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You are on an archaeological dig, and your colleague hands you a small piece of bone.
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It may have come from a cave or an ancient grave.
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Then they ask, how old is it?
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For most of human history, there was no reliable way to answer.
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People could compare the bone with other objects found nearby, they could decide that one object was older than another, but they couldn't measure its age in years.
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That changed in the 1940s when chemist Willard Libby discovered a way to estimate the age of something that had been once alive.
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His method measured how much of a particular type carbon remained in the material.
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This is our story problem.
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I'm Behi Rabani.
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Today we'll learn how carbon-14 acts like a clock and how exponential decay and logarithms let us read that clock.
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The last two episodes gave us the two mathematical tools we need today.
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In episode six, we used exponents to describe repeated multiplication.
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A quantity can grow when we repeatedly multiply by a number greater than one.
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It can also shrink when we repeatedly multiply by a number between zero and one.
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In episode seven, we used logarithms to reverse an exponential calculation.
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If an exponent tells us the result after repeated multiplication, a logarithm tells us how many rounds of multiplication produced that result.
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Today we're going to put those ideas together.
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Carbon-14 decreases exponentially after an organism dies.
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By measuring how much remains, we can work backward and estimate how much time has passed.
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Let's return to the bone.
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Old objects don't usually carry a date.
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Archaeologists can often determine which object came first by looking at where it was found.
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In an undisturbed site, deeper layers are generally older than the layers above them.
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Pottery styles, coins, and written records can also help place an object within a general period.
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Those clues provide relative dates.
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They can show that one object is older than another, or that an object probably belongs to a certain period.
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But they don't always tell us that a bone is 9,000 years old rather than 7,000 years old.
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To estimate an actual age, scientists need something that changes at a known rate.
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That change must begin at a meaningful moment, continue while the object is buried, and still be measured when the object is found.
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Carbon-14 provides that kind of clock for material that was once alive.
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Willard Libby developed radiocarbon dating while working at the University of Chicago in the late 1940s.
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High in the atmosphere, cosmic rays help create a rare, unstable form of carbon called carbon-14.
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It mixes with carbon dioxide in the air, plants take in that carbon dioxide, animals eat the plants, and carbon moves through the food chain.
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Most of the carbon in living things is stable carbon-12.
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Only a very small amount is carbon-14.
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As long as an organism is alive, it keeps exchanging carbon with its surroundings.
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Because new carbon continues to enter, the ratio of carbon-14 to stable carbon stays close to the ratio in the atmosphere.
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When the organism dies, that exchange stops.
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No new carbon-14 enters the organism.
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The carbon-14 already present begins to decay into nitrogen-14.
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As time passes, less and less carbon-14 remains.
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A laboratory doesn't need to know exactly how many carbon-14 atoms the organism had at the moment of its death.
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It measures the carbon-14 relative to stable carbon and compares that ratio with the expected starting ratio.
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That comparison tells us what fraction of the original carbon-14 remains.
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The rate of decrease is predictable.
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Once scientists know the fraction that remains, they can estimate how long the carbon-14 has been decaying.
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Now, carbon-14 does not lose the same number of atoms during every equal period of time.
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Instead, it loses the same fraction, one half during each half-life.
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The half-life of carbon-14 is about 5,730 years.
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That means that after 5,730 years, about half of the original carbon-14 remains.
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Suppose a sample begins with 100 units of carbon-14.
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After one half-life, 50 units remain.
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After a second half-life, half of 50 remains, so the amount falls to 25.
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After a third half-life, it falls to 12.5.
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No one can predict the exact moment when a particular carbon-14 atom will decay, but in a large sample containing enormous numbers of atoms, the overall pattern is remarkably consistent.
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About half of the carbon-14 remains after each half-life.
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So after one half-life, the sample is one half of its original carbon-14.
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After two half-lives, it's one quarter.
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After three, it has one eighth.
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The amount gets closer and closer and closer to zero, but the model never reaches exactly zero.
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This is exponential decay.
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Exponential growth repeatedly multiplies by a number greater than one.
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Exponential decay repeatedly multiplies by a fraction.
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For carbon fourteen, that fraction is one half every five thousand seven hundred thirty years.
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So first let's consider the easy direction.
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If a fresh sample sits for two half-lives, how much carbon fourteen will remain?
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After the first half-life, one half remains.
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After the second, one half of that remains.
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The answer is one quarter.
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So to break this down a bit more, after one half-life, the fraction remaining is one half.
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After two half-lives, it is one half times one half or one half squared.
