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Chapter 38: Problem 22

How many half-lives will it take for the activity of a radioactive sample to diminish to \(10 \%\) of its original level?

Short Answer

Expert verified
The number of half-lives needed to reduce the radioactivity to 10% of its original level is \(n=\log_2{10}\), which is approximately 3.32.

Step by step solution

01

Understanding the Problem

A half-life is the time taken for the radioactivity of a specified isotope to fall to half its original value. Each half-life reduces the remaining radioactivity by half, thus, after \(n\) half-lives, the remaining radioactivity is \( \frac{1}{2^n}\) of the original.
02

Setting Up the Equation

We have to find out how many half-lives will reduce the radioactivity to 10% of its original level. Therefore, we set up the equation \( \frac{1}{2^n} = 0.10\) where \(n\) is the number of half-lives.
03

Solve the Equation

To solve the equation \(0.10 = \frac{1}{2^n}\), we'll first rewrite \(0.10\) as \( \frac{1}{10}\). The equation then becomes \( \frac{1}{10} = \frac{1}{2^n}\). The only way for this equation to hold true is if \( 2^n = 10\). Solving for \(n\) gives us \(n=\log_2{10}\)

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