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 15th Edition

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Young & Freedman Calc 15th Edition

Ch 43: Nuclear Physics

Problem 27

Chapter 42, 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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