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After three half-lives, it's one half cubed, giving us this very simple rule.
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Fraction remaining is one half raised to the power of the number of half-lives, because you're multiplying by one half by itself for the number of half-lives you're interested in.
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Radiocarbon dating asks the question in reverse.
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Scientists measure the amount present now and use it to determine how many half-lives have passed.
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For example, suppose a sample has one quarter of its original carbon-14.
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One half squared equals one quarter, so two half-lives have passed.
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Two times five thousand seven hundred thirty years is eleven thousand four hundred sixty years.
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The sample is about eleven thousand five hundred years old.
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That example works easily because one quarter is an exact power of one half.
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Real measurements usually fall between those simple values.
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A sample might have 30%, 53%, or 68% of its original carbon 14.
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Before using the model, write the measured percentages as a decimal.
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If 30% remains, the fraction is 0.3.
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If 53% remains, the fraction is 0.53.
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We now have our general equation.
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Fraction remaining equals one half raised to the time divided by 5730.
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Alright, so let's go through how we came up with that equation a little bit more slowly.
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Carbon 14 has a half-life of 5,730 years.
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That means that every 5,730 years, half of the carbon 14 remains.
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After one half, one half remains.
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After two half-lives, one half of that half remains, which is one quarter.
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After three half-lives, one eighth.
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So the one half in the equation tells us what happens during each half-life.
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The exponent tells us how many half-lives have passed.
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But the age of the sample is measured in years, not in half-lives.
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To find the number of half-lives, we divide the sample's age by 5,730.
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We are simply asking how many 5,730 year periods fit into the total amount of time.
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For example, if a sample is 11,460 years old, we divide 11,460 by 5,730.
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The answer is 2, so two half-lives have passed.
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That's why the exponent in the equation is time divided by 5730.
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So in other words, the number of half-lives equals time, in our case for carbon 14 divided by 5,730.
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Now let's use the equation in the other direction.
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In carbon dating, we can measure the fraction of carbon-14 that remains, but we don't yet know the age.
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Suppose the sample has 30% of its original carbon-14 remaining.
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We first write 30% as the decimal point three.
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Now our equation becomes 0.3 equals 1 half raised to the power of the total age of our sample divided by 5,730.
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The unknown time is inside the exponent.
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So we first ask a simpler question.
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That power tells us how many half-lives have passed.
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Well, if you remember from our from episode seven, a logarithm answers exactly that kind of question.
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It finds an unknown exponent.
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If we call the number of half-lives h, we can write point three equals one half raised to h.
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We want to find h.
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And in words, we want to find the logarithm of point three using one half as the base.
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That's episode seven speak.
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Most calculators do not have a button for a logarithm with a base of one half.
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So we use the change of base formula.
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The change of base formula says that we can calculate a logarithm with any base by dividing the logarithm of the number by the logarithm of the base.
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In this case, we divide the logarithm of 0.3 by the logarithm of one half.
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This gives us the exponent we're looking for, the number of half-lives that have passed.
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And if you press the logarithm button on your calculator or the natural log button on your calculator, log just assumes log base 10, natural log just assumes log base e.
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It doesn't matter because the log is going to cancel anyway.
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This just makes it much easier because calculators just don't have buttons for log of any base.
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They just have buttons for log, which is assumed log base 10.
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So now that we we have the equation h equals log point three divided by log of one half.
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That calculation gives us approximately 1.74.
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So about 1.74 half-lives have passed.
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Now we convert half-lives into years.
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One half-life is 5,730 years.
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So we multiply 1.74 by 5,730.
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That gives an age of approximately 9,950 years.
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Now let's combine those two steps into one general formula.
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Age equals 5,730 times the log of the fraction remaining divided by the log of 1 half.
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You may use either the log button or the natural log button as we discussed.
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Both give the same answer, as long as you use the same kind of logarithm on the top and the bottom.
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So the full process is this.
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First, measure the fraction of carbon 14 remaining.
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Next, use a logarithm to determine how many half-lives have passed.
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Finally, multiply that number by 5730 to convert half-lives into years.
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So once again, you find the fraction of the carbon 14 remaining.
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We use that change of base formula to use a logarithm to determine how many half-lives have passed.
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That's the log of fraction remaining over the log of one half.
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And then you multiply that number by 5730 to convert the half-lives into years.
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A new dating method has to be tested.
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Before using radiocarbon dating on objects of unknown age, Libby and his team tested it on samples whose approximate ages were already known.
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They studied wood from ancient Egyptian sites and other historical dated material.
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They measured the carbon-14 and compared the calculated ages with dates established by historians and archaeologists.
