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Ch 43: Nuclear Physics
Young & Freedman Calc - University Physics 14th Edition
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook
All textbooksYoung & Freedman Calc 14th EditionCh 43: Nuclear PhysicsProblem 20
Chapter 43, Problem 20

Radioactive isotopes used in cancer therapy have a 'shelf-life,' like pharmaceuticals used in chemotherapy. Just after it has been manufactured in a nuclear reactor, the activity of a sample of 60Co^{60}Co is 50005000 Ci. When its activity falls below 35003500 Ci, it is considered too weak a source to use in treatment. You work in the radiology department of a large hospital. One of these 60Co^{60}Co sources in your inventory was manufactured on October 6, 2011. It is now April 6, 2014. Is the source still usable? The half-life of 60Co^{60}Co is 5.2715.271 years.

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Step 1: Understand the concept of radioactive decay and half-life. The half-life of a radioactive isotope is the time it takes for half of the radioactive nuclei in a sample to decay. The activity of the sample decreases exponentially over time according to the formula: \( A = A_0 \cdot e^{-\lambda t} \), where \( A \) is the activity at time \( t \), \( A_0 \) is the initial activity, \( \lambda \) is the decay constant, and \( t \) is the elapsed time.
Step 2: Calculate the decay constant \( \lambda \) using the relationship between half-life and decay constant: \( \lambda = \frac{\ln(2)}{T_{1/2}} \), where \( T_{1/2} \) is the half-life of the isotope. Substitute \( T_{1/2} = 5.271 \) years into the formula to find \( \lambda \).
Step 3: Determine the elapsed time \( t \) between the manufacturing date and the current date. Subtract October 6, 2011, from April 6, 2014, to find \( t \) in years. Ensure the time is expressed in the same units as the half-life (years).
Step 4: Use the radioactive decay formula \( A = A_0 \cdot e^{-\lambda t} \) to calculate the activity \( A \) of the sample after \( t \) years. Substitute \( A_0 = 5000 \) Ci, \( \lambda \) (calculated in Step 2), and \( t \) (calculated in Step 3) into the formula.
Step 5: Compare the calculated activity \( A \) to the threshold activity of 3500 Ci. If \( A \geq 3500 \) Ci, the source is still usable; otherwise, it is too weak for treatment.

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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 determining how the activity of a radioactive isotope changes over time.
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Half-Life

The half-life of a radioactive isotope is the time it takes for half of the radioactive atoms in a sample to decay. For 60Co, the half-life is 5.271 years, meaning that after this period, the activity of the sample will have decreased to half its original value. This concept is essential for calculating the remaining activity of the isotope after a given period.
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Activity Measurement

Activity is a measure of the decay rate of a radioactive substance, typically expressed in curies (Ci). It indicates how many decays occur per second in a sample. In the context of cancer therapy, knowing the activity level is vital to ensure that the radioactive source is effective for treatment, as it must remain above a certain threshold to be considered usable.
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Related Practice
Textbook Question

The atomic mass of 14C14C is 14.00324214.003242 u. Show that the β−\(\beta\)^{-} decay of 14C14C is energetically possible, and calculate the energy released in the decay.

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Textbook Question

What nuclide is produced in the following radioactive decays?

(a) α\(\alpha\) decay of 94239Pu_{94}^{239}Pu

(b) β−\(\beta\)^{-} decay of 1124Na_{11}^{24}Na

(c) β+\(\beta\)^{+} decay of 815O_8^{15}O

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Textbook Question

Measurements on a certain isotope tell you that the decay rate decreases from 83188318 decays/min to 30913091 decays/min in 4.004.00 days. What is the half-life of this isotope?

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Textbook Question

What particle (a particle, electron, or positron) is emitted in the following radioactive decays?

(a) 1427Si→1327Al_{14}^{27}Si\(\rightarrow\)_{13}^{27}Al

(b) 92238U→90234Th_{92}^{238}U\(\rightarrow\)_{90}^{234}Th

(c) 3374As→3474Se_{33}^{74}As\(\rightarrow\)_{34}^{74}Se

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Textbook Question

The common isotope of uranium, 238U^{238}U, has a half-life of 4.47×1094.47\(\times\)10^9 years, decaying to 234Th^{234}Th by alpha emission.

(a) What is the decay constant?

(b) What mass of uranium is required for an activity of 1.001.00 curie?

(c) How many alpha particles are emitted per second by 10.010.0 g of uranium?

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Textbook Question

The unstable isotope 40K^{40}K is used for dating rock samples. Its half-life is 1.28×1091.28\(\times\)10^9 y.

(a) How many decays occur per second in a sample containing 1.63×10−61.63\(\times\)10^{-6} g of 40K^{40}K?

(b) What is the activity of the sample in curies?

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