User:Vsmith/Dating calc

For radioactive decay
The relationship between fraction remaining (f) and the number of half-lives (n) elapsed can be shown by:

n	1	2	3	4	5	6	7	...	n
f	1/2	1/4	1/8	1/16	1/32	1/64	1/128	...	...
	1/21	1/22	1/23	1/24	1/25	1/26	1/27	...	1/2n

Therefore the fraction remaining after time n is: f=1/2ⁿ which is equivalent to f=(1/2)ⁿ or f=0.5ⁿ. This makes the calculation of the age if the fraction remaining is known quite simple. Solving the above relationship for n using the properties of logarithms:

f=0.5ⁿ becomes

ln f = n·ln 0.5 and

$n={\frac {\ln \left({f}\right)}{\ln \left({0.5}\right)}}$

As an example, for a sample that contains 0.06780 of the original C-14:

$n={\frac {\ln \left({0.06780}\right)}{\ln \left({0.5}\right)}}$ solving gives n= 3.88 half lives and

3.883 half lives * 5730 yrs/half life = 22,250 yrs.

A more simple example, for a sample containing 0.25 of the original:

$n={\frac {\ln \left({0.25}\right)}{\ln \left({0.5}\right)}}$ solving gives n= 2 half lives and

2 half lives * 5730 yrs/half life = 11460 yrs.