Final answer:
After the first half-life, 20 grams of the radioactive isotope would remain, and after the second half-life, 10 grams would remain, indicating that two half-lives have passed since the initial 40-gram sample.
Step-by-step explanation:
To determine how many half-lives have passed for a radioactive isotope when its mass decreases from 40 grams to 10 grams, we can follow a step-by-step process. Initially, there are 40 grams of a radioactive isotope. After the first half-life, half of this would have decayed, leaving 20 grams. After the second half-life, another half (which is half of 20 grams) would decay, leaving 10 grams. Therefore, the number of half-lives that have passed is two. The concept of radioactive decay and half-life is critical to understanding this process. Here, 10 grams represents a quarter of the initial amount, which signifies that two half-lives have occurred because with each half-life, the quantity of the radioactive substance is halved.
The number of half-lives that have passed can be calculated by dividing the initial mass of the radioactive isotope by the final mass and taking the logarithm base 2 of that ratio. In this case, the initial mass is 40 grams and the final mass is 10 grams. So, using the formula:
{{{number_of_half_lives}}} = log2(initial_mass/final_mass)
{{{number_of_half_lives}}} = log2(40/10)
{{{number_of_half_lives}}} = log2(4)
{{{number_of_half_lives}}} = 2
Therefore, 2 half-lives have passed.