Carbon dating of small bits of charcoal used in cave...

Carbon dating of small bits of charcoal used in cave paintings has determined that some of the paintings are from 10000 to 30000 yr old. Carbon- 14 has a half-life of 5730 yr. In a $1 \mu \mathrm{g}$ -sample of carbon from a live tree, the activity of carbon- 14 is $6.4 \mu \mathrm{Ci}$. If researchers determine that $1 \mu \mathrm{g}$ of charcoal from a prehistoric cave painting in France has an activity of $0.80 \mu \mathrm{Ci}$, what is the age of the painting? $(5.4)$

Key Concepts

Carbon-14 Dating

Carbon-14 dating is a form of radiometric dating that utilizes the radioactive isotope carbon-14, which is incorporated into living organisms. When the organism dies, the carbon-14 begins to decay with a known half-life, and by measuring the remaining activity in a sample, scientists can estimate the time elapsed since the death. This method is especially useful for dating carbon-based materials up to tens of thousands of years old.

Exponential Decay Equation

The exponential decay equation mathematically describes how the quantity of a radioactive substance decreases over time. It is typically expressed as N(t) = N0 * exp(-?t), where N0 is the initial quantity, N(t) is the quantity at time t, and ? is the decay constant. This equation is pivotal in calculating the age of a sample by relating the current measured activity to its original activity.

Half-life

The half-life of a radioactive element is the time required for half of its initial atoms to decay. It is a fundamental concept in nuclear physics that characterizes the rate of decay in exponential processes. The half-life allows for the calculation of how much of a radioactive isotope remains after a given period and is essential for interpreting radiometric dating results.

Radioactive Decay

Radioactive decay is the process by which unstable atomic nuclei lose energy by emitting radiation. This process occurs randomly and at a rate proportional to the number of atoms present, leading to an exponential decrease over time. Radioactive decay forms the basis for dating techniques in which the remaining quantity of a radioactive isotope, relative to its original amount, can be used to determine the age of a sample.

Transcript

00:01 Carbon dating is a technique used to identify how old things are, specifically within human history.

00:10 So up to maybe 10 ,000 or 15 ,000 years at the most.

00:15 We don't do dinosaur bones with carbon dating.

00:17 It's not super accurate.

00:18 We have to use something with a longer half -life.

00:20 But for artifacts, this is usually what we go to is carbon.

00:23 Now, you and i have a certain level of radioactive carbon 14.

00:29 It doesn't really do us any harm.

00:31 Its half -life is 5 ,000 years.

00:34 So the odds of a single carbon -14 atom disintegrating and emitting radiation into you is very low in your lifetime.

00:44 All right.

00:45 It takes 5 ,000 years.

00:47 Okay, but we all have a very small amount of carbon 14.

00:50 And since things that are living are carbon -based, we live off of, say, plants and meat, which, you know, the meat we eat also eats plants.

01:00 This level of carbon in our body stays pretty constant.

01:05 So for a living thing, the amount of carbon 14 in the body, the activity of that, which is one way that we measure it, how fast is disintegrating, is 6 .4 microcurease, which is a very, very, very small number in terms of radiation.

01:22 So no need to fret about this.

01:24 Now, when something dies, well, when something that was living dies, that kind of goes hand in hand.

01:31 But when something dies, it's no longer refreshing its carbon intake because it's not eating anymore.

01:37 And so the amount of carbon 14 in the body or in the tree or in the whatever we're talking about, it starts, it stops getting refreshed.

01:46 And so it slowly starts to decrease over time.

01:50 So when we find an artifact, we check to see how much carbon 14 it has or what its activity is.

01:55 And then we can compare that and kind of backtrack to say, okay, that means you've been dead for this long.

02:02 So they have a painting.

02:04 They took the carbon 14 in it and saw that the activity is 0 .80 microcuries.

02:11 Okay, the activity.

02:12 So it's gone down a bit.

02:15 Now, how much it's gone down.

02:17 Luckily, this is a very kind of, it's a certain number of half -lives.

02:21 So if we started with 6 .4 microcuries, after one half -life, we would be down to 3 .2 because it would cut in half.

02:34 That's the point of a half -life.

02:37 Okay.

02:37 So this is one half -life, which would be 5 ,000, it gave us 730 years...

Please give Ace some feedback