Calculation of the Decay of a Radionuclide

#### Introduction

These two exercises are meant to give you practice in calculating the parameters associated with radioactive decay. This is a two variable problem involving the time which has elapsed and the remaining fraction. If you know either one you can calculate the other. In the first exercise you are given the time and are asked to calculate the remaining fraction. In the second you are given the remaining fraction and are asked to calculate the remaining time. In both cases you can check your answers against the computer, which will respond by telling you if you got the answer correct, or if not, what the correct answer should be. You can also elect to try a new problem, either using the same isotope, or by selecting a different isotope with a new half life.

#### Formulae

Both of these problems rely on the same formula. However it looks a little different depending on whether you are given the time and are asked to calculate the remaining fraction, or whether you are given the remaining fraction and are asked to calculate the remaining time.
• If you are given the time then you can calculate the remaining fraction by setting y = N/No = e-kt.
• If you are given the remaining fraction then you can calculate the time by setting x = -kt = ln(y)
In both cases you also need to calculate the decay constant k. Its value in both cases is ln(2)/T, where T is the half life.

For sample solutions see the page on the uses of exponentials and logarithms

#### The questions

 Given that the half life of an isotope is 1. What is the value of the decay constant k = ln2/T? 2. After what fraction of the initial radioactive sample remains?
 For the same isotope, calculate the fraction at a different time Start a brand new problem with a new isotope
 Given that the half life of an isotope is 1. What is the value of the decay constant k = ln2/T? 2. How much time has elapsed if the remaining fraction is ? Choose the correct unitssecondsminuteshoursdaysyearsthousand yearsmillion years
 For the same isotope, calculate the time for a different fraction Start a brand new problem with a new isotope

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