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The results were close enough to show that carbon-14 was changing at a useful, predictable rate.
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This is an important part of science.
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Test a new method on known examples before relying on it to answer unknown questions.
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Radiocarbon dating also depends on the amount of carbon-14 that was in the atmosphere when an organism was alive.
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That amount has changed somewhat over time, so a raw radiocarbon result does not always match a calendar date exactly.
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Scientists correct for those changes by comparing radiocarbon measurements with records of known calendar age.
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Tree rings are especially useful because a tree normally adds one ring each year.
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Researchers can measure the carbon-14 in a ring and know the calendar year which that ring formed.
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Repeating that comparison across many rings produces a calibration curve.
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Scientists use the curve to convert a raw radiocarbon age into a more accurate range of calendar dates.
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That distinction actually does matter.
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A radiocarbon age is a measurement that still needs a little bit of calibration, not automatically an exact calendar year.
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The result is usually reported as a range because every measurement has uncertainty.
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The range depends on the sample, the laboratory measurement, and the calibration curve as we just discussed.
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Scientists also have to guard against contamination from newer or older carbon, which can shift the result a little bit more.
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Well, Libby received the 1960 Nobel Prize in Chemistry for developing radiocarbon dating.
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The method gives archaeologists and other scientists a very practical way to assign dates to material from human past.
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Radiocarbon dating changed archaeology because it allowed researchers to test timelines that had previously depended mainly on comparison and interpretation.
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It did not replace archaeological evidence, it added an independent measurement that could support a proposed timeline, challenge it, or make it more precise.
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The Dead Sea Scrolls are one example.
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Radiocarbon tests of material associated with the scrolls, including parchment and linen, supported dates around the last few centuries before the Common Era and the first century of the Common Era.
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The measurements helped confirm when those texts were produced.
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Another example is Utsi the Iceman.
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In 1991, hikers found a body emerging from the melting ice in the Alps.
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At first, people thought he might have died relatively recently.
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His equipment and clothing suggested something much, much older, and radiocarbon dating confirmed it.
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He lived about 5,300 years ago during the copper age.
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Here is a simplified calculation that shows how an age like that appears.
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Suppose a sample has 53% of its original carbon-14 remaining.
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So remember from our earlier discussion, divide the log of 0.53 by the log of one half.
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The result is about 0.92 half-lives.
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Now multiply 0.92 by 5,730.
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The result is about 5,250 years.
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In real radiocarbon work, scientists also account for measurement uncertainty and use a calibration curve, so the final result is usually reported as a bit of a range of calendar dates.
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It's important to note now that radiocarbon dating can't measure every age.
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After each half-life, less carbon-14 remains.
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After about 10 half-lives, a sample is roughly 50,000 to 60,000 years old, and the remaining carbon-14 is really extremely difficult to distinguish from background and contamination.
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At that point, a tiny amount of newer carbon can have a large effect on the apparent age.
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That's why carbon-14 is useful for many archaeological samples, but not for really old samples like dinosaur fossils.
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Non-avian dinosaurs disappeared about 66 million years ago, far beyond the useful range of radiocarbon dating.
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In addition, most fossils no longer contain the organism's original carbon.
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For much, much older material, scientists use isotopes with much longer half-lives.
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Uranium lead dating, for example, can date certain minerals in rock.
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Those rock layers can help establish the ages of nearby fossils.
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The mathematical idea is the same: a radioactive isotope decays at a predictable rate, and the remaining amount is used to calculate elapsed time.
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The isotope must simply have a half-life appropriate for the age being measured.
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An age such as 5,300 years may sound impossible to know without a written record.
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Radiocarbon dating provides another source of evidence.
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A living organism takes in carbon-14.
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When it dies, that isotope stops.
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The carbon-14 then decreases by half every 5,730 years.
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Scientists measure the fraction remaining and work backward to find the time.
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If the fraction is an exact value, such as one half or one quarter, we may recognize the number of half-lives immediately.
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If it falls between those values, a logarithm finds the exponent for us.
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That's the full connection.
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Exponential decay describes how the amount changes as the time passes.
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A logarithm reverses that relationship, allowing us to calculate the time from the amount that remains.
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Well, that's the episode.
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In Study Hall this week, you'll find 10 problems that begin with simple half-life calculations and then move into real dating examples.
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The last problem also shows that what happens when a sample is too old for carbon-14 dating.
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The answer key is included so you can check each step.
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You can find it at storyproblem.org.
