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 15th Edition

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Young & Freedman Calc 15th Edition

Ch 43: Nuclear Physics

Problem 28

Chapter 42, Problem 28

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

Verified step by step guidance

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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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 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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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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Calculate the energy released in the fusion reaction: $_{2}^{3} H e +_{1}^{2} H \to_{2}^{4} H e +_{1}^{1} H$

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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?

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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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