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It was a teaspoon that helped Benjamin Franklin measure something he couldn't possibly see.
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And not a distant star or the depths of the ocean.
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I mean the size of a single molecule.
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Franklin was standing beside a pond outside London on a windy day.
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He poured a teaspoon of oil onto the water and watched as it spread across an enormous area of the pond.
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That gave him two measurements, the volume of the oil and the area it covered.
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And when you divide the volume by the area, you get the thickness of the oil.
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Volume divided by area equals thickness.
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This was a different way to think at the time and was something revolutionary.
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And all you had to do was divide the volume of the teaspoon of oil with the area of the pond that it covered.
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The experiment would eventually give scientists a way to estimate a distance far too small for anyone to see.
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But numbers that small create a new problem.
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Write them as ordinary decimals, and you quickly end up with a long row of zeros.
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The number becomes difficult to read, difficult to compare, and very easy to get wrong.
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So we need a better way to describe the scale.
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This is an amazing story and one that really illustrates how great narratives can help us understand math and science concepts so much better.
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So wherever you are, on a jog, in the car, doing the dishes, I hope you enjoy this episode as much as I enjoyed researching and making it.
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I'm Becky Rabani, and this is Story Problem.
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Let's get to it! The story begins in 1757.
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Franklin was at sea in a fleet of 96 ships headed toward Louisburg.
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Most of the ships left the kind of broken, choppy wake you would expect, but two of them didn't.
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Behind those two ships, the water looked strangely smooth.
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Franklin asked the captain, What's going on?
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The captain gave him a very ordinary answer.
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The cooks had probably just emptied greasy water through the ship's drains.
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A little fat or oil had reached the sea and changed the surface.
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The captain said this as though everybody knew it.
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Franklin didn't.
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Why would greasy water make a ship's wake look smooth?
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Why should such a small amount of matter and could you recreate this effect yourself?
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Franklin remembered that ancient writers had described sailors pouring oil on rough water.
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He also heard stories about fishermen and divers using oil to calm small ripples so they could see more clearly through the surface.
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He kept thinking about thinking about it for years, actually.
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That part's important because Franklin didn't simply show up at a pond one day and decide to pour oil into it.
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He was following up on something he had noticed years earlier.
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By 1773, Franklin was living in London.
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One windy day he was near a large pond on Clapham Common.
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The surface was rough and he finally had the setting he wanted.
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He brought out a small container of oil and dropped a little wool onto the water.
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The first attempt didn't work.
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He had started on the leeward side, the side the wind was blowing toward.
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The wind simply pushed the oil back to shore.
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So Franklin walked around to the windward side where the ripples were beginning.
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Then he tried again.
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This time, less than a teaspoon of oil spread out with surprising speed.
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The surface immediately began to calm.
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The smooth region grew across the pond until, by Franklin's estimate, it covered nearly half an acre.
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Franklin wrote that the part of the pond covered by the oil became as smooth as a looking glass.
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Now think about how far that oil spread.
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Franklin used only about a teaspoon, but it covered roughly half an acre, about 2,000 square meters.
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That's an area larger than four basketball courts.
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The oil didn't stop the deeper motion of the pond, it didn't turn the water into glass.
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What it suppressed were the tiny surface ripples, the wrinkles that let wind keep transferring energy into the water.
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Franklin didn't have our modern explanation of surface tension and thin films.
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He was trying to understand the wave calming effect.
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But he had carefully recorded two numbers that would matter enormously later.
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How much oil he used and how much surface it covered, volume and area.
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Franklin never used the pond to calculate the size of a molecule.
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It's important to be clear about what Franklin actually did.
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He showed how far a small amount of oil could spread across the water.
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He didn't use the experiment to calculate the size of a molecule.
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That part came much later.
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At the time, scientists were still figuring out what molecules actually were.
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Franklin had performed the experiment, but it would take a later generation to realize it could be used to estimate the size of one.
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More than a century later, the physicist Lord Riley returned to the problem.
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He used carefully measured quantities of oil and measured how far the film spread.
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Then he made one crucial assumption.
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Once the oil had spread as far as it could, the film was approximately one molecule thick.
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Now the geometry becomes a lot more simple.
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Imagine the oil film as a very wide, very flat box.
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Every box obeys the same relationship.
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Volume equals area times thickness.
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So if you know the volume and the area, you can rearrange it.
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Thickness equals volume divided by area.
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And if the film is one molecule thick, the thickness of the film is an estimate of the molecule's length.
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No microscope required.
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The molecule is invisible, but its geometry is math.
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Riley's experiment put the result in the neighborhood of two billionths of a meter, depending on the oil and the experimental assumptions.
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That is the right scale for the length of an oil molecule.
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This is where the math comes in.
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To understand what happened next, we need to work with measurements so small that ordinary decimals become difficult to read and easy to get wrong.
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Let's use Franklin's rough pond numbers.
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A teaspoon is about five cubic centimeters.
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In cubic meters, that is 0.00005 cubic meters.
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Half an acre is about 2,000 square meters.
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Divide the oil's volume by the area it covered, and the estimated thickness is about 0.
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Then take 80.25 meters.
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The first non-zero digit does not appear until the ninth decimal place.
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You can write it that way.