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If this episode helped make exponential decay or logarithms easier to understand, share it with someone who is curious about math but has never felt comfortable with it.
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This has been Story Problem.
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I'll see you next time.
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You are on an archaeological dig, and your colleague hands you a small piece of bone.
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It may have come from a cave or an ancient grave.
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Then they ask, how old is it?
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For most of human history, there was no reliable way to answer.
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People could compare the bone with other objects found nearby, they could decide that one object was older than another, but they couldn't measure its age in years.
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That changed in the 1940s when chemist Willard Libby discovered a way to estimate the age of something that had been once alive.
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His method measured how much of a particular type carbon remained in the material.
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This is our story problem.
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I'm Behi Rabani.
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Today we'll learn how carbon-14 acts like a clock and how exponential decay and logarithms let us read that clock.
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The last two episodes gave us the two mathematical tools we need today.
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In episode six, we used exponents to describe repeated multiplication.
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A quantity can grow when we repeatedly multiply by a number greater than one.
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It can also shrink when we repeatedly multiply by a number between zero and one.
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In episode seven, we used logarithms to reverse an exponential calculation.
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If an exponent tells us the result after repeated multiplication, a logarithm tells us how many rounds of multiplication produced that result.
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Today we're going to put those ideas together.
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Carbon-14 decreases exponentially after an organism dies.
00:01:42.959 --> 00:01:49.280
By measuring how much remains, we can work backward and estimate how much time has passed.
00:01:49.599 --> 00:01:51.599
Let's return to the bone.
00:01:51.920 --> 00:01:55.120
Old objects don't usually carry a date.
00:01:55.359 --> 00:02:01.680
Archaeologists can often determine which object came first by looking at where it was found.
00:02:01.920 --> 00:02:07.519
In an undisturbed site, deeper layers are generally older than the layers above them.
00:02:07.840 --> 00:02:14.319
Pottery styles, coins, and written records can also help place an object within a general period.
00:02:14.639 --> 00:02:17.039
Those clues provide relative dates.
00:02:17.199 --> 00:02:24.400
They can show that one object is older than another, or that an object probably belongs to a certain period.
00:02:24.639 --> 00:02:31.039
But they don't always tell us that a bone is 9,000 years old rather than 7,000 years old.
00:02:31.280 --> 00:02:37.599
To estimate an actual age, scientists need something that changes at a known rate.
00:02:37.840 --> 00:02:46.719
That change must begin at a meaningful moment, continue while the object is buried, and still be measured when the object is found.
00:02:47.039 --> 00:02:52.479
Carbon-14 provides that kind of clock for material that was once alive.
00:02:52.800 --> 00:02:59.520
Willard Libby developed radiocarbon dating while working at the University of Chicago in the late 1940s.
00:02:59.759 --> 00:03:07.039
High in the atmosphere, cosmic rays help create a rare, unstable form of carbon called carbon-14.
00:03:07.520 --> 00:03:17.039
It mixes with carbon dioxide in the air, plants take in that carbon dioxide, animals eat the plants, and carbon moves through the food chain.
00:03:17.280 --> 00:03:21.439
Most of the carbon in living things is stable carbon-12.
00:03:21.680 --> 00:03:24.800
Only a very small amount is carbon-14.
00:03:25.199 --> 00:03:30.639
As long as an organism is alive, it keeps exchanging carbon with its surroundings.
00:03:30.960 --> 00:03:40.319
Because new carbon continues to enter, the ratio of carbon-14 to stable carbon stays close to the ratio in the atmosphere.
00:03:40.960 --> 00:03:44.400
When the organism dies, that exchange stops.
00:03:44.639 --> 00:03:47.520
No new carbon-14 enters the organism.
00:03:47.680 --> 00:03:52.800
The carbon-14 already present begins to decay into nitrogen-14.
00:03:53.280 --> 00:03:57.840
As time passes, less and less carbon-14 remains.
00:03:58.319 --> 00:04:05.759
A laboratory doesn't need to know exactly how many carbon-14 atoms the organism had at the moment of its death.
00:04:06.000 --> 00:04:13.919
It measures the carbon-14 relative to stable carbon and compares that ratio with the expected starting ratio.
00:04:14.159 --> 00:04:19.199
That comparison tells us what fraction of the original carbon-14 remains.
00:04:19.839 --> 00:04:22.240
The rate of decrease is predictable.
00:04:22.399 --> 00:04:29.279
Once scientists know the fraction that remains, they can estimate how long the carbon-14 has been decaying.
00:04:29.600 --> 00:04:35.439
Now, carbon-14 does not lose the same number of atoms during every equal period of time.
