NateRiver wrote:
Pottassium decays into argon. The half life of pottassium is 1.3 billion years. A sample of rock is found on mars and contains 3 argon atoms for every pottassium atom. How old is the rock?
Its got to do with logarithms. In this case though its pretty simple.
At the start you have a certain number of pottassium atoms and no argon atoms.
After 1.3 billion years, half the pottassium has decayed, so now there are an equal number of pottassium and argon atoms.
After another 1.3 billion years, another half of the pottassium has decayed. Put another way, now there are only 1/4 as many pottassium atoms as there were at the start. The other 3/4 of the atoms are now argon. So now the ratio of argon to pottassium atoms is 3:1, which is the condition in the problem. So the answer is 2.6 billion years old.
From a mathematical standpoint, what you have to do is realize that for the ratio of argon to pottassium to be 3:1, that means that you've only got 1/4 as many pottassium atoms as you did to start (3 parts argon + 1 part pottassium = 4 total parts). So, if pottassium's half life is 1.3 billion years, that means after every 1.3 billion years, there's half as many pottassium atoms. So the question is, how many 1.3 billion-year cycles must you go through to end up with 1/4 of the original pottassium? In other words, what power must you raise 1/2 to to get 1/4? This is a logarithm - specifically
log 1/2 (1/4) = 2
note: the 1/2 in the above formula here should be subscripted, but don't think I can do so in this forum.
Normally you'd use a calculator or at worst a log table to find the logarithm, but in this case its an easy one.
So the answer is 2 1.3 billion year cycles, or 2.6 billion years.
26 Feb 2013, 1:55 pm
