Assume a sample of material must undergo a single radioactive decay process (beta decay for example). How many nuclei in the sample must undergo this decay in order for the sample to be considered stable and completely non-radioactive?
To determine the number of nuclei that must undergo radioactive decay for the sample to be stable and completely non-radioactive, we need to understand the concept of half-life.
The half-life of a radioactive substance is the time it takes for half of a given quantity of nuclei to decay. During each half-life, the number of remaining radioactive nuclei reduces by half. The process continues until there are no more radioactive nuclei present.
To consider the sample stable and non-radioactive, we should have a negligible number of radioactive nuclei remaining. This typically means that the number of remaining nuclei should be less than 0.1% of the initial number of nuclei.
To calculate the number of nuclei that need to decay, we can use the equation:
Final number of nuclei = Initial number of nuclei × (1/2)^(number of half-lives)
By rearranging the equation, we can solve for the number of half-lives:
Number of half-lives = log(Final number of nuclei / Initial number of nuclei) / log(1/2)
Let's assume the initial number of nuclei in the sample is N0. If we want the sample to be stable and non-radioactive, we can set the final number of nuclei to be 0.001% (0.00001) of the initial number:
Final number of nuclei = 0.00001 × N0
Now we substitute these values into the equation and solve for the number of half-lives:
Number of half-lives = log(0.00001 × N0 / N0) / log(1/2)
Simplifying the equation:
Number of half-lives = log(0.00001) / log(1/2)
Using logarithmic properties, we can determine that log(1/2) is approximately equal to -0.3010. Substituting this value into the equation:
Number of half-lives = log(0.00001) / -0.3010
Now, let's calculate the number of half-lives:
Number of half-lives = -4.0 / -0.3010
Number of half-lives ≈ 13.29
Since the number of half-lives must be a whole number, we round it up to 14.
Therefore, at least 14 nuclei in the sample must undergo radioactive decay for the sample to be considered stable and completely non-radioactive.