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Ch 42: Nuclear Physics
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoKnight Calc 5th EditionCh 42: Nuclear PhysicsProblema 25
Capitolo 42, Problema 25

The radioactive isotope 230Th has a density of 11,700 kg/m3 and a half-life of 75,000 yr. What is the radius of a 230Th sphere that has an activity of 1.0 Ci?

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Step 1: Understand the relationship between activity, number of nuclei, and decay constant. Activity (A) is given by \( A = \lambda N \), where \( \lambda \) is the decay constant and \( N \) is the number of radioactive nuclei. The decay constant \( \lambda \) can be calculated using the half-life \( T_{1/2} \) with the formula \( \lambda = \frac{\ln(2)}{T_{1/2}} \).
Step 2: Convert the given activity from curies (Ci) to becquerels (Bq). Recall that \( 1 \, \text{Ci} = 3.7 \times 10^{10} \, \text{Bq} \). Use this conversion to express the activity in SI units.
Step 3: Calculate the number of nuclei \( N \) using the formula \( N = \frac{A}{\lambda} \). Substitute the values for \( A \) (in Bq) and \( \lambda \) (calculated from the half-life).
Step 4: Relate the number of nuclei \( N \) to the mass of the sphere. The number of nuclei \( N \) is given by \( N = \frac{m}{M} \times N_A \), where \( m \) is the mass of the sphere, \( M \) is the molar mass of \( ^{230}Th \) (230 g/mol), and \( N_A \) is Avogadro's number \( 6.022 \times 10^{23} \, \text{mol}^{-1} \). Rearrange this formula to solve for \( m \).
Step 5: Use the density of \( ^{230}Th \) to find the radius of the sphere. The mass \( m \) is related to the volume \( V \) by \( m = \rho V \), where \( \rho \) is the density. The volume of a sphere is \( V = \frac{4}{3} \pi r^3 \). Substitute \( m \) and \( \rho \) into this formula and solve for the radius \( r \).

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Radioactive Decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. This decay occurs at a predictable rate characterized by the half-life, which is the time required for half of the radioactive atoms in a sample to decay. Understanding this concept is crucial for calculating the activity of a radioactive substance.
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Amplitude Decay in an LRC Circuit

Activity of a Radioactive Sample

The activity of a radioactive sample refers to the number of decays per unit time, typically measured in curies (Ci) or becquerels (Bq). One curie is defined as 3.7 x 10^10 disintegrations per second. This concept is essential for determining how much of a radioactive substance is present and how it relates to the mass and volume of the material.
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Phase Diagrams

Volume and Density Relationship

The relationship between volume, mass, and density is described by the formula density = mass/volume. In the context of a sphere, the volume can be calculated using the formula V = (4/3)πr³, where r is the radius. This relationship is important for determining the radius of the sphere based on its mass, which can be derived from the activity and density of the radioactive isotope.
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Problems with Mass, Volume, & Density
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