Question

Tc-99 is often used in medicine and has a half-life of about \(6 \mathrm{~h}\). If a patient is given \(20.0\) micrograms of Tc99 , and assuming that none is lost due to other processes besides radioactive decay, how much Tc-99 is left after \(24 \mathrm{~h}\) ?

Short answer

After 24 hours, there will be 1.25 micrograms of Tc-99 left.

Step-by-step solution

Determine the number of half-lives elapsed

Calculate the number of half-lives that have elapsed after 24 hours. A half-life is the time required for half of a sample of a radioactive isotope to decay. The number of half-lives () is found by dividing the total time elapsed by the half-life of the substance. In this case: = 24 hours / 6 hours per half-life = 4 half-lives.

Apply the half-life formula

The amount of a radioactive substance remaining after a certain number of half-lives can be calculated using the formula: Remaining amount = Initial amount × (1/2)^n. Substitute the initial amount of Tc-99 (20.0 micrograms) and the number of half-lives (4) into the formula to get the remaining amount.

Calculate the remaining Tc-99

Using the formula from Step 2, the calculation is: Remaining amount = 20.0 µg × (1/2)^4 = 20.0 µg × 1/16 = 1.25 µg. This is the amount of Tc-99 that remains after 24 hours.

Key concepts

Radioactive Half-Life

Understanding the concept of a radioactive half-life is key to nuclear chemistry. The half-life of a radioactive isotope is the time it takes for half of the sample to decay into another element or isotope. For instance, the isotope Tc-99 mentioned in our exercise has a half-life of approximately 6 hours. This means that if you start with a 20.0 microgram sample of Tc-99, after 6 hours only 10.0 micrograms would remain; the rest has transformed due to radioactive decay.
In a medical setting, knowing the half-life of radioactive materials is crucial for patient safety and treatment efficacy. It allows precise dosage calculations so that the benefits can be maximized while minimizing potential radiation exposure risks. When applying this concept to homework problems, think of it as a ticking clock where every tick is a new half-life interval, and with each interval, the remaining amount of substance reduces by half.
It's also interesting to note that half-life is a constant value; it doesn't change based on the amount of the substance or its concentration. This constancy forms the backbone of calculations involving decay in nuclear chemistry.

Nuclear Chemistry

At the heart of nuclear chemistry is the study of the changes in an atom's nucleus. Radioactive decay, which our exercise focuses on, is one of the most pivotal concepts in this field. Atoms of radioactive elements such as Tc-99 contain nuclei that are unstable, and as they strive to reach a stable state, they emit radiation, transforming into different atoms over time.
This transformation through decay can produce alpha particles, beta particles, gamma rays, and other forms of radiation, each with different properties and implications for science and medicine. For example, the decay of Tc-99 is utilized for diagnostic imaging in the medical field, making it a practical application of nuclear chemistry.

Real-World Impact of Nuclear Chemistry

• Energy Production: Nuclear reactors use controlled reactions to generate electricity.
• Medicine: As with Tc-99, radioactive isotopes are used in diagnosis and treatment.
• Archaeology: Carbon dating allows us to determine the age of ancient artifacts.

This discipline isn't limited to the lab—it affects many aspects of our daily lives.

Exponential Decay

The process of radioactive decay follows a pattern known as exponential decay. This is characterized by a quantity decreasing at a rate that is proportional to its current value, resulting in a rapid decline that slows over time. The decay of Tc-99 is an excellent example of this principle.
Exponential decay isn't linear; it's a curve that swoops downwards steeply at first and then gradually approaches zero without ever touching it. The mathematical formula for exponential decay, as used in our exercise, helps us determine the amount of Tc-99 left after any number of half-lives. The formula looks like this: Remaining amount = Initial amount × (1/2)^n, where 'n' is the number of half-lives that have passed.
In simple terms, if you have a piece of paper and fold it in half over and over again, each fold represents a 'half-life.' Just like you can never fold the paper infinitely, radioactive substances never truly reach zero; they just get incredibly close after many half-lives have elapsed. Practical understanding of exponential decay is essential for those working in fields impacted by nuclear chemistry and is a core concept that can extend to other areas like finance and biology.