Nuclear Reactions and Radioactivity PDF

Chapter 2 Nuclear Reactions and Radioactivity Compiled by Md. Shamim Hossain Mathematical Problems Curated by Md. Roni Islam Abdullah Al Sakib This book is a compilation of publicly available information from various online and academic sources, intended solely for educational use in PHY102: Phy...

Chapter 2 Nuclear Reactions and Radioactivity Compiled by Md. Shamim Hossain Mathematical Problems Curated by Md. Roni Islam Abdullah Al Sakib This book is a compilation of publicly available information from various online and academic sources, intended solely for educational use in PHY102: Physics II course at Daffodil International University and similar courses at other institutions. It should not be redistributed or republished without proper attribution to the original sources  Chapter 2 Nuclear Reactions and Radioactivity SMHN | AAS | RIM 2.1 Nuclear Reactions: Nuclear reactions are processes in which one or more nuclides are produced from the collisions between two atomic nuclei or one atomic nucleus and a subatomic particle. The nuclides produced from nuclear reactions differ from the reacting nuclei (commonly called parent nuclei). In short, a chain reaction refers to a process in which neutrons released in fission produce an additional fission in at least one further nucleus. This nucleus in turn produces neutrons, and the process repeats. The process may be controlled (nuclear power) or uncontrolled (nuclear weapons). 2.1.1 Chain Reaction: A chain reaction is such a process in which once started the reaction continues without requiring further energy and a tremendous amount of energy is released because of the fission of all nuclei. A nuclear chain reaction occurs when the output of one nuclear reaction causes more nuclear reactions to occur. These chain reactions are almost always a series of fission events, which give off excess neutrons. It is these excess neutrons that can go on to cause more fission events to occur, hence the name chain reaction. Nuclear chain reactions are essential to the operation of nuclear power plants. To sustain a nuclear chain reaction, every fission event must lead to one more fission event. The most convenient nuclear species for nuclear chain reactions is a fissile isotope of uranium, 235U. When 235U undergoes fission, it gives off, on average, ~2.5 neutrons per fission event. Careful engineering must go into having those neutrons go on to create more fission events. Contrary to what one may expect, difficulties arise in getting enough neutrons to go on and make a sustained nuclear reaction, rather than having too many nuclear reactions. If every fission event leads to exactly one more fission event, the nuclear chain reaction is said to be critical. Figure 1 shows a simplification of the fission chain reaction. Figure 1. A nuclear fission chain reaction of uranium-235 atoms. In a real nuclear reactor, most of the released neutrons are lost, rather than leading to another fission event. Two notable types of nuclear reactions are nuclear fission reactions and nuclear fusion reactions. 1  Chapter 2 Nuclear Reactions and Radioactivity SMHN | AAS | RIM 2.1.2 Nuclear Fission Reaction: Nuclear fission refers to the splitting of an atomic nucleus into two or lighter nuclei. This process can occur through a nuclear reaction or radioactive decay. Nuclear fission reactions often release a large amount of energy, which is accompanied by the emission of neutrons and gamma rays (photons holding huge amounts of energy, enough to knock electrons out of atoms). Nuclear fission was first discovered by the German chemists Otto Hahn and Fritz Strassmann in the year 1938. The energy produced from fission reactions is converted into electricity in nuclear power plants. This is done by using the heat produced from the nuclear reaction to convert water into steam. Steam is used to rotate turbines in order to generate electricity. Examples: An important example of nuclear fission is the splitting of the uranium-235 nucleus when it is bombarded with neutrons. Various products can be formed from this nuclear reaction, as described in the equations below: 235 U + 1n → 141Ba + 92Kr + 3 1n 235 U + 1n → 144Xe + 90Sr + 2 1n 235 U + 1n → 146La + 87Br + 3 1n 235 U + 1n → 137Te + 97Zr + 2 1n 235 U + 1n → 137Cs + 96Rb + 3 1n 2.1.3 Nuclear Fusion Reaction: In nuclear fusion reactions, at least two atomic nuclei combine/fuse into a single nucleus. Subatomic particles such as neutrons or protons are also formed as products in these nuclear reactions. An illustration of the nuclear fusion reaction between deuterium (2H) and tritium (3H) that yields helium (4He) and a neutron (1n) is provided above. Such fusion reactions occur at the core of the sun and other stars. The fusion of deuterium and tritium nuclei is accompanied by a loss of approximately 0.0188 amu of mass (which is completely converted into energy). Approximately 1.69*109 kilojoules of energy are generated for every mole of helium formed. 