Question

Atoms of AA decay to atoms of BB with a half-life of 100,000 years. If there are 20,000 atoms of AA to begin with (and 0 atoms of BB), how long will it take for there to be 2,500 atoms of AA?

Answer 1

Answer:

300,000 years

Explanation:

Data provided:

A₀ = 20,000 atoms

Half life = 100,000 years

Now,

we know, Atoms after time 't' is given as

A = $A_0e^{-kt}$

k is the decay constant

thus,

for half life

0.5A₀ = $A_0e^{-k\times100,000}$

or

0.5 = $e^{-k\times100,000}$

taking natural log both sides

-0.693 = -k × 100,000

or

k = 0.693 × 10⁻⁵ / year

Therefore,

we have

2,500 = 20,000 $e^{-((0.693\times10^{-5})\times t)}$

or

0.125 = $e^{-((0.693\times10^{-5})\times t)}$

taking natural log both sides

-2.0794 = - 0.693 × 10⁻⁵ × t

or

t = 300,000 years