Question

Technetium-99 is used in medicine as a source of radiation. A typical dose is \(25 \mathrm{mCi}\). How long does it take for the activity to reduce to \(0.100 \mathrm{mCi}\) ? The half-life of \({ }^{99}\) Tc is 210,000 y.

Short answer

It takes about 1,675,797 years for the activity to reduce to 0.100 mCi.

Step-by-step solution

Understanding Half-Life

The half-life of a substance is the time it takes for half of the radioactive atoms to decay. For Technetium-99, its half-life is given as 210,000 years. This information will be essential in calculating how long it takes for an initial amount of radiation to decrease to a specified amount.

Using Decay Formula

The activity of a radioactive substance can be described using the formula: \[A(t) = A_0 \times \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}}\] where \(A(t)\) is the activity at time \(t\), \(A_0\) is the initial activity, and \(T_{1/2}\) is the half-life. Here, \(A_0 = 25\) mCi and \(T_{1/2} = 210,000\) years.

Setting Up the Equation

Plug in the known values into the decay formula: \(0.100 = 25 \times \left(\frac{1}{2}\right)^{\frac{t}{210,000}}\). We need to solve this equation for \(t\).

Solving for Time, t

First, divide both sides of the equation by 25 to isolate the decay term:\[\frac{0.100}{25} = \left(\frac{1}{2}\right)^{\frac{t}{210,000}}\]Which simplifies to:\[0.004 = \left(\frac{1}{2}\right)^{\frac{t}{210,000}}\]Next, take the logarithm of both sides to solve for \(t\).

Applying Logarithms

Taking natural logarithms on both sides:\[\ln(0.004) = \frac{t}{210,000} \times \ln\left(\frac{1}{2}\right)\]Solve for \(t\) by first evaluating the logarithms, and then multiply both sides by 210,000.

Calculating Time

First, calculate the natural logarithms: \(\ln(0.004) \approx -5.521\) and \(\ln\left(\frac{1}{2}\right) \approx -0.693\).Now use these in the equation:\[t = \frac{210,000 \times (-5.521)}{-0.693}\]

Computing Final Answer

Calculate the final result: \[ t = \frac{210,000 \times 5.521}{0.693} \approx 1,675,797 \text{ years} \]

Key concepts

Half-Life Calculation

In radioactive decay, the concept of half-life is crucial. Half-life refers to the period required for half of the radioactive atoms in a sample to decay. It is a constant that characterizes the rate of decay of radioactive substances. Understanding this helps in predicting how a radioactive material behaves over time.

For example, if we begin with 100 atoms of a radioactive substance with a half-life of 1 year, in 1 year only 50 will remain. After 2 years, just 25, and so on.

• Half-life is independent of the initial amount of the substance.
• It provides a simple way to predict the decay process, without requiring complex computations.

This concept is widely used in fields like medicine, archeology, and geology, providing critical insights into things like the age of ancient objects or the duration needed for medical isotopes to lose potency.

Technetium-99

Technetium-99, often represented as (^{99m}Tc) , is an isotope commonly used in medical imaging procedures. Its radioactive properties make it useful, as it emits gamma rays which are detectable by specialized medical imaging scanners.

Some key points about Technetium-99 include:

• It is used primarily due to its ideal half-life of about 6 hours, which is not too short and not too long for medical applications.
• The reduced half-life allows it to decay quickly enough to minimize radiation exposure but still long enough to perform detailed imaging studies.
• Usually administered as part of a radiopharmaceutical, Tc-99m helps doctors diagnose various conditions, enhances imaging clarity, and allows for the assessment of organ function.

Given its attributes, Technetium-99 remains one of the most utilized isotopes in nuclear medicine today, balancing efficiency with patient safety.

Exponential Decay Formula

Radioactive decay can be mathematically modeled using the exponential decay formula. This formula is a powerful tool to calculate the decay of substances over time. The formula used is:

\[A(t) = A_0 \times \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}}\]

Here, \(A_0\) represents the initial quantity or activity of the substance, and \(A(t)\) is the remaining activity after time \(t\). This formula explicitly relies on the half-life \(T_{1/2}\), illustrating that the substance loses activity at a predictable, decaying rate.

Breaking down the components:

• \(A_0\) is the starting activity, like 25 mCi for Technetium-99 in medical doses.
• \(\frac{1}{2}\) reflects the halving nature of half-life.
• \(t/T_{1/2}\) conveys the number of half-lives that have elapsed.

By using logarithms, we can solve for \(t\) if the current activity and half-life are known. This makes it easy to assess how long it will take for a radioactive sample to reach a desired level, demonstrating real-world applications in fields such as medicine and environmental science.