2  Chapter 2 Nuclear Reactions and Radioactivity SMHN | AAS | RIM 2.1.4 Nuclear Reactors: Nuclear reactors are the heart of a nuclear power plant. They contain and control nuclear chain reactions that produce heat through a physical process called fission. Nuclear reactors are used at nuclear power plants for electricity generation and in the propulsion of ships. The heat from nuclear fission is passed to a working fluid (water or gas), which runs through turbines to create electricity. Nuclear reactors are used as research tools, as systems for producing radioactive isotopes, and most prominently as energy sources for nuclear power plants. Figure: Nuclear Reactor The main components of Nuclear Reactors are: ⦁ The Core: It contains all the fuel and generates the heat required for energy production. ⦁ The Coolant: It passes through the core, absorbing the heat and transferring into turbines ⦁ The Turbine: Transfers energy into the mechanical form ⦁ The Cooling Tower: It eliminates the excess heat that is not converted or transferred ⦁ The Containment: The enveloping structure that separates the nuclear reactor from the surrounding environment. 3  Chapter 2 Nuclear Reactions and Radioactivity SMHN | AAS | RIM 2.2 Radioactivity Radioactivity, also known as, nuclear decay or radioactive decay, is the process by which a nucleus of an unstable atom loses energy by emitting ionizing radiation. A material that spontaneously emits this kind of radiation which includes the emission of alpha particles, beta particles, gamma rays and conversion electrons is considered radioactive. Radioactivity is of two types: 1. Natural Radioactivity 2. Artificial Radioactivity Some heavier elements in the periodic table exhibited radiation as found in nature is called natural radioactivity. Applying modern techniques of artificial transmutation of elements has made it possible to produce radioactivity. Such type of radioactivity is known as artificial or induced radioactivity. There are two units of radioactivity: 1. Curie (C) 2. Becquerel (Bq) Radioactivity was first discovered in 1896 by Henri Becquerel when a photographic plate wrapped in black paper was exposed when placed close to a uranium salt. Later, experiments by Marie and Pierre Curie uncovered other radioactive substances and eventually, it was shown that the radiation from radioactive materials could be classified into three fundamentally different groups: alpha rays or particles (α) — ⬚42 𝐻𝑒 nuclei beta rays or particles (β) — electrons (created in and emitted from the nucleus) gamma rays (γ) — high-energy photons 2.2.1 Alpha particle: An alpha particle consists of two neutrons and two protons ejected from the nucleus of an atom. The alpha particle is identical to the nucleus of a helium atom. Examples of alpha emitters are radium, radon, thorium, and uranium. Because alpha particles are charged and relatively heavy, they interact intensely with atoms in materials they encounter, giving up their energy over a very short range. In the air, their travel distances are limited to no more than a few centimeters. As shown in the following illustration, alpha particles are easily shielded against and can be stopped by a single sheet of paper. Since alpha particles cannot penetrate the dead layer of the skin, they do not present a hazard from exposure external to the body. 4  Chapter 2 Nuclear Reactions and Radioactivity SMHN | AAS | RIM However, due to the very large number of ionizations they produce in a very short distance, alpha emitters can present a serious hazard when they are close to cells and tissues such as the lung. Special precautions are taken to ensure that alpha emitters are not inhaled, ingested or injected. 2.2.2 Beta Particle: A beta particle is an electron emitted from the nucleus of a radioactive atom. Examples of beta emitters commonly used in biological research are: hydrogen-3 (tritium), carbon-14, phosphorus-32, phosphorus-33, and sulfur-35. Beta particles are much less massive and less charged than alpha particles and interact less intensely with atoms in the materials they pass through, which gives them a longer range than alpha particles. Some energetic beta particles, such as those from P-32, will travel up to several meters in air or tens of mm into the skin, while low energy beta particles, such as those from H-3, are not capable of penetrating the dead layer of the skin. Thin layers of metal or plastic stop beta particles. 