life of 235U is 7.04 × 108 y. A sample of rock, which solidified with the Earth 4.55 × 109 years ago, contains N atoms of 235U. How many 235U atoms did the same rock have at the time it solidified? 88.2 N
Find the kinetic energy of the alpha particle emitted in the alpha decay process 226Ra → 4He. Nuclear masses of 226Ra, 222Rn and 4He are 226.025410 u, 222.017578 u, 4.002603 u, respectively.
222
Rn +
4.785 MeV
The nucleus 226Ra decays by alpha emission with a half-life of 1600 y. It also decays by emitting 14 C. Find the Q value for 14C emission and compare with that for alpha emission. 28.197 MeV 14. What fraction of the original number of nuclei present in a sample will remain after ( a) 2 half-lives; (b) 4 half-lives; (c) 10 half-lives? 1 4 ,
1 16 ,
1 =0.000997 1024
15. A certain sample of a radioactive material decays at a rate of 548 per second at t = 0. At t = 48 minutes, the counting rate has fallen to 213 per second. (a) What is the half-life of the radioactivity? (b) What is its decay constant? (c) What will be the decay rate at t = 125 minutes? 35 min, 0.020 min-1, 46 s-1 16. What is the decay probability per second per nucleus of a substance with a half-life of 5.0 hours?
3.9 x 10-5 s 17. Tritium, the hydrogen isotope of mass 3, has a half-life of 12.3 y. What fraction of the tritium atoms remains in a sample after 50.0 y? 0.0598 18. Suppose we have a sample containing 2.00 mCi of radioactive 131I (t1/2 = 8.04 d). (a) How many decays per second occur in the sample? (b) How many decays per second will occur in the sample after four weeks? 7.40×10 s-1, 6.62 ×10 s-1 19. Ordinary potassium contains 0.012 percent of the naturally occurring radioactive isotope 40K, which has a half-life of 1.3 x 109 y. (a) What is the activity of 1.0 kg of potassium? (b) What would have been the fraction of 40K in natural potassium 4.5 × 109 y ago? 0.85 Ci, 0.13% 20. For which of the following nuclei is alpha decay permitted? (a) 210Bi; (b) 203Hg; (c) 211At. +5.04 MeV, −0.30 MeV, +5.98 MeV 22. Find the kinetic energy of the alpha particle emitted in the decay of 234U. 4.776 MeV
1. The nuclear force (1) has infinite range, like the electromagnetic or gravitational force.
(2) becomes infinite in strength as the distance between two particles approaches zero. (3) is exerted by each proton or neutron on only its nearest neighbors. (4) is exerted by each proton or neutron on all other protons or neutrons in the nucleus. 2. The binding energy of a nucleus is: (1) the energy needed to remove one proton or neutron. (2) the energy needed to take apart a nucleus into its constituent protons and neutrons. (3) the energy with which the nucleus attracts the atomic electrons. (4) the energy equivalent of the mass of the nucleus. 3. The number of neutrons in a nucleus is: (1) always equal to the number of protons. (2) usually greater than the number of protons. (3) usually smaller than the number of protons. (4) always equal to the number of electrons in the atom. 4. In which type of decay process is the total number of protons before the decay not equal to the total number of protons after the decay? (1) Alpha decay (2) Beta decay (3) Gamma decay (4) All of the decay processes (5) None of the decay processes 5. Which of the following is allowed to change in a radioactive decay process? (1) Total energy (2) Total number of nucleons (protons plus neutrons) (3) Total electric charge (4) Total number of electrons 6. Suppose we combine two nuclei of Ca to make a single nucleus of Zr. (a) Compared with one of the original Ca nuclei, the newly formed nucleus of Zr will have: (1) twice the radius (2) twice the surface area (3) twice the volume (b) If B represents the total binding energy of a Ca nucleus, then the total binding energy of a Zr nucleus is approximately (1) 0.5B (2) B (3) 2B (4) 4B (5) 8B 7. For two protons separated by a distance of about 1 fm (a typical separation in a nucleus), the attractive nuclear (strong) force is stronger than the Coulomb repulsion force. Why then do nuclei need neutrons? Why don’t we find nuclei with Z protons and zero neutrons? 8. Light nuclei generally have N ≈ Z, but more massive nuclei have N ≈ 1.5Z. That is, as nuclei become more massive the number of neutrons increases more rapidly than the number of protons. Why?
