Anthropologists can estimate the age of a bone or other sample of organic matter by its carbon-14 content. The carbon-14 in a living organism is constant until the organism dies, after which carbon- 14 decays with first-order kinetics and a half-life of 5730 years. Suppose a bone from an ancient human contains 19.5% of the C-14 found in living organisms. How old is the bone?

Video transcript

All right. Hi, everyone. So this question says that anthropologists can estimate the age of a bone or other sample of organic matter by its carbon 14 content. The carbon 14 in a living organism is constant until the organism dies. After which carbon 14 decays with first order kinetics and a half life of 5730 years. Suppose a bone from an ancient human contains 19.5% of the carbon 14 found in living organisms. How old is the bone? And here we have four different answer choices labeled A through D proposing different ages for the bone in years. All right. So here, because we are dealing with first order kinetics, we can recall first, the first order integrated law in which the natural logarithm of the concentration of A at a certain time is equal to negative KT added to the natural logarithm of the initial concentration K refers to the rate constant and T is the amount of time. So our first step is to calculate the rate constant in this case. And to do that, we can recall an expression for the value of the half life because the half life or T, one half is equal to 0.693 divided by the rate constant K. And so this means that the value of K is equal to 0.693 divided by the half life. So for our scenario, K is equal to 0.693 divided by 5730 years. And so this equals 1.2094 multiplied by 10 to the power of negative four inverse years. So now that we have an expression for the rate constant, we can go ahead and plug in this value to find the amount of time. Now, in this case, we do not have exact concentrations, but we do have percentages. So that is what we're going to plug in or a sub T and a sub zero respectively. Wait. So here the natural logarithm of 19.5% which is the current concentration is equal to negative K or negative 1.2094 multiplied by 10 to the negative fourth inverse years multiplied by T and added to the natural logarithm of 100% since that's the original concentration. Mhm So substituting or simplifying some of these values, the natural logarithm of 19.5% is equal to 2.9704. And this equals the negative 1.2094 multiplied by 10 to the negative fourth inverse years multiplied by T add it to the natural logarithm of 100 or 100% which is four point 6052. So in this case, we can rearrange this equation at the same time that I'm moving the expression or the term that contains T to the left side, I am subtracting both sides by 2.9704. This means that 1.2094 multiplied by 10 of the negative fourth inverse years multiplied by T is equal to 4.6052 subtracted by 2.9704. And this yields 1.6348. So now T which is the amount of time and therefore the age of the bone is equal to 1.6348 divided by 1.2094 multiplied by 10 to the negative fourth inverse years. And this after rounding 23 significant figures equals 1.35 multiplied by 10 to the fourth power years and there you have it. So this corresponds to option B in the multiple choice. So with that being said, thank you so very much for watching. And I hope you found this helpful.