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Nuclear Reactions AP Physics B Montwood High School R. Casao Nuclear Reactions Nuclear reactions are rearrangements of nuclear components that result from bombardment by a particle rather than a spontaneous natural process. Nuclear reactions are subject to several conservation laws: – – – – Charge Momentum Energy Number of nucleons When two nuclei interact, charge conservation requires that the sum of the initial atomic numbers must equal the sum of the final atomic numbers. Because of conservation of nucleon number, the sum of the initial mass numbers must also equal the sum of the final mass numbers. In general, these are not elastic collisions, and therefore, the total initial mass does NOT equal the total final mass. Reaction Energy The difference between the masses before and after the reaction corresponds to the reaction energy, according to the mass-energy relationship E = m·c2. If initial particles A and B interact to produce final particles C and D, the reaction energy Q is defined as: Q = (MA + MB – MC – MD) ·c2 To balance the electrons, use the neutral atomic masses. Kinetic energy can be used if C and D are in the ground state: Q = (KC + KD) – (KA – KB) Reaction Energy When Q is positive, the total mass decreases and the total kinetic energy increases. – Exothermic reaction. When Q is negative, the total mass increases and the kinetic energy decreases. – Endothermic reaction. – Reaction cannot occur at all unless the initial kinetic energy is at least equal to Q. – Threshold energy: the minimum kinetic energy needed to make the reaction take place. Example When Li-7 is bombarded by a proton, two a particles (He-4) are produced. Determine the reaction energy. 1 7 4 4 H Li He 1 3 2 2 He MeV Q 1.007825 u 7.016003 u (2 4.002603 u) 931 .5 u Q 17.35 MeV This is an exothermic reaction because the final total kinetic energy is 17.35 MeV greater than the initial total kinetic energy. Example Determine the reaction energy for the nuclear reaction: 4 14 17 1 He N O 2 7 8 1H Q 4.002603 u 14.003074 u 16.999131 u 1.007825 u 931 .5 MeV u Q 1.191 MeV This is an endothermic reaction. The minimum initial kinetic energy (threshold energy) for this reaction to occur is 1.191 MeV. The relationship between the threshold kinetic energy KEth and the absolute magnitude of the reaction energy Q can be determined using conservation of energy and momentum: KE th m 1 Q M – where m = mass of incoming particle and M = mass of stationary target particle. Ordinarily, the stationary N-14 nuclei would be bombarded with a particles from an accelerator. The a particles kinetic energy must be greater than 1.191 MeV. When a particle with mass m and kinetic energy K collides with a stationary particle with mass M, the total kinetic energy Ktotal (the energy available to cause reactions) is: K total K total MK Mm 14.003074 u 1.191 MeV 14.003074 u 4.002603 u K total 1.532 MeV The kinetic energy of the a particles must be at least 1.532 MeV. For a charged particle such as a proton or an a particle to penetrate the nucleus of another atom and cause a reaction, it must have enough initial kinetic energy to overcome the potential-energy barrier caused by the repulsive electrostatic forces. k q1 q 2 U r For the reaction of the proton and Li-7, if we treat the proton and the Li-7 nucleus as spherically symmetric charges with radii given by R = Ro·A1/3, their centers will be 3.5 x 10-15 m apart when they touch. The repulsive potential energy of the proton and the Li-7 nucleus at this separation r is: 2 N m 19 19 8.9875 x 10 9 3 1 . 602 x 10 C 1 . 602 x 10 C 2 C U 3.5x10 15 m U 2 x 10 13 J 1.2 MeV Even though the reaction is exothermic, the proton must have a minimum kinetic energy of 1.2 MeV for the reaction to occur. Neutron Absorption Absorption of neutrons by nuclei forms is an important class of nuclear reactions. Heavy nuclei bombarded by neutrons in a nuclear reaction undergo a series of neutron absorptions alternating with beta decays, in which the mass number A increases by as much as 25. Some of the trans-uranic elements, elements having an atomic number larger than 92, are produced in this way. – These elements have not been found in nature. Disintegration Energy (a Decay) Alpha decay: parent daughter + a – Disintegration energy Q = (mp – md – ma)·c2 – Disintegration energy appears in the form of kinetic energy in the daughter nucleus and the alpha particle. – Alpha decay requires Q > 0; if Q < 0, the parent nucleus is stable with respect to alpha decay and alpha decay does not occur. Disintegration Energy (a Decay) Of the energy available to the daughter and alpha particle as kinetic energy: – larger mass has smaller kinetic energy. – smaller mass has greater kinetic energy. KE alpha Q mdaughter mparent KE daughter Q malpha mparent Disintegration Energy (-b Decay) Negative – beta decay: parent daughter + bDisintegration energy Q = (mp – md)·c2 – Example: 14 14 6 C 7 0 N 1e v – C-14 emits a negatively charged beta particle (an ordinary electron) and the resulting nucleus is N-14. – Might expect to determine Q using: Q = (mp – md – mb)·c2 This does not correctly represent the mass of all the electrons in the parent and daughter atoms. – mp and md are atomic masses and include the electron masses. – The mass for C-14 would be accurate because it would include the mass of 6 electrons. – The mass of N-14 would be incorrect because it would include the mass of 7 electrons. – In the N-14 formed by the beta decay there are only the 6 electrons from the parent C-14 atom. – To compensate for the fact that the daughter atom contains the mass of 1 more electron than we need, we do not write the mb term in the energy equation. Correct equation for disintegration energy for negative beta decay: Q = (mp – md)·c2 The nitrogen created in the beta decay of carbon has only 6 orbiting electrons because the parent carbon had only six electrons. The daughter is nitrogen by definition because it contains 7 protons. The N-14 is positively ionized and will attract an electron to complete its outer electron shell. Disintegration Energy (+b Decay) Positive beta decay: parent daughter + b+ – Example: 13 13 0 7 N 6 C 1e v – N-13 emits a positively charged beta particle and the resulting nucleus is C-13. – The mass for the parent N-13 atom will be correct. – The mass for the daughter C-13 atom includes the mass for only the 6 electrons for a normal C-13 atom. In our daughter C-13 atom, there are 7 electrons from the parent N-13 atom. The daughter C-13 is positively ionized. We have to include an me = mb term in the disintegration equation to account for this positive ion. The mass of the positron is also mb. Disintegration energy: Q = (mp – md – 2·mb)·c2 The decay of a free proton into a neutron, a b+, and a neutrino is not possible because the combined mass of the neutron and beta particle is greater than the mass of the proton. p 1 0 0 n 1e ve However, the same reaction is possible if the proton is bound within the nucleus because the required energy is provided by the motion of the nuclei in the nucleus.