User:Jclerman/Dating calc

From Wikipedia, the free encyclopedia

Contents 1 Note 1.1 Computations of ages and dates 1.1.1 Method 1. Comparison table 1.1.2 Method 2. Simple equation 1.1.3 Method 3. From the laboratory to a calendrical date

[edit] Note

[edit] Computations of ages and dates

Equivalent methods to calculate ages are described in the following.

[edit] Method 1. Comparison table

For radioactive decay, the relationship between fraction remaining (f) and the number of half-lives (n) elapsed is:

n	1	2	3	4	5	6	7	8	...	n
f	1/2	1/4	1/8	1/16	1/32	1/64	1/128	1/256	...	...
	1/21	1/22	1/23	1/24	1/25	1/26	1/27	1/28	...	1/2n
years	5730	11460	17190	22920	28650	34380	40110	45840	...	...

Given any intermediate f value, its comparison with the n and age rows provides a rough idea of the age range, sidestepping any calculations. For example, if f = 1/10, n lies between 3 and 4 half lives and the age t lies between 17,190 and 22,920 yrs.

For precise age calculations, see the next methods.

[edit] Method 2. Simple equation

From the above it follows that 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

Example 1: for a sample containing 0.25 of the original C-14:

; solving this gives n = 2 half lives and

2 half lives * 5730 yrs/half life = 11,460 yrs.

Example 2: for a sample that contains 0.06780 of the original C-14:

; solving this gives n = 3.883 half lives and

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

[Suggestion, include Example 3, for an age near 50,000 yrs]

[edit] Method 3. From the laboratory to a calendrical date

Radiocarbon dates are obtained, in the laboratory, after the following succesive steps:

• laboratory measurements of three specific (radio)activities: activity of the sample to be dated (S), activity of a modern standard (M), and activity of a background sample (B),
• calculation of the net sample activity, as fraction of modern, i.e. (S-B)/M; this is the value denoted below as
• calculation of the sample raw age (t) using the formula given below, and
• evaluation of the calibrated calendrical date by inputting t into a calibration curve.

The radioactive decay of carbon-14 follows an exponential decay. A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value. Symbolically, this can be expressed as the following differential equation, where N is the quantity and λ is a positive number called the decay constant:

The solution to this equation is:

,

where C is the initial value of N.

For the particular case of radiocarbon decay, this equation is written:

,

where, for a given sample of carbonaceous matter:

N₀ = number of radiocarbon atoms at t = 0, i.e. the origin of the disintegration time,

N = number of radiocarbon atoms remaining after radioactive decay during the time t,

λ = radiocarbon decay or disintegration constant.

Two related times can be defined:

• half-life: time lapsed for half the number of radiocarbon atoms in a given sample, to decay,
• mean- or average-life: mean or average time each radiocarbon atom spends in a given sample until it decays.

It can be shown that:

t_{1 / 2} = = radiocarbon half-life = 5568 years (Libby value)

t_avg = = radiocarbon mean- or average-life = 8033 years (Libby value)

Notice that dates are customarily given in years BP which implies t(BP) = -t because the time arrow for dates runs in reverse direction from the time arrow for the corresponding ages. From these considerations and the above equation, it results:

For a raw radiocarbon date:

and for a raw radiocarbon age:

One of these t values is then inputted into one of the calibration curves, thus obtaining a range of calendrical dates.