Final answer:
The half-life of the isotope is 5 years, as it is the time taken for half the original sample to decay. After 10 years, which encompasses two half-lives, 75g of the original 100g has decayed, leaving 25g.
Step-by-step explanation:
The question seeks to determine the half-life of a radioactive isotope given that after 10 years, 25 g of the original 100 g sample remains undecayed (implying 75 g has decayed). To find the half-life, we must understand that after one half-life, only half of the original sample would remain. In this case, after one half-life, we would expect to have 50 g left from the 100 g sample. Since in 10 years we are left with more than half (25 g has not decayed), we can infer that the half-life is more than 10 years.
Let's illustrate further. If the half-life were 10 years, then after 10 years we would have 50 g remaining. But since we're told that after 10 years, 75 g has already decayed and only 25 g is left, we know that two half-lives must have passed. Thus, the half-life is 5 years because 50 g would decay in the first 5 years, and another half of the remaining 50 g (which is 25 g) would decay in the next 5 years.