Carbon 14 dating assumes that the carbon dioxide on Earth...

Carbon $ 14 $ dating assumes that the carbon dioxide on Earth today has the same radioactive content as it did centuries ago. If this is true,the amount of $ ^{14} C $ absorbed by a tree that grew several centuries ago should be the same as the amount of $ ^{14} C $ absorbed by a tree growing today. A piece of ancient charcoal contains only $ 15\% $ as much radioactive carbon as a piece of modern charcoal. How long ago was the tree burned to make the ancient charcoal if the half-life of $ ^{14} C $ is $ 5715 $ years?

Key Concepts

Carbon Dating

Carbon dating is a dating method that utilizes the radioactive decay of carbon-14 to estimate the age of organic materials. By comparing the current amount of carbon-14 in a sample with the assumed constant historical level found in contemporary organic matter, it becomes possible to calculate the time elapsed since the organism's death.

Exponential Decay Law

The exponential decay law mathematically models the reduction in the number of radioactive nuclei over time. This law states that the remaining quantity of a substance diminishes exponentially relative to time, with the half-life serving as the constant of proportionality in this decay process.

Radioactive Decay

Radioactive decay is the process by which unstable atomic nuclei lose energy by emitting radiation. This transformation occurs at a rate that is statistically predictable and is described by an exponential decay law, making it possible to relate the remaining quantity of a substance to the time elapsed since its formation.

Half-Life

The half-life of a radioactive isotope is the time required for half of its nuclei to decay. It serves as a fundamental parameter in quantifying the decay process, allowing scientists to use measurements of remaining material to determine the elapsed time since the sample was formed or altered.

Transcript

00:01 All right, this is a really nice half -life problem.

00:04 So carbon 14 is something, a way to measure the age of an item.

00:13 So we need the exponential equation y equals a e to the kt power.

00:19 And a is going to represent that initial amount.

00:22 K is going to be that decay rate and t is your time.

00:27 So they tell you that a piece of ancient charcoal contains only 15 % as much radioactive carbon as a piece of modern charcoal.

00:36 And they want to go, how long ago was the tree burned to make the ancient charcoal? if the half -life of carbon 14 is 5 ,715 years.

00:45 Well, we don't know how much original charcoal we have, right? so we're just going to start with one gram.

00:53 And we know that we want that to be cut in half.

00:57 That's the whole concept of half -life, right? how long it takes for half of the substance to disappear.

01:03 We don't know what rate that is occurring at.

01:06 That's what we have to find first, but we do know that it takes 5 ,100, i lied, 5 ,715 years to do that.

01:15 So what i need to be able to do is undo this exponential function.

01:22 So because i have a natural exponential base, i have to undo that with a natural logarithm.

01:29 So i'll take the natural log of each side.

01:33 I do that so that i can apply some log properties here that i'm allowed to bring the exponent down in front and multiply by the natural log of e, but the natural log of e is one.

01:46 So really all i need to do on my calculator is evaluate the natural log of 0 .5 divided by the half -life of 5 ,715 years.

01:58 So the natural log, let me punch it in the calculator here, the natural log of 0 .5 divided by 5 ,715 is, i'll write it, it's negative 1 .21285596 times 10 to the negative fourth power...

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