Cesium-137, a component of radioactive waste, has a half-life of 30.2 yr. If a sample of waste has an initial activity of 15.0 Ci due to cesium-137, how long will it take for the activity due to cesium-137 to drop to 0.250 Ci?

It takes 178.4 years to drop the activity from 15.0 Ci to 0.25 Ci

Explanation:

Step 1: Data given

Half-life time = 30.2 years

Initial activity = 15.0 Ci

Final activity = 0.250 Ci

Step 2: Calculate the time needed

N / No = e^ (-0.693 * t / T1/2)

⇒ with N = The activity after dropped = 0.250 Ci

⇒ with N0 = the initial activity = 15.0

⇒ with t = the time (in years) needed to drop the activity from 15.0 to 0.250

⇒ with t1/2 = the half - life time is 30.2 years

0.250 / 15.0 = e^ (-0.693 * t / 30.2)

ln (0.250/15.0) = (-0.693 * t / 30.2)

-4.09 * 30.2 = - 0.693t

-123.65 = - 0.693t

t = 178.4 years

It takes 178.4 years to drop the activity from 15.0 Ci to 0.25 Ci