The half-life of Molybdenum-93 is 4000 years. A sample of Molybdenum-93 has a mass of 10 mg. When will the mass be reduced to 1 mg?
Image Source: Virtual museum of chemical elements.
The mass of a radioactive element decays at a rate proportional to the mass.
If m(t) is the mass of the radioactive element at time t then
dm/dt = -λm (1)
where λ is a decay constant. The minus sign means that the amount of the radioactive material decreases over time.
The only solutions of the differential equation (1) are the exponential functions
m(t) = m(0)e-λt (2)
We know that half-life is the time it takes for half a given mass of an element to decay. In order to determine the value of λ, we use the fact that
m(4000) = 5
5 = 10e-λ4000
1/2 = e-λ4000
Now we take the natural logarithm of both sides
ln (1/2) = -λ4000
– ln2 = -λ4000
λ = ln2 /4000.
Therefore, we can rewrite the equation (2) as
m(t) = m(0)e-t*(ln2)/4000
We want to find the value of time such that m(t) = 1, so
1 = 10e-t*(ln2)/4000
0.1 = e-t*(ln2)/4000
We solve this equation for t by taking the natural logarithm of both sides
ln0.1 = -t*(ln2)/4000
t = -4000*ln0.1/ln2 ≈ 13287.7 years