Question

A radioactive isotope has a half-life of 5700. years. Given a 100.0 gram sample of the isotope initially, after a number of years have passed, we find that only 6.250 grams of the original isotope remain. How many years old is the sample?

a 2.280 x 104 yr
b 1.140 x 104 yr
c 1.710 x 104 yr
d 2.850 x 104 yr

Answer 1

Using the concept of radioactive decay and knowing the half-life of the isotope is 5700 years, we find that four half-lives have passed to reduce the 100.0 gram sample to 6.250 grams. This results in an age of 22800 years for the sample.

Step-by-step explanation:

To calculate the age of the sample based on the remaining mass of the original isotope and its half-life, we use the concept of radioactive decay. Given a half-life of 5700 years and only 6.250 grams remaining from a 100.0 gram sample, we need to determine how many half-lives have passed. Each half-life reduces the remaining amount of the isotope by half:

• First half-life: 100.0 g to 50.0 g
• Second half-life: 50.0 g to 25.0 g
• Third half-life: 25.0 g to 12.5 g
• Fourth half-life: 12.5 g to 6.25 g

This means that four half-lives have passed. To find the total number of years, we multiply the number of half-lives by the half-life duration:

4 (half-lives) × 5700 years/half-life = 22800 years

Therefore, the sample is 22800 years old, which corresponds to answer option a.