How many minutes must pass before the mass of the radioactive element decreases to 15.195g?

Question

You have 98.569g of a radioactive element with a half-life of 87.75minutes. How many minutes must pass before the mass of the radioactive element decreases to 15.195g?

Answer 1

Radioactive decay is considered to be first order. As such, we can use the formul

FR = 0.5ⁿ

FR = fraction remaining = 15.195 g / 98.569 g = 0.1542

n = number of half lives that have elapsed = ?

0.1542 = 0.5ⁿ

log 0.1542 = n log 0.5

-0.8119 = -0.3010 n

n = 2.697 half lives

Since each half life = 87.75 minutes, we can calculate the total time needed as follows:

2.697 half lives x 87.75 min / half life = 236.7 minutes

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You could also use the integrated rate equation for a first order decay to solve this problem

[A] = [A]_oe^-kt

Find k from the relationship that t1/2 = ln 2 / k = 0.693 / 2

substitute the values and solve for A