In one hour the decay rate of a small amount of a radioactive isotope drops from 16000 decays/second to 2000 decays/second. What is the half life? Can you solve this without a formula?

Question

In one hour the decay rate of a small amount of a radioactive isotope drops from 16000 decays/second to 2000 decays/second.  What is the half life?  Can you solve this without a formula?

anonymous · Answer

yes. what is the ratio of the final rate to the initial rate?

anonymous · Answer

It did not give an initial or final rate, but i have the answers just not sure how to start the problem.  all we know is how much the isotope decays in a second for the hour given.

anonymous · Answer

final rate: 2000 per sec, initial rate: 16000 per sec.

anonymous · Answer

rate at later time over the rate at earlier time.

anonymous · Answer

That does not give the right answer options are 30 min, 20min, 15min, 7.5 min, and 20min

anonymous · Answer

right, the ratio is not the answer. i am trying to guide you to the answer because openstudy says so. what answer did you come up with and how?

anonymous · Answer

I got to .125 and then i multiplied by 60 seconds and that gave me 7.5 which was the wrong answer.

anonymous · Answer

hmm why did you decide to multiply by 60 minutes? (i suppose you meant minute not second). what do you think of "half life"--how is it relevant here?

anonymous · Answer

Yeah im sorry i meant 60 seconds.

anonymous · Answer

Half life is where is the isotope at half empty.

anonymous · Answer

half of the original material

anonymous · Answer

Is there another step that I am missing because i managed to get 7.5min as an answer, but the real answer is 20min.

anonymous · Answer

yes there is. if it takes 20 minutes for half of something to go away, how much remains after another 20 minutes? and how much remains after one more period of 20 minutes?

anonymous · Answer

say i start with one gram of radioactive isotope. 20 minutes later, i have half a gram. 40 minutes later, how much do i have left?

anonymous · Answer

i tried...for radioactive decay, the rate is proportional to the number of nuclei remaining. so:
$$rate (t) / rate (t_0) = n(t) / n(t_0)$$
so the ratio of the rates is the same as the ratio of the numbers. we know that the ratio of the number changes like this:

after one half life or 20 min, 0.5 
after two half lives (40 min): 0.25
after three half lives (one hour): 0.125 

and the rate ratio changes the same way. so the rates ratio will only be .125 after one hour if the half life is 20 minutes.