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Chapter 42: Q. 53 (page 1237)
A sample contains radioactive atoms of two types, A and B. Initially there are five times as many A atoms as there are B atoms. Two hours later, the numbers of the two atoms are equal. The half-life of A is . What is the half-life of B?
Half life of B is,
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Use the potential-energy diagram in Figure to sketch an approximate graph of the strong force between two nucleons versus the distance between their centers.
The uranium isotope is naturally present at low levels in many soils. One of the nuclei in the decay series of is the radon isotope , which decays by emitting a localid="1650487447445" alpha particle with localid="1650487458663" days. Radon is a gas, and it tends to seep from soil into basements. The Environmental Protection Agency recommends that homeowners take steps to remove radon, by pumping in fresh air, if the radon activity exceeds of air.
a. How many atoms are there in of air if the activity is ?
b. The range of alpha particles in air is . Suppose we model a person as a -tall, -diameter cylinder with a mass of . Only decays within of the cylinder can cause exposure, and only , of the decays direct the alpha particle toward the person. Determine the dose in for a person who spends the entire year in a room where the activity is .
c. Does the EPA recommendation seem appropriate? Why?
The plutonium isotope 239 Pu has a half-life of 24,000 years and decays by the emission of a 5.2 MeV alpha particle. Plutonium is not especially dangerous if handled because the activity is low and the alpha radiation doesn鈥檛 penetrate the skin. However, there are serious health concerns if even the tiniest particles of plutonium are inhaled and lodge deep in the lungs. This could happen following any kind of fire or explosion that disperses plutonium as dust. Let鈥檚 determine the level of danger.
a. How do we know the strong force exists?
b. How do we know the strong force is short range?
Alpha decay occurs when an alpha particle tunnels through the Coulomb barrier. FIGURE CP42.63 shows a simple one-dimensional model of the potential-energy well of an alpha particle in a nucleus with A 鈮� 235. The 15 fm width of this one-dimensional potential-energy well is the diameter of the nucleus. Further, to keep the model simple, the Coulomb barrier has been modeled as a 20-fm-wide, 30-MeV-high rectangular potential-energy barrier. The goal of this problem is to calculate the half-life of an alpha particle in the energy level E = 5.0 MeV. a. What is the kinetic energy of the alpha particle while inside the nucleus? What is its kinetic energy after it escapes from the nucleus? b. Consider the alpha particle within the nucleus to be a point particle bouncing back and forth with the kinetic energy you found in part a. What is the particle鈥檚 collision rate, the number of times per second it collides with a wall of the potential? c. What is the tunneling probability Ptunnel ? d. Ptunnel is the probability that on any one collision with a wall the alpha particle tunnels through instead of reflecting. The probability of not tunneling is 1 - Ptunnel. Hence the probability that the alpha particle is still inside the nucleus after N collisions is 11 - Ptunnel 2N 鈮� 1 - NPtunnel , where we鈥檝e used the binomial approximation because Ptunnel V 1. The half-life is the time at which half the nuclei have not yet decayed. Use this to determine (in years) the half-life of the nucleus.
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