Question

The "Chemistry in Focus" segment PET, the Brain's Best Friend discusses the use of radiotracers to monitor the functioning of organs or to trace the path and final destination of a drug. The isotope \({ }^{18} \mathrm{~F}\) is cited as a possible radiotracer and has a half-life of 110 minutes. From a sample of 1 mole of \({ }^{18} \mathrm{~F}\), about how many atoms are left after a day?

Short answer

After one day, there are approximately \( 7.347 \times 10^{20} \) atoms of \({ }^{18} \mathrm{~F}\) left in the sample.

Step-by-step solution

Determine the number of half-lives in one day

To find the number of half-lives in one day, we need to convert one day into minutes and then divide by the half-life (110 minutes). 1 day = 24 hours/day * 60 minutes/hour = 1440 minutes There are 1440 minutes in a day. Now, we can find the number of half-lives: Number of half-lives = Total time / Half-life time Number of half-lives = 1440 minutes / 110 minutes/half-life

Calculate the number of half-lives

Divide 1440 minutes by 110 minutes to find the number of half-lives. Number of half-lives = \( \frac{1440}{110} \approx 13.09 \) After rounding down, since we cannot have a fraction of a half-life, 13 half-lives have passed in one day.

Calculate the remaining fraction of the sample after 13 half-lives

We will use the decay formula to find the remaining fraction of the sample after 13 half-lives: Remaining fraction = \( (\frac{1}{2})^n \) where n is the number of half-lives. Remaining fraction = \( (\frac{1}{2})^{13} \approx 0.000122 \)

Calculate the initial number of atoms

We know that 1 mole of any substance contains Avogadro's number of particles (in this case, atoms). Avogadro's number is approximately \( 6.022 \times 10^{23} \) atoms/mole. Initial number of atoms = 1 mole * \( 6.022 \times 10^{23} \) atoms/mole = \( 6.022 \times 10^{23} \) atoms

Calculate the remaining number of atoms

Multiply the initial number of atoms by the remaining fraction to find the remaining number of atoms after one day: Remaining number of atoms = Initial number of atoms * Remaining fraction Remaining number of atoms = \( (6.022 \times 10^{23}) \times 0.000122 \approx 7.347 \times 10^{20} \) atoms After one day, there are approximately \( 7.347 \times 10^{20} \) atoms of \({ }^{18} \mathrm{~F}\) left in the sample.

Key concepts

Radiotracers

Radiotracers are radioactive isotopes used in various fields of medicine and research to track and visualize different processes occurring in the body. These specially chosen isotopes undergo radioactive decay, helping scientists and doctors to detect and monitor the pathways of substances inside the body. As they decay, radiotracers emit radiation, which can be captured using imaging techniques, such as PET scans.

In a PET (Positron Emission Tomography) scan, radiotracers like \({ }^{18}F\) are administered to the patient. This isotope is particularly effective due to its moderate half-life of 110 minutes. This time frame allows sufficient durations for detailed imaging without posing significant harm from radiation exposure.

The ideal radiotracer will effectively travel to the desired part of the body, bind to the target tissues or molecules, and provide clear signals that can be used to compose detailed images. These characteristics make radiotracers invaluable in researching various medical conditions and assessing the effectiveness of new drugs.

Half-Life Calculation

The concept of half-life is essential in understanding radioactive decay. It is the time taken for half of the radioactive nuclei in a sample to decay. For example, the half-life of \({ }^{18}F\) is 110 minutes, meaning every 110 minutes, half of the remaining \({ }^{18}F\) atoms will have decayed.

To calculate how much of a radioactive substance remains after a certain period, we use the concept of half-lives passed during that period. In the context of a day, which consists of 1440 minutes, you can determine how many 110-minute intervals (i.e., half-lives) fit into this time frame.

Knowing 13 half-lives elapse in a day (as calculated), we then apply the decay formula: \[\text{Remaining fraction} = \left(\frac{1}{2}\right)^n\,\text{where }n\]\ is the number of half-lives. After 13 half-lives, the fraction of the substance remaining is approximately 0.000122, indicating a significant reduction from the original quantity.

Avogadro's Number

Avogadro's Number is a fundamental concept in chemistry, referring to the number of atoms, molecules, or other particles in one mole of a substance. This constant is approximately \(6.022 \times 10^{23}\) per mole.

In the context of the problem with \(^{18}F\), Avogadro’s Number allows us to determine the starting quantity of atoms when given a mole's worth of the isotope. Initially, one mole of \(^{18}F\) would contain approximately \(6.022 \times 10^{23}\) atoms.

Having established the initial quantity, Avogadro's Number is an instrumental part of further calculations to determine remaining atoms based on decay over time. By multiplying the initial number of atoms by the remaining fraction after a given number of half-lives, one can accurately compute the leftover atoms. In this example, after a day, about \(7.347 \times 10^{20}\) atoms of \(^{18}F\) remain, a substantial but predictable decrease facilitated by the understanding of Avogadro's constant.