Question

You are trying to determine the half-life of a new radioactive element you have isolated. You start with 1 gram, and 5 days later you determine that it has decayed down to 0.6 grams. What is its half-life? (Round your answer to three significant digits.)

Answer 1

The remaining mass mass of the substance at time t can be modeled with an exponential equation in this form:

m = ab^t,
where m is the remaining mass in grams, and t is the time in days.

We are given the mass at time zero and at time two.

Let t = 0.
10 = ab^0
a = 10
m = 10b^y

Let t = 2.
7 = 10b²
b² = 0.7
b ≈ 0.84
m ≈ 10(0.84)^t

Now to find the half-life, let m = 5 g, which is half of the initial mass. Solve for t.

5 = 10(0.84)^t
(0.84)^t = 0.5
t ln(0.84) = ln(0.5)
t = ln(0.5) / ln(0.84) ≈ 3.9 days