Equation of a Line
So, how does this relate to how scientist can date old materials? One of the interesting features of reactions whose concentrations decrease with exponential decay, which we call first order reactions, is that they have a constant half life. As you might have guessed, the half life is the time it takes to lose one half of the carbon-14. You might be surprised to learn that it takes the same amount of time to go from 100% of the original amount of C-14 to 50% as it does to go from 50% to 25% the original amount, or from 25% to 12.5%.
We can calculate the half life as:
Remember, k is the rate constant and the slope of the best fit line you found on the last page. Be careful though. When you are calculating half-lives, remember that the curve of carbon-14 concentration v time is an exponential decay. You lose carbon-14 more slowly has time goes on. This means you can't, for example, calculate a "forth-life" that is half the length of time as a half life because it takes less than 50% of a half life to lose 1/4 of your carbon-14. Check this out on the graph you made in part 1 and convince yourself this is true. How long does it take to lose half of your C-14? How long to lose 1/4? I've reproduced the graph below the answer keys.
Try applying this information for yourself! I've listed some questions below. Try to answer them before looking at the answers underneath.
1. What is the half life of carbon 14?
2. How long would it take to lose 87.5% of the original C-14? Does this match what your graph showed in plot one?
3. Scientists have trouble carbon dating things more than 50,000 years old. Can you think of why that might be? How can we know how old things are that are hundreds of thousands or millions of years old?