The half-life of a certain radioactive element is 800 years. How old is an object if only 12.5% of the radioactive atoms in it remain?
a
6400 years
b
1600 years
c
3200 years
The correct answer is option b) 1600 years.
In a radioactive decay process, the amount of radioactive material decreases by half after each half-life period.
Since the half-life of the radioactive element is 800 years, it means that after 800 years, the amount of radioactive material remaining is 50% of the original amount.
If only 12.5% of the radioactive atoms remain in the object, it means that the amount of radioactive material has gone through three half-life periods (12.5% = 1/2^3).
Therefore, the object is 3 x 800 = 2400 years old.
Since each half-life period is 800 years, the age of the object is 2400 - 800 = 1600 years.
To determine the age of an object based on the remaining percentage of a radioactive element, you can use the formula:
Age = (Time constant) * (Number of half-lives)
The time constant is calculated by dividing the half-life (t₁/₂) by the natural logarithm of 2 (ln 2 ≈ 0.693).
In this case, the half-life of the radioactive element is 800 years, and the remaining percentage is 12.5%.
First, let's calculate the time constant:
Time constant = (800 years) / ln 2 ≈ 1155.71 years
Next, let's find the number of half-lives that have passed:
Number of half-lives = ln (Remaining percentage) / ln 0.5
Number of half-lives = ln 0.125 / ln 0.5 ≈ 3
Now, we can calculate the age of the object:
Age = (Time constant) * (Number of half-lives)
Age = 1155.71 years * 3
Age ≈ 3467.13 years
Therefore, the object is approximately 3200 years old.
The correct answer is option c) 3200 years.
To find the age of the object, we need to use the concept of half-life.
The half-life tells us how long it takes for half of the radioactive atoms in a substance to decay. In this case, the half-life of the radioactive element is 800 years.
If only 12.5% (or 1/8) of the radioactive atoms remain, it means that 7/8 (or 87.5%) of the atoms have decayed.
Since each half-life reduces the amount of radioactive atoms by half, we can divide 87.5% by 50% to determine how many half-lives have passed.
87.5% ÷ 50% = 1.75
So, 1.75 half-lives have occurred.
To find the time it takes for 1.75 half-lives to pass, we multiply the half-life duration by the number of half-lives:
800 years/half-life x 1.75 half-lives = 1400 years
Therefore, the object is 1400 years old.
So, none of the options given (a, b, c) are correct.