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?

Ch 43: Nuclear Physics

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 43: Nuclear Physics

Problem 27

Chapter 43, Problem 27

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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Identify the given quantities: the initial decay rate $R_0 = 8318$ decays/min, the final decay rate $R = 3091$ decays/min after a time interval $t = 4.00$ days.

Recall that radioactive decay follows an exponential decay law given by $R = R_0 \times e^{-\lambda t}$, where $\lambda$ is the decay constant.

Rearrange the decay law to solve for the decay constant $\lambda$: $\lambda = -\frac{1}{t} \ln\left(\frac{R}{R_0}\right)$.

Calculate the decay constant $\lambda$ using the given values of $R$, $R_0$, and $t$ (make sure to convert time $t$ into consistent units, such as minutes or days, depending on your preference).

Use the relationship between the half-life $T_{1/2}$ and the decay constant: $T_{1/2} = \frac{\ln 2}{\lambda}$ to find the half-life of the isotope.

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Key Concepts

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

Radioactive Decay and Decay Rate

Radioactive decay is a random process where unstable nuclei lose energy by emitting radiation. The decay rate, or activity, is the number of decays per unit time and decreases exponentially over time as the number of undecayed nuclei diminishes.

Exponential Decay Law

The exponential decay law describes how the quantity of a radioactive substance decreases over time: N(t) = N0 * e^(-λt), where λ is the decay constant. The decay rate is proportional to the number of undecayed nuclei, so it also follows this exponential decrease.

Half-Life and Decay Constant Relationship

The half-life is the time required for half of the radioactive nuclei to decay. It is related to the decay constant by t½ = ln(2)/λ. Knowing the decay rate at two times allows calculation of λ, and thus the half-life.

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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

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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Textbook Question

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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Textbook Question

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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