Derivation of the Carbon-14 Test

Carbon-14 dating depends on a differential equation. The reason is straightforward: archaeologists wanted a reliable method for estimating the age of ancient organic material such as charcoal, wood, or bone, but measuring how much carbon remained in a sample was not enough on its own. A model was needed to describe how that amount changes over time.

Why a Decay Model Was Needed

Living organisms contain both stable carbon-12 and radioactive carbon-14:

$${}^{12}\mathrm{C}, \qquad {}^{14}\mathrm{C}$$

While the organism is alive, it continuously exchanges carbon with its environment, so the ratio of carbon-14 to carbon-12 remains approximately stable. Once the organism dies, that exchange stops. Carbon-14 is no longer replenished and begins to decay.

If a sample now contains only a fraction of its original carbon-14, the scientific question becomes:

What function describes the amount of radioactive carbon remaining after time (t)?

That is the point of the decay equation. It gives a mathematical rule for working backward from the amount observed today to the time of death.

Setting Up the Differential Equation

Let (x(t)) denote the amount of carbon-14 present at time (t). Then (x) is a function of time:

$$x = f(t)$$

The key physical assumption is that the rate of decay at any instant is proportional to the amount currently present. So for some positive constant (k),

$$\frac{dx}{dt} \propto x$$

which we write as

$$\frac{dx}{dt} = -kx.$$

The negative sign indicates that the amount is decreasing over time.

Solving the Differential Equation

To solve for (x) explicitly as a function of (t), separate variables:

$$\frac{dx}{x} = -k\,dt.$$

Integrating both sides gives

$$\int \frac{1}{x}\,dx = \int -k\,dt$$

so

$$\ln x = -kt + C.$$

Exponentiating both sides,

$$x = e^{-kt + C}.$$

Using exponent rules,

$$x = e^C e^{-kt}.$$

Since (e^C) is just a constant, let

$$A = e^C.$$

Then the decay law becomes

$$x(t) = A e^{-kt}.$$

Determining the Constants A and k

The constant (A) is the initial amount of carbon-14 present at the moment of death. Since (t=0) at that moment,

$$x(0) = A e^0 = A.$$

So (A = x(0)).

The constant (k) is the decay rate. For carbon-14, (k) is obtained from the half-life, which is approximately (5730) years. After one half-life, half the original carbon-14 remains, so

$$\frac{A}{2} = A e^{-5730k}.$$

Dividing by (A),

$$\frac{1}{2} = e^{-5730k}.$$

Taking logarithms,

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

and therefore

$$k = \frac{\ln 2}{5730}.$$

Dating Ancient Cave Artifacts

Suppose archaeologists discover charcoal from a cave fire and determine that only (25%) of the original carbon-14 remains. Then

$$x = 0.25A.$$

Using the decay equation,

$$0.25A = A e^{-kt}.$$

Dividing by (A),

$$0.25 = e^{-kt}.$$

Taking the natural logarithm,

$$\ln(0.25) = -kt.$$

Solving for (t),

$$t = \frac{-\ln(0.25)}{k}.$$

Substituting (k = \frac{\ln 2}{5730}),

$$t \approx 11460 \text{ years}.$$

So the charcoal, and therefore the cave dwellers who made the fire, lived roughly 11,460 years ago.

Conclusion

Carbon-14 dating comes down to one equation:

$$x(t) = A e^{-kt}.$$

Once (A) and (k) are known, the amount of carbon-14 left in a sample can be used to estimate how much time has passed since the organism died.

References

• Tenenbaum, Morris and Harry Pollard. Ordinary Differential Equations. Dover Publications, 2008.