Carbon-14 dating, also called radiocarbon dating, estimates the age of once-living material by comparing
its remaining carbon-14 content with the amount found in a modern reference sample. Since carbon-14 is radioactive, its
amount decreases over time after the organism stops exchanging carbon with the environment. The decay follows the same
exponential law used for any radioactive isotope, but in this case the isotope is fixed: carbon-14.
Carbon-14 decay law
If \(N_0\) is the initial carbon-14 amount and \(N\) is the amount still present after time \(t\), then the radioactive
decay law is
Exponential decay relation.
\[
\begin{aligned}
N &= N_0 e^{-\lambda t}.
\end{aligned}
\]
Dividing both sides by \(N_0\) gives a very useful normalized form:
\[
\begin{aligned}
\frac{N}{N_0} &= e^{-\lambda t}.
\end{aligned}
\]
This is why many carbon-14 problems are expressed directly in terms of a remaining fraction or a
percent modern carbon value. You do not need the absolute number of atoms if you already know the
fraction relative to the modern standard.
Half-life and decay constant
Carbon-14 has a half-life of about
\[
\begin{aligned}
T_{1/2} &= 5730\ \mathrm{years}.
\end{aligned}
\]
The decay constant is therefore
Carbon-14 decay constant.
\[
\begin{aligned}
\lambda &= \frac{\ln 2}{T_{1/2}} \\
&= \frac{\ln 2}{5730}.
\end{aligned}
\]
Numerically, this gives a very small value in year\(^{-1}\), which reflects the slow decay of carbon-14.
Age formula
To estimate the age, solve the decay equation for \(t\). Starting from
\[
\begin{aligned}
\frac{N}{N_0} &= e^{-\lambda t},
\end{aligned}
\]
take the natural logarithm of both sides:
\[
\begin{aligned}
\ln\!\left(\frac{N}{N_0}\right) &= -\lambda t.
\end{aligned}
\]
Rearranging gives the radiocarbon age formula
Carbon-14 dating age formula.
\[
\begin{aligned}
t
&= \frac{1}{\lambda}\ln\!\left(\frac{N_0}{N}\right) \\
&= \frac{1}{\lambda}\ln\!\left(\frac{1}{N/N_0}\right).
\end{aligned}
\]
This is the main equation used in the calculator. If the measured quantity is already the remaining fraction
\(N/N_0\), then the computation becomes very direct.
Percent modern carbon
Sometimes the measurement is given as percent modern carbon, abbreviated pMC. This is simply the
remaining fraction expressed as a percentage:
\[
\begin{aligned}
\mathrm{pMC} &= 100 \cdot \frac{N}{N_0}.
\end{aligned}
\]
So if a sample has 50 pMC, then its remaining fraction is \(0.50\). If it has 25 pMC, the remaining fraction is \(0.25\).
Activity ratio
The radioactive activity is proportional to the number of undecayed nuclei:
\[
\begin{aligned}
A &= \lambda N.
\end{aligned}
\]
Since the same decay constant applies to both the sample and the modern reference, the activity ratio is simply
\[
\begin{aligned}
\frac{A}{A_0} &= \frac{N}{N_0}.
\end{aligned}
\]
This is why the remaining fraction and the relative activity tell the same story for carbon-14 dating.
Sample calculation
Suppose the remaining carbon-14 fraction is
\[
\begin{aligned}
\frac{N}{N_0} &= 0.5.
\end{aligned}
\]
Then the sample has one half of its original carbon-14 left, so physically we already expect it to be one half-life old.
Using the formula confirms this:
Step 1. Write the age formula.
\[
\begin{aligned}
t &= \frac{1}{\lambda}\ln\!\left(\frac{1}{N/N_0}\right).
\end{aligned}
\]
Step 2. Substitute the fraction.
\[
\begin{aligned}
t
&= \frac{1}{\lambda}\ln\!\left(\frac{1}{0.5}\right) \\
&= \frac{1}{\lambda}\ln(2).
\end{aligned}
\]
But since \(\lambda = \ln 2 / 5730\), this becomes
\[
\begin{aligned}
t &= 5730\ \mathrm{years}.
\end{aligned}
\]
So a sample with exactly half the modern carbon-14 content has an estimated radiocarbon age of 5730 years.
Interpretation of the decay curve
The decay curve is steep at first and then flattens out. That shape is characteristic of exponential decay. Each full
half-life multiplies the remaining fraction by \(1/2\), not by subtracting a fixed amount. For example:
| Elapsed half-lives |
Age |
Remaining fraction |
| 1 |
\(5730\) years |
\(1/2\) |
| 2 |
\(11460\) years |
\(1/4\) |
| 3 |
\(17190\) years |
\(1/8\) |
| 4 |
\(22920\) years |
\(1/16\) |
Practical note
Real radiocarbon dating is more subtle than the ideal formula alone. The atmospheric carbon-14 level has not always been
exactly constant, and laboratory measurements require corrections. This is why advanced work uses
calibration curves. Even so, the basic exponential formula remains the starting point and gives the
essential first estimate of the sample’s age.
Advanced note
At university level, one may study calibration datasets, reservoir effects, isotopic fractionation corrections, and
statistical uncertainty propagation. Those refinements improve historical accuracy, especially for old samples. The
calculator here deliberately focuses on the clean single-isotope exponential model so the age estimate remains easy to
understand and compute from the measured C-14 fraction.
| Concept |
Main relation |
Meaning |
| Decay law |
\(N = N_0 e^{-\lambda t}\) |
Remaining carbon-14 after time \(t\) |
| Normalized fraction |
\(N/N_0 = e^{-\lambda t}\) |
Remaining fraction relative to modern carbon |
| Decay constant |
\(\lambda = \ln 2 / 5730\) |
Carbon-14 decay rate in year\(^{-1}\) |
| Age formula |
\(t = (1/\lambda)\ln(N_0/N)\) |
Radiocarbon age estimate |
| Percent modern carbon |
\(\mathrm{pMC} = 100(N/N_0)\) |
Fraction written as a percentage |
| Activity ratio |
\(A/A_0 = N/N_0\) |
Relative decay activity of the sample |