Why does the rate of radioactive decay follow a decaying exponential function?

[onqun]

Hello, I am curious. Eventhough, the half-time of an element is always the same. Why not we can't find quarter of elements vanish time, because; half life is directly proportional, 10gr 10 min -> 5gr 20min. if 10 gr vanishes in 10 min, 2.5gr vanishes in 2.5 min for 20gr radioactive material.

 

[kurros]

I am a little unclear what you are asking, but your numbers are wrong. Your numbers imply some kind of linear decay, which is weird, i.e. that everything would be gone after 20 minutes.
We are talking about something that undergoes exponential decay, so to find the amount left after a certain amount of time you need this formula (see http://en.wikipedia.org/wiki/Exponential_decay for info about where this comes from):

N(t)=N_0 2^{-t/t_{1/2}}

where N0 is the original number of "things" you have which are decaying, N(t) is the number left after time t and t_1/2 is the half life. For a 20 gram sample with half life of 10 minutes, i.e. N0=20 (mass is proportional to particle number) and t_1/2=10, then after 2.5 minutes you have 16.82 grams left, after 10 minutes you have 10 grams (duh) and after 20 minutes you have 5 grams left (two half lives), 30 minutes->2.5 grams, etc.

 

[Astronuc]

Hello, I am curious. Eventhough, the half-time of an element is always the same. Why not we can't find quarter of elements vanish time, because; half life is directly proportional, 10gr 10 min -> 5gr 20min. if 10 gr vanishes in 10 min, 2.5gr vanishes in 2.5 min for 20gr radioactive material.

See the discussion on half-life here.
http://hyperphysics.phy-astr.gsu.edu/Hbase/nuclear/halfli2.html

As kurros indicated, it represents a decaying exponential function based on a first order differential equation which indicates that the rate of decay is proportional to the amount of the particular substance (radionuclide) existent (still remaining).

0	1	
1	0.933	
1.9265	0.87500	1/8 initial amt decayed
2	0.871	
3	0.812	
4	0.758	
4.1504	0.75000 1/4 initial amt decayed
5	0.707	
6	0.660	
7	0.616	
8	0.574	
9	0.536	
10	0.500	1/2 initial amt decayed
                          one half-life