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?

Ch 42: Nuclear Physics

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 42: Nuclear Physics

Problem 25

Chapter 42, Problem 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?

Verified step by step guidance

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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.

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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