Final answer:
After two half-lives, three-quarters of the original parent isotope would have decayed into the daughter isotope. If you start with 2 micrograms of the parent isotope, 1.5 micrograms will have decayed by the end of the second half-life, plus the 1 microgram from the first half-life equals 2.5 micrograms, which could be approximated to 3 micrograms of daughter isotope.
Step-by-step explanation:
When dealing with the decay of radioactive isotopes, it's essential to understand the concept of half-life. One half-life is the time period required for half of the radioactive isotope ("parent") in a sample to decay into its product ("daughter"). After one half-life, half the parent isotope is transformed into the daughter isotope. After two half-lives, one quarter of the original amount of the parent isotope remains, while the remainder has decayed to the daughter isotope.
So, for the question at hand, after two half-lives, one quarter of the original parent isotope would be left and three quarters would have decayed into the daughter isotope. If the original amount of a parent isotope is given as 2 micrograms, after one half-life, there would be 1 microgram of the parent isotope and 1 microgram of the daughter isotope. After another half-life, one half of the remaining parent isotope (0.5 micrograms) would remain, and additional 0.5 microgram would have decayed, making a total of 1.5 micrograms of the daughter. Adding this to the first 1 microgram from the previous half-life, we would have a total of 2.5 micrograms of the daughter isotope, which could round to 3 micrograms if we are following the answer options provided by the student.