# Radioactivity half life carbon dating

(See In general, whenever a quantity \(x(t)\) obeys an exponential decay equation \[ x(t) = C e^, \] where the continuous decay rate \(k\) is negative, then the quantity \(x\) has a half-life \(T\).After any time period of length \(T\), the quantity \(x\) decreases by half. As \(k\) is negative, the factor \(e^\) decreases from 1 (at \(t=0\)) towards 0 (as \(t\) approaches \(\infty\)).Let's investigate what happens to the sample over time. Since \(m\) has a continuous decay rate of \(-0.000121\), a general solution to the differential equation is \[ m(t) = C e^, \] where \(C\) is a constant.

Samples from the past 70,000 years made of wood, charcoal, peat, bone, antler or one of many other carbonates may be dated using this technique.

Therefore there is a time \(t=T\) such that \[ e^ = \dfrac.

\] We now solve for \(T\) and obtain \begin k T &= \log_e \dfrac \ &= - \log_e 2, \end so \[ T = - \dfrac \log_e 2. From time \(t=0\) to time \(t=T\), the factor \(e^\) decreases from \(e^0 = 1\) to \(e^ = \dfrac\), that is, decreases by half.

Carbon-14 is also passed onto the animals that eat those plants.

After death the amount of carbon-14 in the organic specimen decreases very regularly as the molecules decay.