Expert: Paul Klarreich - 8/2/2009

Question
The half life of polonium-210 is 138.4 days. How many milligrams of polonium-210 remain after 415.2 days if you start with 2.0 mg of the isotope?

Answer
Questioner: Lucie Thompson
Country: United States
Category: Advanced Math
Private: No
Subject: math + chemistry
Question: The half life of polonium-210 is 138.4 days. How many milligrams of polonium-210 remain after 415.2 days if you start with 2.0 mg of the isotope?
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Hi, Lucie,

The usual thing is to write  x = x0 e^(-kt), where
x = amount at time t.
x0 = amount at time 0.

But maybe we can get away with writing:

x = x0 2^-kt

Now at t = half-life = 138.4,  x = x0/2:

x0/2 = x0 2^(-138.4 k)

1/2 = 2^(-138.4 k)

2^-1 = 2^(-138.4 k)

-138.4 k = -1

k = 1/138.4

So x = x0 2^(- t/138.4)

Now put  x0 = 2.0,  t = 415.2:

x = 2.0 2^(-415.2/138.4), and just use your calculator:

x = 0.25

Of course, you might have noticed that 138.4 * 3 is exactly 415.2.

Which means it is exactly three half-lives, so it should be exactly 1/8 the original amount.