Understanding Radioactive Decay and Exponential Decay Using a 16 Gram Isotope Sample

The process of radioactive decay is governed by the principles of exponential decay, which can be analyzed using mathematical models. In this article, we will explore how to calculate the amount of a radioactive isotope remaining in a sample after a specific period. A detailed example will be provided, focusing on a 16 gram sample with a half-life of 1.25 years.

The Exponential Decay Formula and Its Application

The formula for calculating the remaining quantity of a radioactive isotope is given by:

N N_0 left(frac{1}{2}right)^{frac{t}{t_{1/2}}}

Where,

N is the remaining quantity of the substance. N_0 is the initial quantity of the substance (16.0 grams in this case). t is the total time elapsed (5 years). t_{1/2} is the half-life of the substance (1.25 years).

Step-by-Step Calculation

First, we need to determine how many half-lives have passed in 5 years:

Number of half-lives frac{t}{t_{1/2}} frac{5 text{ years}}{1.25 text{ years}} 4

Now, using the decay formula:

N 16.0 left(frac{1}{2}right)^{4}

We calculate left(frac{1}{2}right)^{4}:

left(frac{1}{2}right)^{4} frac{1}{16}

Finally, we find N:

N 16.0 times frac{1}{16} 1.0 text{ gram}

Thus, after 5 years, 1.0 gram of the radioactive isotope will remain in the sample.

Additional Clarification on Radioactive Decay Products

It's important to note that radioactive decay doesn't just vanish the original isotope. Instead, the decay process produces a new isotope, often referred to as a decay product. This newly formed isotope could be:

Stable and non-radioactive. Radioactive, with a different half-life and decay mode.

The new isotope arises depending on the type of decay undergone by the original atom, such as alpha, beta, positron, or gamma emission. For instance, if the original 16 grams of 'this isotope' are reduced to 1 gram after 5 years, the remaining 15 grams will generally transform into another isotope.

Therefore, the significant takeaway is that initially, we have 16 grams of the isotope, and after 5 years, 1.0 gram remains unchanged. The rest of the sample, 15 grams, transforms into a different isotope, which may or may not be radioactive, depending on the specific decay process.

Conclusion

Radioactive decay is a fascinating and crucial phenomenon in various scientific fields. Through the use of the exponential decay formula, we can accurately predict the amount of a radioactive substance remaining after a given period. Understanding this concept is essential for fields such as nuclear physics, radiology, and environmental science.

Related Keywords

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