Final answer:
Using the half-life of cesium-137, which is 30 years, it would take two half-lives (60 years) for a 10-g sample to be reduced to slightly more than 2 g. To determine the exact time when only 2 g remains requires a more complex calculation, but it is slightly less than 60 years.
Step-by-step explanation:
To calculate the time it will take for a sample to reduce from 10 grams to 2 grams, given the half-life of cesium-137 is 30 years, we can use the concept of radioactive decay. Radioactive decay is an exponential process, and after each half-life, half of the remaining radioactive nuclei decay to daughter elements. Since we start with a 10-g sample and want 2 g to remain, this corresponds to ¼ of the original amount, or two half-lives (since (1/2) × (1/2) = 1/4).
Therefore, after one half-life (30 years), 5 g remains, and after two half-lives (2 × 30 years = 60 years), 2.5 g would remain. Since we want exactly 2 g to remain, it will take slightly less than two full half-lives. To find this exact time, we could set up a more complex logarithmic equation. However, it's important to note that after 60 years, slightly more than 2 g will still be present. Consequently, the time to reach exactly 2 g is slightly less than 60 years.