Final answer:
It will take approximately 79852 years for 100 grams to decay to 10 grams with a half-life of 24100 years.
Step-by-step explanation:
In radioactive decay, the quantity of radioactive nuclei decreases by half in one half-life. For example, if there are 100 grams of a radioactive substance with a half-life of 24100 years, after 24100 years there will be 50 grams remaining, after 48200 years there will be 25 grams remaining, and so on.
To calculate the time it will take for 100 grams to decay to 10 grams, we need to find out how many half-lives it would take.
We divide the initial mass (100 grams) by the final mass (10 grams) and take the logarithm base 2 of that value. This gives us the number of half-lives it takes for the decay.
Using the formula: number of half-lives = log2(initial mass/final mass), we can calculate the number of half-lives.
In this case, number of half-lives = log2(100/10)
= log2(10)
= 3.32 (approximately).
Since each half-life corresponds to a time of 24100 years, we multiply the number of half-lives (3.32) by the half-life time (24100 years) to obtain the total time it will take for 100 grams to decay to 10 grams.
The calculation is: total time = number of half-lives x half-life time
= 3.32 x 24100 = 79852 years (approximately).