The soil in Chernobyl

Question

In soil near Chernobyl, Cesium-137 in decomposing slower that it does under laboratory conditions. Find the ecological half-life of Cesium-137 if the soil in the areas affected by the accident in 1986 still contains 81.48% of the initial amount of Cesium-137. (For comparison, laboratory half-life of Cesium-137 is 30 years.)

Answer 1

The first job is to find k in the radioactive decay formula A = P e^-kt. Let the initial amount P be 100 grams for convenience; the current amount A is then 81.48 grams. Elapsed time t would be the year T at which the amount is 81.48% (T is not provided in the problem summary above) minus the starting year 1986 for the Ce-137 decay. We have

81.48 = 100 e^-k(T-1986)

Divide both sides by 100 and take the natural log of both sides:

ln(0.8148) = -k(T-1986) ln e, but ln e = 1.

So k = - ln(0.8148) / (T-1986) = 0.2048 / (T-1986).

If you can find out what the present year T is for this problem, plug it in to find the value of k ( if T =2019, then k = 0.006206).

Finally, a half-life is when A = 50% of P, so 0.5 P = P e^-kt, ln(0.5) = -kt = -0.6931, and t = 0.6931 / k

(if T = 2019, then half-life t is 111.7 years, much longer than the lab value).