Expert: Josh - 10/19/2007

Question
carbon 14 dating. If the initial sample contains 4 grams of carbon 14, how much will be left after 1000 years?

Answer
Debbie,

The amount of carbon 14 at time t is governed by an exponential equation. The equation is given by A(t)=A(0)*exp(-kt), where A(0) is the initial amount at time t=0. Here, k represents a decay constant. You need to do a bit of research yourself to find out the half life of Carbon 14. i.e., the amount of time it takes for the amount of radioactive carbon 14 to halve.

The relationship between k and the half life T is as follows. Suppose the amount falls to half the initial value at time T. Then, we have A(T)=0.5*A(0)=A(0)*exp(-kT). Canceling the common factor A(0) on both sides of the equation leads to 0.5=exp(-kT). Taking the natural logarithm on both sides to undo the exponential function, we arrive at T=-ln(0.5)/k. Alternatively, the half life may be expressed as T=ln(2)/k.

e.g., Suppose the half life of a chemical element is 2000 years. We have T=2000 in this example. Using k=ln(2)/T, the decay constant k=0.69314718/2000=0.0003465735. Thus, the amount of this chemical present in a sample is given by A(t)=A(0)*exp(-0.0003465735*t), where A(0) is the initial amount at time t=0. Now, it's your turn to find the value of k for Carbon 14 yourself. Using an internet search engine, once the half life of C14 (the value T) is found, your can work out the amount remaining at t=1000 -- after one thousand years. In this question, we are told the initial amount A(0)=4 grams.

Just recapping: You need to find T from the internet for carbon 14, then compute k=ln(2)/T. Put this value of k into the formula A(t)=A(0)*exp(-k*t) and solve for A(t) where t=1000.

Good luck:)