2.2.3 Gamma Ray: A gamma ray is a packet (or photon) of electromagnetic radiation emitted from the nucleus during radioactive decay and occasionally accompanying the emission of an alpha or beta particle. Gamma rays are identical to electromagnetic radiations such as light or microwaves but are of much higher energy. Examples of gamma emitters are cobalt-60, zinc-65, cesium-137, and radium-226. Like all forms of electromagnetic radiation, gamma rays have no mass or charge and interact less intensively with matter than ionizing particles. Because gamma radiation loses energy slowly, gamma rays can travel significant distances. Depending upon their initial energy, gamma rays can travel tens or hundreds of meters in the air. Gamma radiation is typically shielded using very dense materials (the denser the material, the more chance that a gamma ray will interact with atoms in the material) such as lead or other dense metals. Gamma radiation particularly can present a hazard from exposures external to the body. The difference between these 3 particles: The below table describes the characteristics of beta, alpha and gamma radiations and compares the masses and charges of the three rays: 5 Chapter 2 Nuclear Reactions and Radioactivity SMHN | AAS | RIM Property α ray β ray γ ray Nature Positively charged Negatively charged Uncharged particles, 42𝐻𝑒 nucleus particles (electrons). Charge +2e –e 0 Mass 6.6466 × 10–27 kg 9.109 × 10–31 kg 0 Range ~10 cm in air, Up to a few m in air, Several m in air, Can be stopped by 1mm Can be stopped by a thin Can be stopped by a of Aluminium layer of Aluminium thick layer of Lead Natural By natural radioisotopes By radioisotopes Excited nuclei 236 68 Sources e.g. 𝑈92 e.g. 𝐶𝑜29 formed because of Gamma decay 6  Chapter 2 Nuclear Reactions and Radioactivity SMHN | AAS | RIM 2.3 Radioactive Decay Law Consistent with the nature of quantum mechanics, one cannot predict precisely when a radioactive nucleus will decay. One can, however, calculate precisely the probability that any one nucleus will decay within a given time interval. This law states that “At any moment the number of radioactive atoms that disintegrate in unit time is directly proportional to the number of unchanged radioactive atoms remaining.” If the rate of radioactive disintegration of atoms is 𝑑𝑁 and if N is the number of unchanged 𝑑𝑡 atoms at time t, then the mathematical representation of the radioactive decay law is: 𝑑𝑁 𝑑𝑡 ∞-N 𝑑𝑁 𝑑𝑡 =-λN 𝑑𝑁 𝑁 = - λ dt Where, λ is the radioactive decay constant or disintegration constant. 𝑑𝑁 ∫ = - λ ∫ dt 𝑁 Log e N = - λ t + C................................. (1) Where, C is the constant of integration. Suppose, N = N0 at time t = 0 Then, Log e N0 = C From equation number (1) Log e N = - λ t + LogeN0 𝑁 Loge = -λt 𝑁𝑜 𝑁 𝑁𝑜 = e – λt N = N0 e – λt This is the law of radioactive decay or disintegration. 7 Chapter 2 Nuclear Reactions and Radioactivity SMHN | AAS | RIM 2.3.1 Half Life The half-life of a radioactive element is defined as the time during which the number of atoms remaining unchanged becomes half of its initial value. 2.3.2 Mean Lifetime Mean life, in radioactivity, average lifetime of all the nuclei of a particular unstable atomic species. This time interval may be thought of as the sum of the lifetimes of all the individual unstable nuclei in a sample, divided by the total number of unstable nuclei present. 2.3.3 Rate of Decay (Activity) Since radioactive decay is spontaneous and random, it is useful to consider the average number of nuclei which are expected to decay per unit time. This is known as the average decay rate. As a result, each radioactive element can be assigned a decay constant. The decay constant λ is defined as: “The probability that an individual nucleus will decay per unit of time” 8  Chapter 2 Nuclear Reactions and Radioactivity SMHN | AAS | RIM When a sample is highly radioactive, this means the number of decays per unit time is very high. This suggests it has a high level of activity. Activity, or the number of decays per unit time can be calculated using: Where: A = activity of the sample (Bq) ΔN = number of decayed nuclei Δt = time interval (s) λ = decay constant (s-1) N = number of nuclei remaining in a sample The activity of a sample is measured in Becquerels (Bq). An activity of 1 Bq is equal to one decay per second, or 1 s-1 This equation shows: The greater the decay constant, the greater the activity of the sample. The activity depends on the number of undecayed nuclei remaining in the sample. The minus sign indicates that the number of nuclei remaining decreases with time - however, for calculations it can be omitted. Mathematical Problems 198 1. The half-life of Au is 2.70 days. Find the decay constant of 198 Au. Ans: 0.257d-1 2. The decay constant of a radioactive substance is 3.75 × 10 - 3 y - 1 Determine its half-life. Ans: 184.8 y 3. The half-life of tritium is 12.5 y. After 25 years how many parts of a piece of tritium will remain unchanged? Ans: ¼ will be unchanged. 