What is the original number of atoms in a radioactive sample remaining at the end of one mean life?

To understand the concept of radioactive decay and the number of atoms remaining after one mean life, we first need to clarify what a "mean life" or "mean lifetime" is. In the context of radioactive decay, the mean life is the average time it takes for a radioactive substance to decay. This is closely related to the half-life, which is the time required for half of the radioactive atoms in a sample to decay.

Radioactive Decay Basics

Radioactive decay is a random process where unstable atomic nuclei lose energy by emitting radiation. This decay can be described statistically, and we often use the concept of half-life to express how quickly a substance decays. The half-life is the time it takes for half of the radioactive atoms in a sample to transform into a different element or isotope.

Understanding Mean Life

The mean life (τ) of a radioactive isotope is related to its half-life (t_1/2) by the formula:

• τ = t_1/2 / ln(2)

Here, ln(2) is the natural logarithm of 2, approximately equal to 0.693. This relationship shows that the mean life is longer than the half-life.

Calculating Remaining Atoms

Now, let’s consider a sample of radioactive atoms. If we start with N₀ atoms, after one mean life (τ), we can determine how many atoms remain using the exponential decay formula:

• N(t) = N₀ * e^(-t/τ)

For t = τ, this simplifies to:

• N(τ) = N₀ * e^(-1)

Since e^(-1) is approximately 0.3679, we find that:

• N(τ) ≈ 0.3679 * N₀

This means that after one mean life, about 36.79% of the original number of atoms will remain. So, if you started with 100 atoms, approximately 37 atoms would still be present after one mean life.

Example for Clarity

Let’s say you have a sample of 200 radioactive atoms. After one mean life, the calculation would be:

• N(τ) ≈ 0.3679 * 200 ≈ 73.58

Rounding this, you would have approximately 74 atoms remaining after one mean life.

Key Takeaways

In summary, after one mean life, about 36.79% of the original radioactive atoms in a sample will remain. This concept is crucial in fields such as nuclear physics, radiometric dating, and understanding the behavior of radioactive materials in various applications.