Ch 43: Nuclear Physics

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 43: Nuclear Physics

Problem 25

Chapter 43, Problem 25

The unstable isotope $to the 40th power K$ is used for dating rock samples. Its half-life is $1.28 \times 10^{9}$ y.
(a) How many decays occur per second in a sample containing $1.63 \times 10^{- 6}$ g of $to the 40th power K$ ?
(b) What is the activity of the sample in curies?

Verified step by step guidance

Step 1: Calculate the number of atoms of 40K in the sample. Use the formula $ N = \frac{m}{M} \times N_A $, where $ m $ is the mass of the sample (1.63 $\times$ 10^{-6} $\text{ g}$), $ M $ is the molar mass of 40K (approximately 40 $\text{ g/mol}$), and $ N_A $ is Avogadro's number (6.022 $\times$ 10^{23} $\text{ atoms/mol}$).

Step 2: Determine the decay constant $ \lambda $ using the relationship $ \lambda = \frac{\ln(2)}{T_{1/2}} $, where $ T_{1/2} $ is the half-life of 40K (1.28 $\times$ 10^9 $\text{ years}$). Convert the half-life into seconds before calculating $ \lambda $.

Step 3: Calculate the number of decays per second (activity in becquerels) using the formula $ A = \lambda N $, where $ N $ is the number of atoms calculated in Step 1 and $ \lambda $ is the decay constant from Step 2.

Step 4: Convert the activity from becquerels to curies. Recall that 1 curie (Ci) is equal to $ 3.7 \times 10^{10} \text{ Bq} $. Use the formula $ \text{Activity in Ci} = \frac{\text{Activity in Bq}}{3.7 \times 10^{10}} $.

Step 5: Summarize the process and ensure all units are consistent throughout the calculations. Verify that the activity in curies is correctly derived from the becquerel value.

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Key Concepts

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

Half-life

Half-life is the time required for half of the radioactive nuclei in a sample to decay. For 40K, with a half-life of 1.28 billion years, this concept is crucial for understanding the rate of decay and calculating the number of decays over a given time period. It allows us to determine how long it takes for a certain fraction of the isotope to transform into its decay products.

Radioactive Decay

Radioactive decay is a random process by which unstable atomic nuclei lose energy by emitting radiation. This process can be quantified by the decay constant, which relates to the half-life and indicates the probability of decay per unit time. Understanding this concept is essential for calculating the number of decays occurring in a sample over a specific time frame.

Activity (in Curies)

Activity is a measure of the rate at which a radioactive sample decays, typically expressed in units such as Curies (Ci). One Curie is defined as 3.7 x 10^10 decays per second. This concept is important for quantifying the level of radioactivity in a sample, which can have implications for safety and dating techniques in geology.

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Textbook Question

Measurements on a certain isotope tell you that the decay rate decreases from $8318$ decays/min to $3091$ decays/min in $4.00$ 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) $_{14}^{27} S i \to_{13}^{27} A l$

(b) $_{92}^{238} U \to_{90}^{234} T h$

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Textbook Question

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 $to the 60th power C o$ is $5000$ Ci. When its activity falls below $3500$ 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 $to the 60th power C o$ 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 $to the 60th power C o$ is $5.271$ years.

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Textbook Question

The common isotope of uranium, $to the 238th power U$ , has a half-life of $4.47 \times 10^{9}$ years, decaying to $to the 234th power T h$ by alpha emission.

(a) What is the decay constant?

(b) What mass of uranium is required for an activity of $1.00$ curie?

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Textbook Question

It has become popular for some people to have yearly whole-body scans (CT scans, formerly called CAT scans) using x rays, just to see if they detect anything suspicious. A number of medical people have recently questioned the advisability of such scans, due in part to the radiation they impart. Typically, one such scan gives a dose of $12$ mSv, applied to the whole body. By contrast, a chest x ray typically administers $0.20$ mSv to only $5.0$ kg of tissue. How many chest x rays would deliver the same total amount of energy to the body of a $75$ -kg person as one whole-body scan?

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At an archeological site, a sample from timbers containing $500$ g of carbon provides $2690$ decays/min. What is the age of the sample?

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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?

Verified video answer for a similar problem:

Key Concepts

Half-life

Radioactive Decay

Activity (in Curies)

The unstable isotope $to the 40th power K$ is used for dating rock samples. Its half-life is $1.28 \times 10^{9}$ y.
(a) How many decays occur per second in a sample containing $1.63 \times 10^{- 6}$ g of $to the 40th power K$ ?
(b) What is the activity of the sample in curies?