Question

The activity of a sample of \({ }_{83}^{210} \mathrm{Bi}\) (with a half-life of 5.01 days) is measured to be \(1.000 \mu \mathrm{Ci}\). What will the activity of this sample be after \(1 \mathrm{yr?}\) ?

Short answer

Answer: The remaining activity of the sample after 1 year will be approximately \(1.16 \times 10^{-22}\,\mu\text{Ci}\).

Step-by-step solution

Convert time to the same units

We are given the half-life in days and we need to find the activity after 1 year. Let's convert 1 year into days to have consistent units. There are 365 days in a year.

Find the number of half-lives

We should determine how many half-lives have passed within 1 year. We will divide the total time (1 year in days) by the half-life, which is 5.01 days. $$ \text{number of half-lives} = \frac{365\,\text{days}}{5.01\,\text{days}} $$

Calculate the fraction of the remaining isotopes

To find the remaining fraction of the isotopes, we will use the formula for exponential decay, which is: $$ \text{remaining fraction} = \left( \frac{1}{2} \right)^n $$ where n is the number of half-lives.

Calculate the remaining activity

Now that we know the fraction of isotopes remaining, we can find the remaining activity by multiplying the initial activity by the remaining fraction. $$ \text{remaining activity} = \text{initial activity} \times \text{remaining fraction} $$ Let's perform the calculations: $$ \text{number of half-lives} = \frac{365\,\text{days}}{5.01\,\text{days}} = 72.86 \approx 73 $$ $$ \text{remaining fraction} = \left( \frac{1}{2} \right)^{73} \approx 1.16 \times 10^{-22} $$ $$ \text{remaining activity} = (1.000\,\mu\text{Ci}) \times (1.16 \times 10^{-22}) \approx 1.16 \times 10^{-22}\,\mu\text{Ci} $$ After 1 year, the remaining activity of the sample will be approximately \(1.16 \times 10^{-22}\,\mu\text{Ci}\).

Key concepts

Understanding Half-Life

The concept of half-life is central to the study of radioactive decay. A half-life is the time it takes for half of the radioactive isotopes in a sample to decay into other elements or isotopes. This measurement can be used to predict how long it will take for a certain percentage of the sample to decay.

For example, if a sample has a half-life of 5.01 days, like the Bi isotope mentioned in the exercise, after 5.01 days, only half of the original isotopes would remain undecayed. After another 5.01 days (or one additional half-life), only one quarter of the original isotopes would be left, and so on.

While calculating the activity after a year, as in the exercise, we divide the total time elapsed by the half-life to find out how many half-life periods have passed. Using this information, we are able to determine the remaining fraction and consequently the remaining activity of the sample.

Exponential Decay in Radioactive Materials

Radioactive decay follows a pattern known as exponential decay. It's described by a rapid decrease in the amount of a substance as time progresses. In mathematical terms, if you plot the remaining amount of a radioactive sample over time, you get a curve that steeply drops off as time increases and eventually approaches zero but never quite touching it.

This is formalized in the exponential decay formula, where the remaining fraction after a certain number of half-lives can be calculated using the equation: \[\[\begin{align*}\text{remaining fraction} = \left( \frac{1}{2} \right)^n\end{align*}\]\], where n is the number of half-lives passed. It's worth mentioning that this decay is unaffected by external conditions like temperature or pressure; it's a fixed property of the radioactive isotope. For a given isotope, this rate of decay is consistent and can be used to date ancient objects, understand natural processes, or even treat cancer in the form of radiation therapy.

Activity of Isotopes and Radioactive Decay

In the context of radioactivity, 'activity' refers to the rate at which a radioactive sample undergoes decay. It's commonly measured in units like Curie (Ci) or Becquerel (Bq), where one Curie is defined as 37 billion decays per second, and one Becquerel is one decay per second.

To calculate the new activity after a certain time has passed, you multiply the initial activity by the fraction of isotopes remaining. This is precisely what we do in the exercise—after finding how many half-lives have passed within one year, we use that number to find the remaining fraction and then calculate the remaining activity of the isotope. As seen from the original solution, after a year, the activity of the Bi sample becomes negligibly small, demonstrating just how quickly exponential decay reduces the activity of radioactive samples.