9. Choose from among the following decay processes: (1) Alpha decay (2) All beta decays (3) Negative beta decay (4) Positive beta decay (5) Negative or positive beta decay (6) Electron capture decay (7) Gamma decay (8) Alpha and gamma decays (9) Alpha, electron capture, and gamma decays (10) Alpha, beta, and gamma decays (a) In which type of decay might we expect to see bremsstrahlung? (b) Which type of decay is accompanied by monoenergetic X rays? (c) Which type of decay is accompanied by two 0.511-MeV photons? (d) In which type of decay is a monoenergetic particle emitted? (e) In which type of decay is new matter created? (f) Assuming the same Q value, in which type of decay would the residual nucleus have the largest kinetic energy? 1. A sample contains a large number of radioactive nuclei. At any instant of time, the rate of decay is: (a) directly proportional to the number of nuclei that have already decayed. (b) directly proportional to the number of nuclei that have not yet decayed. (c) constant in time. (d) directly proportional to the half-life of the decay. 2. The nuclear force: (a) has infinite range. (b) is generally stronger than the electromagnetic force. (c) becomes infinite as the distance between particles approaches zero. (d) acts on electrons that may be inside the nucleus. 3. Nucleus A has a half-life T and nucleus B has a half-life 2T. Initially the number of nuclei of type A equals the number of nuclei of type B. After a certain time, 10% of the nuclei of type B remain. At this same time, what fraction of the nuclei of type A remains? (a) 5% (b) 1% (c) 0.01% (d) 20% (e) 50% 4. The energy necessary to remove a proton or a neutron from a nucleus is typically about (a) 1 MeV (b) 10 MeV (c) 100 MeV (d) 1000 MeV 5. 5. Two nuclei of 40Ca (atomic number 20) undergo fusion to form a nucleus of 80Zr (atomic number 40). The total binding energy of 40Ca is B. What would be the best estimate for the total binding energy of 80Zr? (a) 4B (b) 2B (c) B (d) B/2 (e) B/4
1. A nucleus of 4He absorbs a photon of energy E which causes it to split apart into two nuclei of 2H. The two 2H nuclei fly apart with kinetic energies K1 and K2. Is the total final kinetic energy K (which is equal to K1 + K2) greater than, less than, or equal to the photon energy E? EXPLAIN YOUR ANSWER. 2. Suppose we can break apart a nucleus of 48 24 24 Cr in two different ways: 2 nuclei of 24
Mg or 3 nuclei of 16 8 8 O . Is the amount of energy required to break it into two 12 12 24
Mg greater than, less than, or equal to the amount of energy required to break it into three 16 8 8 O ? EXPLAIN YOUR ANSWER. 3. You wish to obtain a supply of 20 free neutrons and 20 free protons. You have available either one nucleus of 40Ca (Z = 20, N = 20) or two nuclei of 20Ne (Z = 10, N = 10). Will the energy necessary to obtain the 20 neutrons and 20 protons from one 40Ca nucleus be less than, greater than, or equal to the energy necessary to obtain the same number of neutrons and protons from the two 20Ne nuclei? EXPLAIN YOUR ANSWER. 4. 224Ra can decay either by alpha emission or by 12C emission. The probability for alpha emission is about 109 greater than the probability for 12C emission. How would you explain this great difference? 12 12
(a) Natural uranium today consists of about 0.7% of the isotope 235U (half life = 7.1 × 108 y) and 99.3 % of 238U (half life = 4.5 × 109 y). At some time in the past, natural uranium would have contained 3.0% of 235U, enough to make natural watermoderated fission reactors. How long before the present time did this occur? (b) Compute the binding energy per nucleon of 238U (atomic number = 92, atomic mass = 238.050784 u).
