Question

Most students have no trouble understanding that \(1.5 \mathrm{~g}\) of a \(24-g\) sample of a radioisotope would remain after \(8 \mathrm{~h}\) if it had \(t_{1 / 2}=2 \mathrm{~h}\). What they don't always understand is where the other \(22.5 \mathrm{~g}\) went. How would you explain this disappearance to another student?

Short answer

The 22.5 g decay into different elements or isotopes, leaving only 1.5 g of the original radioisotope after 8 hours.

Step-by-step solution

Understanding Radioactive Decay

Radioactive decay is a process in which an unstable atomic nucleus loses energy by emitting radiation. During each half-life period, half of the remaining substance decays into another element or isotope, which may or may not be stable.

Determining Number of Half-Lives

The given half-life of the isotope is 2 hours, and the time period considered is 8 hours. Calculate how many half-life periods occur in 8 hours by dividing the total time by the half-life: \[ \frac{8}{2} = 4 \text{ half-lives} \].

Calculating Remaining Mass After Each Half-Life

Initially, we have 24 g of the isotope. With each half-life, the mass of the remaining isotope is halved. After 1 half-life, 24 g becomes 12 g; after 2 half-lives, 12 g becomes 6 g; after 3 half-lives, 6 g becomes 3 g; after 4 half-lives, 3 g becomes 1.5 g.

Explaining Where the Mass Goes

The 22.5 g that 'disappear' are transformed into other materials. As the radioactive isotope decays, it changes into another substance, which may be stable or undergo further decay. This transformation results in less radioactive material corresponding to our initial isotope, but the total atomic mass remains conserved.

Key concepts

Half-life

The concept of half-life is central to understanding radioactive decay. It refers to the time required for half of a radioactive substance to decay into another element or isotope. This process continues through multiple half-lives, steadily reducing the amount of the original radioactive material. For instance, if we start with 24 grams of a radioisotope and its half-life is 2 hours, in 2 hours, only 12 grams will be left. This pattern continues with each subsequent half-life: after 4 hours, 6 grams remain; after 6 hours, 3 grams remain; and finally, after 8 hours, only 1.5 grams are left.

It's important to remember that each half-life reduces the current amount by half, not the original amount. Understanding this can help students grasp why significantly less of the radioisotope remains after a few half-lives. It provides a predictable pattern that can be calculated easily and aids in determining how much of a radioactive substance will remain over a specified period.

Nuclear Transmutation

Nuclear transmutation is another key concept in radioactive decay. This term describes the transformation of one element into another as a result of changes in the nucleus. During radioactive decay, the unstable nucleus releases energy, and often particles, which alters its identity. Thus, a particular radioisotope can become a different element or isotope.

For example, when a radioisotope decays, it may emit an alpha or beta particle. This emission changes the number of protons or neutrons in the nucleus, transforming the element. Understanding nuclear transmutation helps explain what happens to the disappearing 22.5 grams in our scenario. They don't just vanish; instead, they're converted into different substances.

• The number of neutrons and protons changes through decay, resulting in a different configuration.
• The process can lead to the formation of a stable isotope, but it might also result in a series of further decay until stability is reached.

This continual change is a roadmap of transformation from one nuclear state to another.

Mass Conservation

While radioactive decay alters the nucleus of an atom, the principle of mass conservation tells us that total mass and energy in a system remain constant throughout the process. Even though 22.5 grams of the original isotope seem to disappear, they have not been lost. Instead, the mass has been converted and accounted for in the new materials formed through the decay process.

Mass conservation is sometimes difficult to visualize because the mass is redistributed among various elements or isotopes. Additionally, some tiny mass differences convert into energy, as outlined by Einstein's famous equation, \( E=mc^2 \). Despite these transformations, nature ensures that the overall amount of matter and energy remains unchanged.

• The total mass of reactants equals the total mass of products plus any energy emitted.
• This balance confirms that nothing is ever lost, just rearranged or transformed.

Understanding this concept provides clarity on how material transitions occur without violating natural laws.