The half-life of the radioactive element is 10 minutes, with three half-lives passing in the initial 30-minute period.
The process of radioactive decay involves the transformation of a substance over time, typically characterized by a half-life—the duration required for half of the radioactive material to decay. To determine the half-life based on the given information, we can observe the decay process.
In the provided scenario, the initial mass of the radioactive element is 80mg, and after 30 minutes, it decays to 10mg. This implies that during this time, half of the original mass has decayed. Now, we need to identify how many half-lives have passed.
Let's denote the unknown number of half-lives as "n". The relationship between the remaining mass and the number of half-lives can be expressed as:
Remaining mass = Initial mass * (1/2)^n
Given that the remaining mass after 30 minutes is 10mg:
10 = 80 * (1/2)^n
Solving for "n":
(1/2)^n = 10/80 = 1/8
Taking the logarithm base 2 of both sides:
n = log2(1/8) = -3
Therefore, after 30 minutes (one half-life), the radioactive element has passed three half-lives. The half-life of the radioactive element is the time it takes for half of it to decay, so each half-life corresponds to 30/3 = 10 minutes.