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Chapter 42: Q.55 (page 1237)

We’ve noted that fewer than $10 %$ of the known nuclei are stable (i.e., not radioactive). Because nuclei are characterized by two independent numbers, $N$ and $Z$ , it is useful to show the known nuclei on a plot of neutron number $N$ versus proton number $Z$ .

Short Answer

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Earth when it was born 86.1 greater than the number of uranium atoms.

Step by step solution

NU=NU0 12 It/2 Uranium Atoms:

Ratio of NU0/NU

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Most popular questions from this chapter

A chest x ray uses 10 keV photons with an RBE of 0.85. A 60 kg person receives a 0.30 mSv dose from one chest x ray that exposes 25% of the patient’s body. How many x ray photons are absorbed in the patient’s body?

What is the age in years of a bone in which the $^{14} C^{12} C$ ratio is measured to be $1.65 \times 10^{- 13}$ ?

The uranium isotope ${}^{238}U$ is naturally present at low levels in many soils. One of the nuclei in the decay series of ${}^{238}U$ is the radon isotope ${}^{222}R n$ , which decays by emitting a localid="1650487447445" $5.50 M e V$ alpha particle with localid="1650487458663" $t_{1 / 2} = 3.82$ 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 $4 p C i p e r l i t e r$ of air.

a. How many ${}^{222}R n$ atoms are there in $1 m^{3}$ of air if the activity is $4 \frac{p C i}{L}$ ?

b. The range of alpha particles in air is $\approx 3 c m$ . Suppose we model a person as a $180 - c m$ -tall, $25 - c m$ -diameter cylinder with a mass of $65 k g$ . Only decays within $3 c m$ of the cylinder can cause exposure, and only $\approx 50 %$ , of the decays direct the alpha particle toward the person. Determine the dose in $m S v p e r y e a r$ for a person who spends the entire year in a room where the activity is $4 p C i / L$ .

c. Does the EPA recommendation seem appropriate? Why?

Calculate (in $M e V$ ) the binding energy per nucleon for $H e^{2}$ and $H e^{4}$ . Which is more tightly bound?

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 ≈ 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’s 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 ≈ 1 - NPtunnel , where we’ve 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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We’ve noted that fewer than 10%of the known nuclei are stable (i.e., not radioactive). Because nuclei are characterized by two independent numbers, Nand Z, it is useful to show the known nuclei on a plot of neutron number Nversus proton numberZ.

Short Answer

Step by step solution

NU=NU0 12 It/2 Uranium Atoms:

Ratio of NU0/NU

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We’ve noted that fewer than $10 %$ of the known nuclei are stable (i.e., not radioactive). Because nuclei are characterized by two independent numbers, $N$ and $Z$ , it is useful to show the known nuclei on a plot of neutron number $N$ versus proton number $Z$ .