[tashkent]

Hello everyone!

I'm currently preparing for my GRE Chemistry exam and I need help on this half-life problem found on no. 113 of the GRE Practice Book:

113) The half-life of C-14 is 5730 years.  The C-14 activity of living material is approximately 920 decays/hr per gram of carbon.  A fragment of wool fabric from an archaeological site has an activity of 680 decays/hr per gram of carbon.  The approximate date of the sample is:

a) AD 1950
b) 500 BC
c) 3700 BC
d) 5700 BC
e) 10000 BC

The answer is B.  However, I believe you can only solve this by using a calculator, get k from t1/2 = 0.693/k then use the formula t = -1/k ln Nt/No.  The problem is the GRE exam does not make use of a calculator.  Can this problem be solved manually?  I will appreciate any ideas on how to solve this problem in the head and at the quickest possible time...

Hope to receive replies from u soon.  Thanks!

Regards,
Tashkent

[Demotivator]

Well I'll try. You can convert a natural log to log base 10 .
ln X = 2.3logX or maybe 2logX  (make X scientific  notation).
then you can apply the law:
log(n x 10^c) = log n + log 10^c
so c = log 10^c
now need log n where n is between 1 and 10.
But n can be APPROXIMATED to a series of 2 factors (or any number of your choosing).
eg if n=5, log 5= log (2x2 ) + error= log 2 + log 2 + error
Just need to remember what the log of 2 is.

[jdurg]

I've always found half-lives to be really strange.  It makes it seem as if an element will decay much faster if you have more of it in one area, then if you only had a few atoms.  Like if you only had two atoms of Uranium, it would take a few hundred million years for it to decay into one atom due to its half-life.  Weird.  

[Demotivator]

Hi Jdurg,
Exponential decay can be understood a little better when one considers that it is based on probabilty. Natural Decay is the opposite of natural growth. Consider a growing population. It is based on the cumulative probabilities of individuals being able to replicate more than once. Some produce no offspring, some one offspring, others more to varying degrees. The more individuals, the more capability for multireproduction exists. Thus, growth is not linear but exponential.
Same thing with decay. Some individuals atoms never decay. Some will decay faster than others. The more individuals, the greater the number that are available that can  decay at the most probable rate.

Note: In your example of two atoms of uranium, it is not true that one will absolutely decay in so many years. The sample is so small that one, both or none might decay at any time. Predictable Probability is only accurate on large samples (like coin flipping).
 

[tashkent]

Demotivator,

Thanks for taking the time in solving my GRE Problem.  At the moment, I don't understand your technique.  I'm very weak in trigonometry and logarithms.  But I'll get back to u as soon as I figure your method out.

Regards,
Tashkent