00:04:35.680 --> 00:04:40.720
Instead, it loses the same fraction, one half during each half-life.
00:04:40.959 --> 00:04:46.160
The half-life of carbon-14 is about 5,730 years.
00:04:46.399 --> 00:04:53.279
That means that after 5,730 years, about half of the original carbon-14 remains.
00:04:53.600 --> 00:04:57.600
Suppose a sample begins with 100 units of carbon-14.
00:04:58.079 --> 00:05:01.199
After one half-life, 50 units remain.
00:05:01.439 --> 00:05:06.639
After a second half-life, half of 50 remains, so the amount falls to 25.
00:05:06.959 --> 00:05:10.240
After a third half-life, it falls to 12.5.
00:05:10.639 --> 00:05:22.240
No one can predict the exact moment when a particular carbon-14 atom will decay, but in a large sample containing enormous numbers of atoms, the overall pattern is remarkably consistent.
00:05:22.480 --> 00:05:26.480
About half of the carbon-14 remains after each half-life.
00:05:26.720 --> 00:05:32.000
So after one half-life, the sample is one half of its original carbon-14.
00:05:32.079 --> 00:05:34.399
After two half-lives, it's one quarter.
00:05:34.639 --> 00:05:37.120
After three, it has one eighth.
00:05:37.279 --> 00:05:43.439
The amount gets closer and closer and closer to zero, but the model never reaches exactly zero.
00:05:43.759 --> 00:05:45.839
This is exponential decay.
00:05:46.000 --> 00:05:50.399
Exponential growth repeatedly multiplies by a number greater than one.
00:05:50.879 --> 00:05:54.480
Exponential decay repeatedly multiplies by a fraction.
00:05:54.720 --> 00:06:01.360
For carbon fourteen, that fraction is one half every five thousand seven hundred thirty years.
00:06:02.399 --> 00:06:04.959
So first let's consider the easy direction.
00:06:05.199 --> 00:06:10.639
If a fresh sample sits for two half-lives, how much carbon fourteen will remain?
00:06:10.879 --> 00:06:13.600
After the first half-life, one half remains.
00:06:13.759 --> 00:06:16.959
After the second, one half of that remains.
00:06:17.120 --> 00:06:18.959
The answer is one quarter.
00:06:19.439 --> 00:06:25.519
So to break this down a bit more, after one half-life, the fraction remaining is one half.
00:06:25.759 --> 00:06:31.680
After two half-lives, it is one half times one half or one half squared.
00:06:31.920 --> 00:06:37.920
After three half-lives, it's one half cubed, giving us this very simple rule.
00:06:38.240 --> 00:06:48.240
Fraction remaining is one half raised to the power of the number of half-lives, because you're multiplying by one half by itself for the number of half-lives you're interested in.
00:06:48.480 --> 00:06:52.079
Radiocarbon dating asks the question in reverse.
00:06:52.319 --> 00:06:58.240
Scientists measure the amount present now and use it to determine how many half-lives have passed.
00:06:58.399 --> 00:07:02.879
For example, suppose a sample has one quarter of its original carbon-14.
00:07:03.360 --> 00:07:08.879
One half squared equals one quarter, so two half-lives have passed.
00:07:09.120 --> 00:07:16.160
Two times five thousand seven hundred thirty years is eleven thousand four hundred sixty years.
00:07:16.399 --> 00:07:20.160
The sample is about eleven thousand five hundred years old.
00:07:20.480 --> 00:07:26.000
That example works easily because one quarter is an exact power of one half.
00:07:26.319 --> 00:07:30.560
Real measurements usually fall between those simple values.
00:07:30.800 --> 00:07:37.759
A sample might have 30%, 53%, or 68% of its original carbon 14.
00:07:38.399 --> 00:07:43.199
Before using the model, write the measured percentages as a decimal.
00:07:43.519 --> 00:07:46.639
If 30% remains, the fraction is 0.3.
00:07:47.040 --> 00:07:51.120
If 53% remains, the fraction is 0.53.
00:07:51.759 --> 00:07:54.560
We now have our general equation.
00:07:54.959 --> 00:08:03.920
Fraction remaining equals one half raised to the time divided by 5730.
00:08:04.160 --> 00:08:09.279
Alright, so let's go through how we came up with that equation a little bit more slowly.
00:08:09.519 --> 00:08:14.319
Carbon 14 has a half-life of 5,730 years.
00:08:14.560 --> 00:08:20.639
That means that every 5,730 years, half of the carbon 14 remains.
