how many mg will remain after 2000 years?

Question

The half-life of Radium-226 is 1590 years. If a sample contains 200 mg, how many mg will remain after 2000 years?

Answer 1

The formula for exponential decay is 

 

f(t) = A*e^k*t  where 

 

A = original amount = 200 mg in this case

k = decay constant

t = time  = 1590 or 2000 in this case

 

If the half-life is 1590 then F(1590) = A/2 so 

 

100 = 200*e^1590*k

e^1590*k = 0.5

ln(e^1590*k) = ln(0.5)

1590*k = ln(0.5)

k = ln(0.5) / 1590 = -0.000435941 

 

Now plug t = 2000 into the formula 

 

f(2000) = 200*e^{(-0.000435941*2000)} 

            = 200*e^(-0.871882) = 200*(0.418163823) ~ 83.633 mg