4. The half-life of a radioactive substance is 12 days. In how many days 85% of that substance will decay? Ans: 32.85 d. 5. The half-life of radium is 1950 year. Find the value of its decay constant and mean life. Ans: 4.36 × 10 – 4 y-1; 2294 y 6. A piece of radium becomes one-fifth part on radiating radioactive radiation for 5000 years. Find the decay constant of radium. Ans: 3.22× 10 – 4 y-1 7. The mean life of Radium is 2341 y. What is the value of its decay constant? Ans: 4.27× 10 – 4 -1 y 8. The mean life of uranium is 6.4935 × 109 years. What is its half-life? Ans: 4.5× 10 9 y 9. In the initial state if 108 number of 198 Au atoms are present in a piece of material then how many atoms will decay in one day? The half-life of 198Au 2.70 d. Ans: 2.27× 10 7 10. Each gram of Ra 226 radiates 3.5 ×1010 number of alpha particles per second. What is the half-life of radium? Ans: 1677.5 y 9  Chapter 2 Nuclear Reactions and Radioactivity SMHN | AAS | RIM 11. How much time is required to decay 60% of the piece of radon? Half-life of radon is 3.82 days. Ans: 5.06 d 12. The decay constants of two radioactive element A and B are 0.181 d-1 and 0.257 d-1 respectively. (a) Find the mean life of element B. Ans: 3.89 d. (b) Will the required time to decay 75% of both the elements same or not -give your opinion with mathematical analysis Ans: will not be same 13. A and B are two radioactive elements. Their half-lives are respectively 6 d and 9 d. 14. (a) Find the average life of element B. Ans: 12.987d 15. (b) Which element will take longer time to decay 60%? Analyze mathematically. Ans: Element B will take longer time. 16. The half-life of a radioactive element is 5 hours. What is the value of its decay constant? Ans: 3.85× 10 – 5 s-1 17. If the half-life of a radioactive element is 3 minutes, then find the value constant. Ans: 3.85× 10 – 3 s-1 18. The decay constant of a radioactive material is 0.00385s-1. What is its half- of its decay? 19. Ans: 180s 20. The value of decay constant of a radioactive material is Determine its half- 3.75× 10 - 3 s - 1 life. Ans: 184.8s 21. The mean life of radium is 2294 years. Find its decay constant and half-life. Ans: 4.36× 10 - 4 -1 y , 1589.45 y 22. A piece of radium becomes one-fifth part on radiating radioactive radiation for 4000 years. Find the decay constant of radium. Ans: 4.02× 10 - 4 y-1 23. The half-life of radon is 4 days. What is its average life? Ans: 5.77d 24. The half-life of uranium is 45 ×10 3 years. Find its average life. Ans: 6.49× 109 y 25. The half-life of a radioactive material is 5 years. After 10 years how many kilograms the material of mass 20 ×10 - 3 kg will remain unchanged? Ans: 5× 10 - 3 kg 26. The half-life of a radioactive material is 15 hours. If the initial mass of that material is 4 g then after 60 hour how much of that material will remain unchanged? Ans: 0.25g 27. The half-life of a radioactive material is 10 days. In how many days 75% of that material will be disintegrated? Ans: 20 d 28. The half-life of a radioactive material is 15 days. In how many days 65% of that material will be disintegrated? Ans: 22.72 d 29. The half-life of a material is 5 years. After 15 years how many parts material will be disintegrated? Ans: 87.5% 30. 27. How much time will be required to decay 75% of a piece of 198 Au? The half-life is 2.7 days. Ans: 5.4 d 31. The half-life of radon is 4 days. What is the value of decay constant and after days 1/20th part of the initial quantity of Radon will remain unchanged? Ans: 0.173 d-1 ; 17.32 d 32. How long will it take to decay 40% of a piece of radon? Half-life of radon is 3.82 days. Ans: 2.82 d 33. If there are 108 no. of atoms of radon initially remains in a piece of substance, then how many atoms will disintegrate in one day? Half-life of radon is 4 days. Ans: 15.9 × 106 34. A piece of radium became one-fifth after emitting radioactive radiation for 5000 years. Calculate the decay constant of radium. Ans: 3.22× 10-4 y-1 10