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Not to be confused with single's night for devilish ham radio enthusiasts.

Radiometric dating involves dating rocks or other objects by measuring the extent to which different radioactive isotopes or nuclei have decayed.

Although the time at which any individual atom will decay cannot be forecast, the time in which any given percentage of a sample will decay can be calculated to varying degrees of accuracy. The time that it takes for half of a sample to decay is known as the half life of the isotope. Some isotopes have half lives longer than the present age of the universe, but they are still subject to the same laws of quantum physics and will eventually decay, even if doing so at a time when all remaining atoms in the universe are separated by astronomical distances.

Various elements are used for dating different time periods; ones with relatively short half-lives like carbon-14 (or 14C) are useful for dating once-living objects (since they include atmospheric carbon from when they were alive) from about ten to fifty thousand years old. See Carbon dating. Longer-lived isotopes provide dating information for much older times. The key is to measure an isotope that has had time to decay a measurable amount, but not so much as to only leave a trace remaining. Given isotopes are useful for dating over a range from a fraction of their half life to about four or five times their half life.

## How it works

Symbolically, the process of radioactive decay can be expressed by the following differential equation, where N is the quantity of decaying nuclei and k is a positive number called the exponential decay constant. The meaning of this equation is that the rate of change of the number of nuclei over time is proportional only to the number of nuclei. This is consistent with the assumption that each decay event is independent and its chance does not vary over time.

$\frac{dN}{dt} = -{k N}.$

The solution to this equation is:

$N = N_0 e^{k t} \!$

For decay, the constant k in the above equation can be calculated as:

$k = \frac{\ln(\frac{1}{2})}{\tau_{1/2}}$

where $\tau_{1/2}$ is the half-life of the element, $t$ is the time expired since the sample contained the initial number $N_0$ atoms of the nuclide, and $N$ is the remaining amount of the nuclide. We can measure $N$ directly, for example by using a radiation detector, and obtain a good estimate of $N_0$ by analyzing the chemical composition of the sample. The half-life $\tau_{1/2}$, specific to each nuclide, can be accurately measured on a pure sample, and is known to be independent of the chemical composition of the sample, temperature and pressure.[1] Solving for $t$ gives us the estimated age of the sample:

$t = \frac{\tau_{1/2}}{\ln(\frac{1}{2})}\ln \frac{N_0}{N}$

### Example Problem

You find a bone fragment and through analysis you determine that it contains 13% of its original carbon-14. The half-life of carbon-14 is approximately 5,730 years. Approximately how old is the bone?

$N = N_0 e^{k t} \!$

Since the quantity $N$ represents 13% (or 13/100ths) of $N_0$, it follows that $N =(\frac{13}{100})N_0$, thus:

$\left(\frac{13}{100}\right)N_0= N_0 e^{k t} \!$

$N_0$ cancels out, leaving us with:

$\left(\frac{13}{100}\right)=e^{k t} \!$

We can take the "inverse property of logarithms" and get:

$\ln\left(\frac{13}{100}\right)= {k t} \!$

Solving for t gives us:

$\left(\frac{1}{k}\right)\ln\left(\frac{13}{100}\right)= {t}\!$

We can now calculate the value of k as follows:

$k = \frac{\ln\left(\frac{1}{2}\right)}{5730}$

Plug into equation and solve:

${t}=\frac{5730}{\ln\left(\frac{1}{2}\right)}\ln\frac{13}{100} \approx 16861.329$

Thus the bone is approximately 17,000 years old. (Our input data had two significant figures, so reporting a more accurate result would be meaningless.)

A important limitation of radiometric dating often overlooked by layman (and not always made clear in scholarly works as well) is that any date is actually a range, following the 68–95–99.7 rule.

A proper radiometric date should read years before present (with 1950 being present) ± range/2 at x standard deviations (Xσ)', but is often reported as a single year or a year range, like 1260–1390 CE (the date for the Shroud of Turin). This leaves out important information which would tell you how precise is the dating result.

Carbon-14 dating has an interesting limitation in that the ratio of regular carbon to carbon-14 in the air is not constant and therefore any date must be calibrated using dendrochronology. Another limitation is that carbon-14 can only tell you when something was last alive, not when it was used.

Note that although carbon-14 dating receives a lot of attention, since it can give information about the relatively recent past, it is rarely used in geology (and almost never used to date fossils). Carbon-14 decays almost completely within 100,000 years of the organism dying, and many fossils and rock strata are hundreds of times older than that. To date older fossils, other methods are used, such as potassium-argon or argon-argon dating.