Lesson Six: Application Half Life
Application: Half Life
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?
Answers
1. The k value you found was about 0.00012 per year. You can plug that k value in to the half life equation above and find that the half life is about 5,600 years.
2. 87.5% of the original C-14 is lost after three half lives. The first half-life takes you from 100-> 50% of your original C-14. The second half life cuts that 50% to 25% and after the third half-life you have only one half of that 25% left, or 12.5%. If you have 12.5% left, that means you have lost 87.5% of your original C-14.
Three half lives is 5600*3=16,800 years.
If you found a substance that had lost 87.5% of its original C-14, it would be from a time when homo sapiens had just recently become the only humans on the planet, but several thousand years before we developed agriculture, domesticated animals, or started to move to the Americas.
Your graph in part 1 looked like this
This graph only went out 12,000 years. However, after 12,000 years we still had a bit over 20% of our original C-14 and the line was starting to flatten out. It makes sense that in 6000 more years we'd have 12.5% of the original C-14 left.
3. After many half lives, very little C-14 is left. The noise from the measurements becomes almost as large as the signal itself. There are many ways of dating older objects, the easiest is perhaps to use a compound that decays with a longer half life.
Extra Challenge: The Smithsonian has a timeline of major events in human history. Can you figure out approximately how many half lives of C-14 have past for different events? How much C-14 is left from the artifacts humans left during these events?
http://humanorigins.si.edu/human-characteristics/humans-change-world