At an archeological site, a sample from timbers containing 500500 g of carbon provides 26902690 decays/min. What is the age of the sample?

Ch 43: Nuclear Physics

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 43: Nuclear Physics

문제 30

43장, 문제 30

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

검증된 단계별 안내

Step 1: Understand the problem. The sample contains carbon, and the decay rate is given. This problem involves radioactive decay, specifically carbon-14 dating. The goal is to calculate the age of the sample using the decay rate and the known properties of carbon-14.

Step 2: Recall the formula for radioactive decay: $ N(t) = N_0 e^{-\lambda t} $, where $ N(t) $ is the number of decays per unit time at time $ t $, $ N_0 $ is the initial decay rate, $ \lambda $ is the decay constant, and $ t $ is the time elapsed. For carbon-14, the half-life $ T_{1/2} $ is approximately 5730 years.

Step 3: Relate the decay constant $ \lambda $ to the half-life using the formula $ \lambda = \frac{\ln(2)}{T_{1/2}} $. Substitute $ T_{1/2} = 5730 $ years to calculate $ \lambda $.

Step 4: Use the given decay rate (2690 decays/min) and the initial decay rate (which can be calculated based on the mass of carbon and the known decay rate per gram for fresh carbon-14) to find $ t $. Rearrange the decay formula to solve for $ t $: $ t = \frac{-\ln\left(\frac{N(t)}{N_0}\right)}{\lambda} $.

Step 5: Substitute the values for $ N(t) $, $ N_0 $, and $ \lambda $ into the formula. Perform the logarithmic and division operations to determine the age of the sample.

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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 determining the age of archaeological samples through techniques like carbon dating.

Carbon-14 Dating

Carbon-14 dating is a method used to determine the age of organic materials by measuring the amount of carbon-14, a radioactive isotope of carbon, remaining in the sample. Living organisms continuously exchange carbon with their environment, but once they die, the carbon-14 begins to decay. By comparing the remaining carbon-14 to the initial levels, scientists can estimate the time since the organism's death.

Decay Rate and Activity

The decay rate, or activity, of a radioactive sample is measured in decays per minute (dpm) and indicates how many atoms decay in a given time frame. This rate decreases over time as the radioactive material diminishes. In carbon dating, the activity of the sample is compared to a standard reference to calculate its age, making it essential to understand how to interpret these measurements.

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In a diagnostic x-ray procedure, $5.00 \times 10^{10}$ photons are absorbed by tissue with a mass of $0.600$ kg. The x-ray wavelength is $0.0200$ nm.

(a) What is the total energy absorbed by the tissue?

(b) What is the equivalent dose in rem?

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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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교과서 질문

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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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?

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교과서 질문

A $67$ -kg person accidentally ingests $0.35$ Ci of tritium.

(a) Assume that the tritium spreads uniformly throughout the body and that each decay leads on the average to the absorption of $5.0$ keV of energy from the electrons emitted in the decay. The half-life of tritium is $12.3$ y, and the RBE of the electrons is $1.0$ . Calculate the absorbed dose in rad and the equivalent dose in rem during one week.

(b) The $β^{-}$ decay of tritium releases more than $5.0$ keV of energy. Why is the average energy absorbed less than the total energy released in the decay?

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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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