CHEMISTRY : If 50% of a radioactive element remains after 4000 years, what is the half-life?

Question

If 50% of a radioactive element remains after 4000 years, what is the half-life?

 please help.

Answer 1

Simply understanding that 50% is mathematically equal to half, and the definition of half life, which is the time it takes to decrease to half the original amount, will immediately lead to the answer of 4000 years.

Answer 2

One of the easiest formulas to use for first order decay is

F = 0.5ⁿ where F is the fraction remaining and n is the number of half lives elapsed.

In the current problem, F, the fraction remaining = 0.5 (since 50% remains and as a fraction that is 0.5)

So, now we find n, the number of half lives that have elapsed. (Of course it will be 1 half life, by definition)

But the math looks like this:

0.5 = 05ⁿ

log 0.5 = n log 0.5

n = 1

Since 1 half life has elapsed, and that took 4000 year, the half life = 4000 years