Final answer:
It will take approximately 180 seconds for 95% of a radium-221 sample to decay, as this is roughly six half-lives of the substance with each half-life being 30 seconds.
Step-by-step explanation:
The student's question is about how long it will take for 95% of a radium-221 sample to decay given that its half-life is 30 seconds. To solve this, we can use the concept of radioactive decay and half-lives.
The half-life is the time it takes for half of a radioactive substance to decay. Since 95% decay means that 5% remains, we need to calculate how many half-lives it takes to get from 100% to 5%. This calculation can be done using the formula for exponential decay:
N(t) = N0 * (1/2)^(t/h)
Where:
- N(t) is the remaining quantity of the substance after time t,
- N0 is the initial quantity of the substance,
- t is the time elapsed,
- h is the half-life of the substance.
Since we want to know when 5% of the original sample is left (N(t) = 0.05 * N0), we can set up the equation 0.05 = (1/2)^(t/30) and solve for t. Taking the logarithm of both sides, we get: t/30 = log2(0.05), which gives us t as approximately six half-lives or 180 seconds.
Thus, it will take approximately 180 seconds for 95% of a radium-221 sample to decay.