Question

Radioactive samples are considered to become nonhazardous after 10 half-lives. If the half-life is less than 88 days, the radioactive sample can be stored through a decay-in-storage program in which the material is kept in a lead- lined cabinet for at least 10 half-lives. What percent of the initial material will remain after 10 half-lives?

Short answer

After 10 half-lives, 0.09765625% of the initial radioactive material will remain.

Step-by-step solution

Understanding Half-Life

A half-life is the time required for half of the radioactive substance to decay. After one half-life, 50% of the original sample remains.

Calculating Remaining Sample After Each Half-Life

The remaining amount of material after each half-life is found by multiplying the remaining material by 0.5 (or 50%).

Determining Remaining Material After 10 Half-Lives

After 10 half-lives, the calculation is as follows: Initial material * (1/2)^10. This represents the remaining percentage of the initial sample.

Performing the Calculation

The calculation for 10 half-lives looks like this: (1/2)^10, which equals 1/1024.

Converting To Percentage

To convert the fraction 1/1024 to a percentage, multiply it by 100. Therefore, 1/1024 * 100 gives the remaining percentage.

Key concepts

Radioactive Decay

Radioactive decay is a fundamental process wherein an unstable atomic nucleus loses energy by emitting radiation. This phenomenon is inherent to several elements, especially those with a high atomic number. During decay, atoms transform into a more stable form, which may be a different element altogether; this process is known as transmutation.

There are various types of decay, such as alpha, beta, and gamma decay, each characterized by the specific particles or energy it emits. The rate of radioactive decay is constant for a given isotope and is quantified by its half-life. Understanding the half-life of an isotope is crucial as it determines the time it takes for half of the radioactive atoms in a sample to decay. This intrinsic property is not affected by external factors such as temperature or pressure, making the half-life a reliable metric for studying radioactive substances.

Nuclear Chemistry

Nuclear chemistry explores the changes in the atomic nucleus and involves the study of radioactive isotopes and their properties. It's a branch of chemistry that overlaps with physics, particularly in areas concerning radioactivity, nuclear reactions, and the changes in chemical and physical properties that arise as nuclei transform. In practice, nuclear chemistry has vital applications ranging from medical treatments using radioisotopes to energy production in nuclear reactors.

In the context of half-lives and radioactive decay, nuclear chemistry provides the theoretical framework for understanding how and why certain elements undergo radioactive decay. For instance, nuclear chemists examine the stability of atomic nuclei and predict the types of radioactive decay that may occur. They also study the resultant by-products, including any radiation released during the process.

Additionally, the principles of nuclear chemistry are applied in determining safe handling and storage protocols for radioactive materials, ensuring that the use of such materials takes health and environmental considerations into account.

Decay-In-Storage

Decay-in-storage is a practical application of our understanding of half-lives in nuclear chemistry. It is a method for managing certain types of radioactive waste that have relatively short half-lives. By storing the radioactive materials securely for a duration of at least 10 half-lives, their radioactivity diminishes to levels that are considered nonhazardous, or at the very least, more manageable.

The use of a lead-lined cabinet for storage is standard practice, providing significant protection from radiation while the material undergoes its natural decay process. Notably, calculating that only about 0.1% of the original material remains after 10 half-lives makes the decay-in-storage approach an efficient solution for certain low-level wastes. This reduces the need for more complex and expensive disposal methods and helps in mitigating the potential risks associated with long-term storage of radioactive materials.