Carbon dating, also known as radiocarbon dating, is a scientific method used to determine the age of an artifact or biological material by measuring the amount of carbon14 it contains. This technique is crucial in archaeology, geology, and other fields where understanding the age of organic materials is essential.
What is Carbon14?
Carbon14 (¹⁴C) is a radioactive isotope of carbon that is formed in the atmosphere when cosmic rays interact with nitrogen14. It is incorporated into carbon dioxide, which is then absorbed by plants during photosynthesis. Animals and humans consume these plants, making carbon14 a part of their biological makeup.
The Decay of Carbon14
The key aspect of carbon dating is the radioactive decay of carbon14. Carbon14 has a halflife of about 5,730 years, meaning that after this time, half of the original amount of carbon14 will have decayed into nitrogen14 (¹⁴N).
The Carbon Dating Formula
The basic formula used in carbon dating is derived from the exponential decay of carbon14. The formula is:
[ N(t) = N_0 \times e^{\lambda t} ]
Where:
 ( N(t) ) = the quantity of carbon14 remaining after time ( t )
 ( N_0 ) = the initial quantity of carbon14 at the time of the organism's death
 ( \lambda ) = the decay constant
 ( t ) = time in years since the organism died
 ( e ) = Euler's number (approximately equal to 2.71828)
Decay Constant Calculation
The decay constant ( \lambda ) can be calculated from the halflife (( t_{1/2} )) using the formula:
[ \lambda = \frac{\ln(2)}{t_{1/2}} ]
Given the halflife of carbon14 is approximately 5,730 years, we can substitute this value to find ( \lambda ):
[ \lambda \approx \frac{0.693}{5730} \approx 1.21 \times 10^{4} \text{ years}^{1} ]
Using the Carbon Dating Formula
When a sample is collected, scientists measure the remaining amount of carbon14 using sensitive instruments like accelerator mass spectrometers or liquid scintillation counters. By plugging this value into the carbon dating formula, researchers can estimate how long it has been since the organism died.
Example Calculation

Measure the Remaining Carbon14: Let's say a sample has ( N(t) = 25 ) grams of carbon14 remaining, and the initial quantity ( N_0 ) was 100 grams.

Calculate the Decay Constant: As calculated, ( \lambda \approx 1.21 \times 10^{4} \text{ years}^{1} ).

Use the Formula to Solve for Time ( t ):
Rearranging the formula gives:
[ t = \frac{1}{\lambda} \ln\left(\frac{N(t)}{N_0}\right) ]
Substituting the known values:
[ t = \frac{1}{1.21 \times 10^{4}} \ln\left(\frac{25}{100}\right) ]
After calculating, you can find the approximate age of the sample.
Limitations of Carbon Dating
While carbon dating is a powerful tool, it has limitations:
 Age Range: Carbon dating is effective for dating materials up to about 50,000 years old. Beyond that, the remaining carbon14 is often too minimal to measure accurately.
 Sample Contamination: Contamination from modern carbon can skew results.
 Assumptions: The method assumes that the ratio of carbon14 to carbon12 has remained constant over time, which may not always be the case.
Conclusion
The carbon dating formula is an essential tool in various scientific fields, allowing researchers to date organic materials accurately. By understanding the decay of carbon14 and applying the relevant formulas, scientists can unlock the secrets of the past and gain insights into historical timelines. As with any scientific method, it is vital to understand its limitations and to interpret the results with caution.