The half-life of radium- \(-224\) is 3.66 days. What was the original mass of radium-224 if 0.0500 g remains after 7.32 days?

Half-life is a term that is crucial in the field of nuclear physics and chemistry. It is the time required for half the atoms of a radioactive substance to undergo decay. It is a measure of the speed at which a radioactive substance transforms.

For example, if we start with a 100g sample of a radioactive isotope and its half-life is 5 years, then after 5 years, only 50g of the substance would remain undecayed. After another 5 years (making a total of 10 years), 25g would remain, and this pattern continues. The concept of half-life applies to all radioactive elements, providing a consistent way to describe decay rates despite the substance involved.

The calculation of the original mass of a radioactive substance using half-life presumes that the decay follows a predictable pattern, which allows us to backtrack from the amount remaining to the amount with which we started. This is quite useful in various applications such as carbon dating, medical diagnosis, and nuclear power generation.

Radium-224 is a radioactive isotope of radium with a relatively short half-life of 3.66 days. The short half-life denotes that radium-224 decays quickly compared to many other radioactive substances. Radium, in general, is historically known for its use in luminescent paints and medical applications before the risks of radiation were fully understood.

Given its rapid decay rate, radium-224 is used today in cancer treatment, specifically in targeted alpha therapy. It is advantageous because it delivers high-energy particles directly to cancer cells, minimizing damage to surrounding healthy cells. The knowledge of its half-life makes it possible to calculate specific dosages for treatments and ensure the safety and effectiveness of the medical procedures in which it is used.

Understanding the half-life of radium-224 is essential when tackling problems involving the decay of this isotope, because it allows one to determine how much of the isotope will remain after a certain period, which is a frequent question in both scientific and educational contexts.

Exponential decay is the process by which a quantity decreases at a rate proportional to its current value. In the context of radioactive substances, the quantity we often refer to is the mass of the substance. The mathematical model for exponential decay is captured by the formula, \( N(t) = N_0(1/2)^{t/T} \) where:

• \( N(t) \) is the amount of substance remaining after time \( t \),
• \( N_0 \) is the original amount of the substance,
• \( T \) is the half-life of the substance.

In the case of radium-224, the exponential decay formula helps us calculate the amount of isotope remaining after a given time period. For example, if you know the half-life and the remaining mass, you can determine the original mass, as in the exercise provided.

Moreover, understanding exponential decay is important not just for radioactive substances but also for other fields that model growth and decay such as finance, biology, and environmental science. The exponential decay equation provides a powerful tool for predicting future values based on current observations.