00:08:20.879 --> 00:08:23.040
After one half, one half remains.
00:08:23.199 --> 00:08:28.079
After two half-lives, one half of that half remains, which is one quarter.
00:08:28.240 --> 00:08:30.720
After three half-lives, one eighth.
00:08:31.040 --> 00:08:36.720
So the one half in the equation tells us what happens during each half-life.
00:08:36.960 --> 00:08:40.960
The exponent tells us how many half-lives have passed.
00:08:41.279 --> 00:08:45.200
But the age of the sample is measured in years, not in half-lives.
00:08:45.600 --> 00:08:51.200
To find the number of half-lives, we divide the sample's age by 5,730.
00:08:51.360 --> 00:08:58.399
We are simply asking how many 5,730 year periods fit into the total amount of time.
00:08:58.639 --> 00:09:09.519
For example, if a sample is 11,460 years old, we divide 11,460 by 5,730.
00:09:09.759 --> 00:09:13.279
The answer is 2, so two half-lives have passed.
00:09:13.519 --> 00:09:19.279
That's why the exponent in the equation is time divided by 5730.
00:09:19.519 --> 00:09:27.600
So in other words, the number of half-lives equals time, in our case for carbon 14 divided by 5,730.
00:09:28.080 --> 00:09:30.879
Now let's use the equation in the other direction.
00:09:31.039 --> 00:09:38.559
In carbon dating, we can measure the fraction of carbon-14 that remains, but we don't yet know the age.
00:09:38.879 --> 00:09:44.799
Suppose the sample has 30% of its original carbon-14 remaining.
00:09:45.039 --> 00:09:48.799
We first write 30% as the decimal point three.
00:09:49.279 --> 00:10:01.279
Now our equation becomes 0.3 equals 1 half raised to the power of the total age of our sample divided by 5,730.
00:10:02.080 --> 00:10:05.679
The unknown time is inside the exponent.
00:10:06.159 --> 00:10:08.720
So we first ask a simpler question.
00:10:13.440 --> 00:10:17.360
That power tells us how many half-lives have passed.
00:10:18.000 --> 00:10:25.120
Well, if you remember from our from episode seven, a logarithm answers exactly that kind of question.
00:10:25.279 --> 00:10:27.919
It finds an unknown exponent.
00:10:28.159 --> 00:10:36.879
If we call the number of half-lives h, we can write point three equals one half raised to h.
00:10:37.919 --> 00:10:39.600
We want to find h.
00:10:40.000 --> 00:10:46.559
And in words, we want to find the logarithm of point three using one half as the base.
00:10:46.879 --> 00:10:48.960
That's episode seven speak.
00:10:49.519 --> 00:10:54.720
Most calculators do not have a button for a logarithm with a base of one half.
00:10:54.879 --> 00:10:58.000
So we use the change of base formula.
00:10:58.639 --> 00:11:10.320
The change of base formula says that we can calculate a logarithm with any base by dividing the logarithm of the number by the logarithm of the base.
00:11:10.799 --> 00:11:17.279
In this case, we divide the logarithm of 0.3 by the logarithm of one half.
00:11:17.759 --> 00:11:23.440
This gives us the exponent we're looking for, the number of half-lives that have passed.
00:11:24.720 --> 00:11:36.720
And if you press the logarithm button on your calculator or the natural log button on your calculator, log just assumes log base 10, natural log just assumes log base e.
00:11:37.039 --> 00:11:40.320
It doesn't matter because the log is going to cancel anyway.
00:11:40.559 --> 00:11:46.960
This just makes it much easier because calculators just don't have buttons for log of any base.
00:11:47.200 --> 00:11:52.159
They just have buttons for log, which is assumed log base 10.
00:11:52.399 --> 00:11:59.679
So now that we we have the equation h equals log point three divided by log of one half.
00:12:00.080 --> 00:12:03.840
That calculation gives us approximately 1.74.
00:12:04.399 --> 00:12:08.559
So about 1.74 half-lives have passed.
00:12:08.799 --> 00:12:11.519
Now we convert half-lives into years.
00:12:11.679 --> 00:12:16.000
One half-life is 5,730 years.
00:12:16.240 --> 00:12:20.879
So we multiply 1.74 by 5,730.
00:12:21.120 --> 00:12:26.080
That gives an age of approximately 9,950 years.
00:12:26.399 --> 00:12:31.200
Now let's combine those two steps into one general formula.
00:12:31.919 --> 00:12:41.919
Age equals 5,730 times the log of the fraction remaining divided by the log of 1 half.
