Calculating the Remaining Fraction of a Radioative Isotope: A Step-by-Step Guide

Understanding Half-Life and Radioactive Decay

The concept of half-life is fundamental in understanding the behavior of radioactive isotopes. A half-life is the time it takes for half of a given amount of a radioactive isotope to decay or 'degrade' into its product. This is an exponential process, meaning the amount of the isotope decreases by a constant factor over regular intervals of time.

Formulas for Calculating Radioactive Decay

There are several ways to calculate the remaining quantity of a radioactive isotope after a certain period. The most common method involves the use of an exponential decay formula based on the half-life of the isotope.

The formula for exponential decay is given by:

N_t N_0 left(frac{1}{2}right)^{frac{t}{T_{1/2}}}

Nt is the quantity remaining after time t. N0 is the initial quantity. T1/2 is the half-life of the isotope. t is the elapsed time.

Example: A Radioactive Isotope with a Half-Life of 8 Days

Let's take the example of a radioactive isotope with a half-life of 8 days. We want to determine how much of the original quantity remains after 72 days.

Step 1: Calculate the Number of Half-Lives

First, we need to find out how many half-lives fit into the elapsed time of 72 days:

Number of half-lives frac{t}{T_{1/2}} frac{72}{8} 9

Step 2: Apply the Exponential Decay Formula

Now, we can use the formula to find the fraction of the original quantity that remains:

N_t N_0 left(frac{1}{2}right)^{9}

Therefore, the fraction of the atoms that will remain after 72 days is:

left(frac{1}{2}right)^{9} frac{1}{512}

This means that roughly 0.1953% or 0.001953 of the original quantity remains after 72 days.

Additional Calculations and Methods

Using logarithms, another method to calculate the remaining fraction of a radioactive isotope is:

T1/2 frac{ln(C_0 / C)}{k}

Where:

T1/2 is the half-life. Co is the initial quantity. C is the current quantity. k is the decay rate constant.

We can also use the y a1 - b^x formula to find the decay rate and then apply it to determine the remaining fraction after a specific time. However, for simple calculations like this example, the first method is more straightforward.

Conclusion

In conclusion, the remaining fraction of a radioactive isotope after 72 days, given a half-life of 8 days, is approximately 0.001953, or roughly 0.1953% of the original quantity. This is a direct result of the exponential decay process, where the quantity is halved in each half-life period.