A radioactive gas with a half-life of 2.4 days accidentally leaks into a room. If the radiation level is now 25% above a safe amount, for how long should the room be emptied?

Radioactive half-life is the time it takes for half an amount of radioactive material to decay into something else. In this case, it is assumed that the decay product is not radioactive or otherwise hazardous.

We must use the radioactive decay formula to determine at what time the radiation reaches a safe level.

A = Ao[e^ (-0.693) (t) (t 1/2) where t 1/2 is the half-life, t is elapsed time, Ao is the original quantity, A is the future quantity.

We are given a half-life of 2.4 days, an Ao of 1.25 and an A of 1.00:

1.00 = (1.25) e^ (-0.693) (2.4) t

1.00/1.25 = e^ (-1.6632) t

0.8 = e^ (-1.6632) t

t = 0.135 days = 3 hrs 15 min

This is the amount of time to a "safe level" using only radioactive decay, not venting or other means.