Half-Life Practice

1. A scientist begins with $100.0$ grams of a radioactive isotope. After exactly $12$ days have passed, only $25.0$ grams remain.

   What is the half-life of this isotope?

2. Many volcanic rocks contain potassium, which allows scientists to date them using the radioactive decay of ${K-40}$. This is how scientists discovered that the Galápagos Islands were only $3$ million years old.

   
   

   La Cumbre volcano on Isla Fernandina

   Potassium-$40$ has a half-life of $1.25$ billion years.

   If you start with $8.0$ grams of ${K-40}$, how many grams will remain after $2.5$ billion years?

   Choose 1 answer:

   1. $0$ grams
   2. $2.0$ grams
   3. $4.0$ grams

   4. $16.0$ grams

3. The following graph shows three different radioisotopes decaying over a period of $10$ seconds.

   
   

   Rank the isotopes from shortest to longest half-life.

   • Isotope C
   • Isotope A
   • Isotope B

4. A nuclear lab has produced $0.50$ moles of a pure radioisotope. The graph below shows it decaying over the course of $36$ hours.

   
   

   What is the approximate half-life of this radioisotope?

   Choose 1 answer:

   1. $1$ hour
   2. $3$ hours
   3. $6$ hours
   4. $12$ hours

Source: Khan Academy, https://www.khanacademy.org/science/hs-chemistry/x2613d8165d88df5e:nuclear-chemistry-hs/x2613d8165d88df5e:half-life-and-radiometric-dating/e/apply-half-life-and-radiometric-dating
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1. Half-life is the time it takes for half of a radioisotope sample to decay.

   To answer the question, let’s first take our initial mass and see how many half lives it takes to reach a value around $25.0$ grams. We started with $100.0$ grams, so after one half-life we'll have $50.0$ grams. After a second half-life we’ll have $25.0$ grams. So two half-lives have passed.

   Now let's figure out how long the half-life is. The total time it took for two half-lives to pass was $12$ days. Dividing $12$ days by $2$ gives us $6$ days. So, every $6$ days, the sample decreased by half.

   So, the half-life of this isotope is $6$ days.

   ---

2. Half-life is the time it takes for half of a radioisotope sample to decay.

   To answer the question, let's first determine how many half-lives have passed after $2.5$ billion years. To do this, we divide the number of years by the half-life:

   $2.5$ billion years / $1.25$ billion years per half-life = $2$ half-lives.

   So, two half-lives have passed.

   How much ${K-40}$ will be left after two half-lives? We started with $8.0$ grams, so after one half-life we'll have $4.0$ grams. After a second half-life we’ll have $2.0$ grams.

   So, $2.0$ grams of ${K-40}$ will remain.

   ---

3. Half-life is the time it takes for half of a radioisotope sample to decay. The faster an isotope decays, the shorter its half-life.

   The graph shows the decay of three different isotopes: A, B, and C. By looking at the y-axis (which shows the fraction of parent isotope remaining), we can see that all three curves start at the same point, $100$%. However, the curves show us that each isotope decays at a different rate.

   To order the isotopes by decreasing half-life, we can compare when each isotope reaches half of its starting amount ($50$%). Starting at $50$% on the y-axis, we can read across the graph to intersect the decay curves. Then, we can read down to find the corresponding times on the x-axis.

   
   

   The curve that reaches $50$% in the shortest amount of time has the shortest half-life. Similarly, the curve that reaches $50$% in the longest amount of time has the longest half-life.

   So, ranking the isotopes in increasing order from shortest to longest half-life gives us C, B, A.

   ---

4. Half-life is the time it takes for half of a radioisotope sample to decay. Let's use the graph to identify this value for the isotope described above.

   The graph shows how the mass of the sample changes over time. It shows that at $0$ hours (when the sample was prepared), there was $0.50$ moles of sample.

   Half of this amount is $0.25$ moles. Starting at $0.25$ moles on the y-axis, we can read across the graph to intersect the decay curve. Then, we can read down to find the time (in hours) on the x-axis. The sample decays to $0.25$ moles at $6$ hours.

   
   

   So, the half-life of the isotope is $6$ hours.