00:12:42.559 --> 00:12:47.360
You may use either the log button or the natural log button as we discussed.
00:12:47.519 --> 00:12:53.600
Both give the same answer, as long as you use the same kind of logarithm on the top and the bottom.
00:12:53.840 --> 00:12:55.679
So the full process is this.
00:12:55.919 --> 00:12:59.360
First, measure the fraction of carbon 14 remaining.
00:12:59.600 --> 00:13:04.080
Next, use a logarithm to determine how many half-lives have passed.
00:13:04.399 --> 00:13:11.519
Finally, multiply that number by 5730 to convert half-lives into years.
00:13:11.679 --> 00:13:15.759
So once again, you find the fraction of the carbon 14 remaining.
00:13:16.000 --> 00:13:22.080
We use that change of base formula to use a logarithm to determine how many half-lives have passed.
00:13:22.320 --> 00:13:26.399
That's the log of fraction remaining over the log of one half.
00:13:26.559 --> 00:13:33.360
And then you multiply that number by 5730 to convert the half-lives into years.
00:13:33.759 --> 00:13:36.159
A new dating method has to be tested.
00:13:36.320 --> 00:13:46.159
Before using radiocarbon dating on objects of unknown age, Libby and his team tested it on samples whose approximate ages were already known.
00:13:46.320 --> 00:13:51.279
They studied wood from ancient Egyptian sites and other historical dated material.
00:13:51.440 --> 00:13:58.320
They measured the carbon-14 and compared the calculated ages with dates established by historians and archaeologists.
00:13:58.559 --> 00:14:04.799
The results were close enough to show that carbon-14 was changing at a useful, predictable rate.
00:14:05.120 --> 00:14:06.879
This is an important part of science.
00:14:07.039 --> 00:14:12.879
Test a new method on known examples before relying on it to answer unknown questions.
00:14:13.120 --> 00:14:20.879
Radiocarbon dating also depends on the amount of carbon-14 that was in the atmosphere when an organism was alive.
00:14:21.120 --> 00:14:30.960
That amount has changed somewhat over time, so a raw radiocarbon result does not always match a calendar date exactly.
00:14:31.279 --> 00:14:39.120
Scientists correct for those changes by comparing radiocarbon measurements with records of known calendar age.
00:14:39.440 --> 00:14:46.080
Tree rings are especially useful because a tree normally adds one ring each year.
00:14:46.320 --> 00:14:52.879
Researchers can measure the carbon-14 in a ring and know the calendar year which that ring formed.
00:14:53.120 --> 00:14:58.320
Repeating that comparison across many rings produces a calibration curve.
00:14:58.559 --> 00:15:06.240
Scientists use the curve to convert a raw radiocarbon age into a more accurate range of calendar dates.
00:15:06.480 --> 00:15:08.559
That distinction actually does matter.
00:15:08.720 --> 00:15:17.039
A radiocarbon age is a measurement that still needs a little bit of calibration, not automatically an exact calendar year.
00:15:17.279 --> 00:15:22.559
The result is usually reported as a range because every measurement has uncertainty.
00:15:22.720 --> 00:15:28.799
The range depends on the sample, the laboratory measurement, and the calibration curve as we just discussed.
00:15:29.039 --> 00:15:37.200
Scientists also have to guard against contamination from newer or older carbon, which can shift the result a little bit more.
00:15:37.600 --> 00:15:43.759
Well, Libby received the 1960 Nobel Prize in Chemistry for developing radiocarbon dating.
00:15:44.000 --> 00:15:51.600
The method gives archaeologists and other scientists a very practical way to assign dates to material from human past.
00:15:52.000 --> 00:16:02.080
Radiocarbon dating changed archaeology because it allowed researchers to test timelines that had previously depended mainly on comparison and interpretation.
00:16:02.320 --> 00:16:13.840
It did not replace archaeological evidence, it added an independent measurement that could support a proposed timeline, challenge it, or make it more precise.
00:16:14.240 --> 00:16:17.360
The Dead Sea Scrolls are one example.
00:16:17.679 --> 00:16:30.320
Radiocarbon tests of material associated with the scrolls, including parchment and linen, supported dates around the last few centuries before the Common Era and the first century of the Common Era.
00:16:30.480 --> 00:16:34.799
The measurements helped confirm when those texts were produced.
00:16:35.200 --> 00:16:38.080
Another example is Utsi the Iceman.
00:16:38.559 --> 00:16:44.399
In 1991, hikers found a body emerging from the melting ice in the Alps.
