Question

The synthetic radioisotope technetium-99, which decays by beta emission, is the most widely used isotope in nuclear medicine. The following data were collected on a sample of \({ }^{99} \mathrm{Tc}\) : $$ \begin{array}{ll} \hline \text { Disintegrations per Minute } & \text { Time (h) } \\ \hline 180 & 0 \\ 130 & 2.5 \\ 104 & 5.0 \\ 77 & 7.5 \\ 59 & 10.0 \\ 46 & 12.5 \\ 24 & 17.5 \\ \hline \end{array} $$ Using these data, make an appropriate graph and curve fit to determine the half-life.

Short answer

To determine the half-life of technetium-99, follow these steps: 1. Create a table with time (hours), Disintegrations Per Minute, and their natural logarithm (ln(DPM)). 2. Plot the semi-logarithmic graph, with time (x-axis) and ln(DPM) (y-axis), representing the exponential decay with a straight line. 3. Draw the best fit line through the plotted points. 4. Calculate the slope of the line using the slope formula (\(slope = \frac{y_2 - y_1}{x_2 - x_1}\)). 5. Obtain the half-life by calculating \(t_{1/2} = \frac{ln(2)}{\lambda}\), where \(slope = -\lambda\).

Step-by-step solution

Create a table with semi-logarithmic values

Prepare a table with time (T) in hours, Disintegrations Per Minute (DPM), and the natural logarithm of DPM values. Note that we'll use the logarithm base e, i.e., ln(DPM).

Plot a semi-logarithmic graph

Plot the semi-logarithmic graph, with time as the x-axis and ln(DPM) as the y-axis. This means, we have a linear relationship between the x and y values, as the exponential decay in activity will be represented as a straight line in a semi-log plot.

Best fit line

Draw the best fit straight line that passes through the plotted points. Ideally, this line should be the one that has the minimal deviation from the plotted points.

Calculate the slope of the line

To determine the half-life, we need the slope of the line. Calculate the slope by selecting any two points on the line, say \((x_1,y_1)\) and \((x_2,y_2)\), and apply the formula for slope: \(slope = \frac{y_2 - y_1}{x_2 - x_1}\)

Find the half-life

The slope of the line, obtained in step 4, is related to the decay constant (\(\lambda\)) through the following relationship: \(slope = -\lambda\) The half-life (t₁/₂) can be calculated using the decay constant: \(t_{1/2} = \frac{ln(2)}{\lambda}\) Replace the calculated slope value to obtain the half-life of technetium-99, and analyze the obtained value.

Key concepts

Nuclear medicine

Nuclear medicine utilizes radioactive substances for the diagnosis and treatment of diseases. One of their primary uses is in medical imaging. Doctors can visualize and diagnose conditions within the body that might not be observable directly. Technetium-99, specifically, plays a vital role because it emits gamma rays, which can be detected by imaging devices. This quality makes it highly valuable for scanning organs such as the brain, heart, and bones.
Because these procedures involve radioactive materials, safety measures are crucial. The isotopes used, like technetium-99, are carefully chosen because they have properties that allow for detailed scanning while posing minimal risk to patients. It's important to note that nuclear medicine provides a non-invasive way to gather diagnostic information, which is sometimes not possible with other imaging techniques like X-rays or CT scans.
In summary:

• Nuclear medicine leverages radioisotopes for diagnostic or therapeutic purposes.
• Technetium-99 is widely used due to its effective imaging capabilities.
• It provides a non-invasive method to study internal organs without surgery or invasive procedures.

Radioisotope decay

Radioisotope decay refers to the process by which an unstable atomic nucleus loses energy by emitting radiation. This decay can result in the transformation of the isotope into a different element or a different isotope of the same element. Technetium-99, for example, undergoes beta decay. This type of decay is crucial in nuclear medicine as it allows isotopes to reach a stable form.
Understanding the decay process is essential for predicting how long a radioisotope will remain active. By knowing the decay pattern and rate, medical professionals can time procedures to make the most of the isotopes while they are useful. In practical terms, once injected into a patient, the decay process helps visualize the target area, since the radiation can be detected.
Here are key points about radioisotope decay:

• It involves transformation due to emission of radiation.
• Helps in calculating the longevity and effectiveness of isotopes in medical applications.
• Involves changes at the nuclear level, which can convert isotopes into different elements.

Exponential decay curve

The exponential decay curve is a crucial concept in understanding radioisotope decay. It describes how the quantity of a radioactive isotope decreases over time. In a graph, the curve shows that initially, the substance decays rapidly, then the rate slows down. This pattern is a direct consequence of the exponential nature of the decay process, where the rate of decay is proportional to the remaining quantity of the substance.
This concept is especially useful when determining the half-life of an isotope like technetium-99. By plotting the natural logarithm of the disintegrations per minute against time, you get a straight line in a semi-log plot, simplifying the analysis. The slope of this line helps in calculating the decay constant, which is then used to determine the half-life.
Key characteristics of the exponential decay curve:

• Starts with a rapid decline, followed by a slower rate of decay.
• Provides a straightforward method to visualize and analyze decay data.
• Helps in calculating important parameters like half-life and decay constant.

Beta decay process

Beta decay is a type of radioactive decay where a beta particle, which is either an electron or a positron, is emitted. In the case of technetium-99, it undergoes beta-minus decay, where a neutron in the nucleus is transformed into a proton, emitting an electron and an antineutrino in the process. This transformation results in a change in the elemental identity of the atom.
This process is fundamental to nuclear medicine as it explains how isotopes change over time and how their emissions can be harnessed for imaging. Beta decay ensures that technetium-99m, used in diagnostics, eventually transforms into the stable isotope technetium-99. This decay is safe and predictable, which is crucial for medical processes.
Understanding the beta decay process involves:

• A transformation at the nuclear level, changing the number of protons and neutrons.
• The emission of beta particles, crucial for imaging techniques in medicine.
• Providing a pathway for an unstable isotope to reach a stable state.