Question

Fluorine-18 is an isotope used in Positron Emission Tomography (PET) to scan the brain. If a researcher has \(1.50 \mu \mathrm{g}\) of \({ }^{18} \mathrm{~F}\), how long before it decays to \(1.0\) ng? The half-life of \({ }^{18} \mathrm{~F}\) is \(109.8\) minutes. a. \(2.9 \times 10^{-2}\) hours b. 91 hours c. 39 hours d. 19 hours

Short answer

The decay time is approximately 91 hours.

Step-by-step solution

Understand the problem

The problem asks how long it will take for a mass of Fluorine-18 to decay from \(1.50 \mu \text{g}\) to \(1.0 \text{ ng}\). The half-life of Fluorine-18 is given as 109.8 minutes. We need to convert both masses to the same units and then use the decay formula.

Convert units

Convert the initial mass from micrograms to nanograms: 1 microgram = 1000 nanograms, so \(1.50 \mu \text{g} = 1500 \text{ ng}\). The final mass is already given in nanograms as 1.0 ng.

Apply the decay formula

The decay of a radioactive substance is given by the formula \(N(t) = N_0 \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}}\), where \(N(t)\) is the remaining mass, \(N_0\) is the initial mass, \(t\) is the time elapsed, and \(T_{1/2}\) is the half-life. Here, \(N_0 = 1500 \text{ ng}\), \(N(t) = 1.0 \text{ ng}\), and \(T_{1/2} = 109.8 \text{ min}\).

Solve for time t

Set up the equation: \(1.0 = 1500 \times \left(\frac{1}{2}\right)^{\frac{t}{109.8}}\). Divide both sides by 1500 to get \(\frac{1}{1500} = \left(\frac{1}{2}\right)^{\frac{t}{109.8}}\). To isolate \(t\), take the natural logarithm of both sides: \(\ln\left(\frac{1}{1500}\right) = \frac{t}{109.8} \ln\left(\frac{1}{2}\right)\).

Calculate t in minutes

Calculate \(\ln\left(\frac{1}{1500}\right)\) and \(\ln\left(\frac{1}{2}\right)\), then solve for \(t\): \(t = \frac{\ln\left(\frac{1}{1500}\right)}{\ln\left(\frac{1}{2}\right)} \times 109.8\).

Convert t to hours

After calculating \(t\) in minutes, convert to hours: \(\text{time in hours} = \frac{t}{60}\). Calculate this to determine which answer fits best.

Key concepts

Half-life Calculation

Radioactive decay happens because unstable isotopes emit particles, turning into different elements or isotopes. An important concept related to radioactive decay is the half-life, which is the time it takes for half of a sample to decay. This means if you begin with a certain amount of a radioactive substance, half of it will have decayed after one half-life period. The half-life is crucial in calculations because it helps predict how long a sample takes to reach a particular state of decay.

For instance, the decay of a substance can be described mathematically as follows: \[N(t) = N_0 \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}}\]where:

• \( N(t) \): Remaining mass at time \( t \)
• \( N_0 \): Initial mass
• \( T_{1/2} \): Half-life of the substance

Using this formula, students can predict the time it takes for a particular amount of substance to decay by rearranging the formula and solving for \( t \). This involves logarithmic calculations, where the natural logarithm is often used to isolate the variable \( t \), allowing us to determine the time at which a particular mass remains.

Fluorine-18

Fluorine-18 is a radioisotope significant in the field of medicine. It is a positron emitter, which makes it highly suitable for imaging techniques like Positron Emission Tomography (PET).

Fluorine-18 has a relatively short half-life of approximately 109.8 minutes. This is ideal for medical imaging because the isotope decays quickly enough to minimize long-term radiation exposure to patients but allows sufficient time to conduct a PET scan. During its decay, Fluorine-18 releases positrons. When these positrons encounter electrons, they annihilate, producing gamma rays that the PET scanner detects to create detailed images of the body's internal structures and functions.

In practice, this means that handling and storing Fluorine-18 requires stringent safety protocols due to its radioactive nature. Nevertheless, its short lifespan demands efficient logistics to ensure that it is utilized in medical scans soon after production.

Positron Emission Tomography (PET)

Positron Emission Tomography (PET) is a powerful imaging technique in medical diagnostics.

PET scans take advantage of the properties of substances like Fluorine-18, which emit positrons during decay. These positrons travel short distances before encountering electrons, resulting in the emission of gamma rays. PET scanners detect these gamma rays to produce comprehensive images of metabolic processes.

PET is predominantly used in oncology, cardiology, and neurology. The scans allow healthcare professionals to detect abnormalities and monitor disease progression by revealing changes in cellular activity. This makes PET a valuable tool for early detection of cancers, brain disorders, and heart diseases, offering insights that other imaging techniques might not reveal.

The precision of PET is remarkable as it doesn't just show structural changes but can illustrate functional activities within tissues. This aspect is pivotal in advancing personalized medicine and treatment plans tailored specifically to a patient's unique physiological conditions.

Unit Conversion in Chemistry

Unit conversion is a fundamental skill in chemistry, essential for solving problems accurately. It allows chemists and students to express quantities in different units that align with the context of the problem.

In the context of radioactive decay, converting units like micrograms to nanograms can be crucial. For instance:

• 1 microgram (\(\mu\text{g}\)) = 1000 nanograms (ng)
• This fact helps standardize units for calculations involving changes in mass

Using correct units is vital. Whether you're calculating the remaining quantity of a substance or solving for time, consistent unit usage avoids errors that could result in incorrect scientific interpretations.

Learning to adeptly navigate between units often involves dimensional analysis, allowing for smooth transitions between complex conversions in chemistry problems. Over time, this skill enhances accuracy in laboratory settings and when working with theoretical problems in scientific research.