00:16:44.720 --> 00:16:49.120
At first, people thought he might have died relatively recently.
00:16:49.360 --> 00:16:55.279
His equipment and clothing suggested something much, much older, and radiocarbon dating confirmed it.
00:16:55.440 --> 00:17:00.320
He lived about 5,300 years ago during the copper age.
00:17:00.559 --> 00:17:05.440
Here is a simplified calculation that shows how an age like that appears.
00:17:05.839 --> 00:17:11.519
Suppose a sample has 53% of its original carbon-14 remaining.
00:17:11.680 --> 00:17:18.480
So remember from our earlier discussion, divide the log of 0.53 by the log of one half.
00:17:18.720 --> 00:17:22.319
The result is about 0.92 half-lives.
00:17:22.640 --> 00:17:27.200
Now multiply 0.92 by 5,730.
00:17:27.359 --> 00:17:31.519
The result is about 5,250 years.
00:17:31.839 --> 00:17:43.599
In real radiocarbon work, scientists also account for measurement uncertainty and use a calibration curve, so the final result is usually reported as a bit of a range of calendar dates.
00:17:43.839 --> 00:17:48.880
It's important to note now that radiocarbon dating can't measure every age.
00:17:49.200 --> 00:17:52.720
After each half-life, less carbon-14 remains.
00:17:52.880 --> 00:18:05.839
After about 10 half-lives, a sample is roughly 50,000 to 60,000 years old, and the remaining carbon-14 is really extremely difficult to distinguish from background and contamination.
00:18:06.160 --> 00:18:12.400
At that point, a tiny amount of newer carbon can have a large effect on the apparent age.
00:18:12.640 --> 00:18:22.559
That's why carbon-14 is useful for many archaeological samples, but not for really old samples like dinosaur fossils.
00:18:22.799 --> 00:18:31.759
Non-avian dinosaurs disappeared about 66 million years ago, far beyond the useful range of radiocarbon dating.
00:18:31.920 --> 00:18:37.119
In addition, most fossils no longer contain the organism's original carbon.
00:18:37.440 --> 00:18:43.759
For much, much older material, scientists use isotopes with much longer half-lives.
00:18:44.000 --> 00:18:49.119
Uranium lead dating, for example, can date certain minerals in rock.
00:18:49.359 --> 00:18:54.480
Those rock layers can help establish the ages of nearby fossils.
00:18:54.720 --> 00:19:05.279
The mathematical idea is the same: a radioactive isotope decays at a predictable rate, and the remaining amount is used to calculate elapsed time.
00:19:05.519 --> 00:19:11.200
The isotope must simply have a half-life appropriate for the age being measured.
00:19:11.519 --> 00:19:17.519
An age such as 5,300 years may sound impossible to know without a written record.
00:19:17.759 --> 00:19:21.599
Radiocarbon dating provides another source of evidence.
00:19:21.920 --> 00:19:24.799
A living organism takes in carbon-14.
00:19:24.960 --> 00:19:27.440
When it dies, that isotope stops.
00:19:27.680 --> 00:19:33.200
The carbon-14 then decreases by half every 5,730 years.
00:19:33.519 --> 00:19:38.480
Scientists measure the fraction remaining and work backward to find the time.
00:19:38.960 --> 00:19:47.279
If the fraction is an exact value, such as one half or one quarter, we may recognize the number of half-lives immediately.
00:19:47.519 --> 00:19:53.200
If it falls between those values, a logarithm finds the exponent for us.
00:19:53.759 --> 00:19:55.519
That's the full connection.
00:19:55.680 --> 00:20:01.200
Exponential decay describes how the amount changes as the time passes.
00:20:01.440 --> 00:20:08.720
A logarithm reverses that relationship, allowing us to calculate the time from the amount that remains.
00:20:14.559 --> 00:20:16.319
Well, that's the episode.
00:20:16.559 --> 00:20:25.359
In Study Hall this week, you'll find 10 problems that begin with simple half-life calculations and then move into real dating examples.
00:20:25.599 --> 00:20:31.440
The last problem also shows that what happens when a sample is too old for carbon-14 dating.
00:20:31.680 --> 00:20:34.799
The answer key is included so you can check each step.
00:20:34.960 --> 00:20:38.079
You can find it at storyproblem.org.
00:20:38.319 --> 00:20:48.240
If this episode helped make exponential decay or logarithms easier to understand, share it with someone who is curious about math but has never felt comfortable with it.
00:20:50.160 --> 00:20:51.759
This has been Story Problem.
00:20:52.079 --> 00:20:53.920
I'll see you next time.