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Scientific notation gives the number a much cleaner name, 2.5 times 10 to the negative ninth meters, or using a meter prefix, 2.5 nanometers.
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Those three forms describe the same length, 0.8025 meters, 2.5 times 10 to the negative 9th meters, or 2.5 nanometers.
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Same quantity, different name.
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That should sound familiar from our last episode on measurement and conversion, but scientific notation adds something new.
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It doesn't merely shorten the number, it makes the scale more visible to us.
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A number in scientific notation has two pieces.
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The first piece is the coefficient.
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That is the ordinary number in the front.
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In our example, it's 2.5.
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The second piece is the power of 10.
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Here it's 10 to the negative 9th.
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The coefficient tells you you are within the scale.
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The exponent tells you which scale you're actually on.
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10 to the 0 is 1.
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10 to the first is 10.
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10 to the second is 100.
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10 to the third is 1000.
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Negative exponents move in the other direction.
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10 to the negative first is 1 tenth, 10 to the negative second is 100th, 10 to the negative third is 1000th.
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And 10 to the negative 9th is 1 billionth.
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So 2.5 times 10 to the negative 9th means 2.5 billionths of a meter.
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So the first step is to rewrite the number in scientific notation.
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Don't stare at a long line of zeros and hope you counted correctly.
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Rewrite the number so the coefficient is between 1 and 10 and let the exponent tell us the scale of the number.
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Now look back at the pond calculation.
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The teaspoon is 5 times 10 to the negative 6 cubic meters.
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The pond area is 2 times 10 to the third square meters.
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We're dividing volume by area.
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The ordinary number gives us 5 divided by 2, which is 2.5.
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The power of 10 tells us how the scale moves.
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When we divide, we subtract the exponents.
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Negative 6 minus positive 3 is negative 9.
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And the units tell the same story.
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Cubic meters divided by square meters leaves meters.
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So the answer is a length, 2.5 times 10 to the negative 9th meters.
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We'll slow that math down in office hours.
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We'll work through more why exponents add when we multiply and subtract when we divide.
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For now, I want you to notice the role they play.
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We handle the 5 and the 2 just as we normally would.
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The exponents tell us how far the result moves us up or down the scale.
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That brings us to the second step, move between scales.
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Scientific notation also tells us to compare things that our eyes can't compare directly.
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Take a human hair, a typical rough value for its width is 1 10,000th of a meter.
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In scientific notation, that is 1 times 10 to the negative 4 meters.
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Our oil molecule is about 2.5 times 10 to the negative 9th meters long.
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Just look at the exponents, negative 4 and negative 9.
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They're five powers of 10 apart.
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So before doing any exact arithmetic, you already know the answer will be in the neighborhood of tens of thousands.
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Finish the division and you get about 40,000.
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Roughly 40,000 of these molecules could line up across the width of one human hair.
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That comparison is much easier to understand than a long decimal.
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The decimal tells us that the molecule is small, the ratio tells us how small it is compared to something familiar.
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That's the third move that we want to learn.
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Compare it.
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When you compare two numbers in scientific notation, start with the exponents.
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The difference between them tells you roughly how many orders of magnitude separates the numbers.
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Then compare the coefficients for the more precise answer.
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The story of the oil film did not end with Riley.
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In Germany, a young woman named Agnes Pockels had been studying the surface of water at home.
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She had wanted formal scientific training, but universities were not open to women at that time.
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She also spent much of her life caring for her family members.
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So she built a laboratory from what she had.
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Pockels made a narrow metal trough, filled it with water, and laid a movable strip across the surface.
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By sliding the strip, she could compress a film of oil or soap into a smaller and smaller area.
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With a small disc and a balance, she could measure how the surface tension changed.
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It was a simple apparatus, but it gave her control over something Franklin could only watch spread across a pond.
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When Pockles learned about Riley's work, she wrote to him.
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She explained that she had already spent years making related measurements but had no straightforward way to publish them.
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Riley recognized the quality of the work and sent a translated version of her letter to Nature.
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It appeared in 1891.
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Pockel's experiments had helped establish that these surface films could be measured as organized layers.
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She could estimate how much surface area an individual molecule occupied.
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Now we can estimate another dimension of the molecule.
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Franklin's pond gives us a rough estimate of the oil layer's thickness.
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Pockel's trough helps us estimate how much surface area each molecule occupies.
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Later, Irving Langmuir built a broader theory of molecules arranged in single layers at surfaces.
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His work in surface chemistry earned the Nobel Prize in Chemistry in 1932.
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So the scientific journey was more of a relay race than one heroic sprint.
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Sailors notice greasy water, Franklin turns the observation into an experiment, Riley turns the experiment into a molecular estimate, Pockles turns the surface into something that can be controlled and measured, and Langmere builds a theory of the molecular layer.
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The same mathematical idea connects all these experiments.
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Start with quantities you can measure and use them to estimate something too small to see.
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Now let's move in the opposite direction.
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A single oil molecule is about a few billionths of a meter long, but a teaspoon of oil contains an enormous number of them.
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Using a simplified molecular shape and the measurements from the thin film, we can estimate the volume of one molecule.
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Then we can divide the total volume of oil by the volume of one molecule.
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For Franklin's teaspoon, the estimate lands around 10 to the 22nd molecules.
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That's about 10 sixtillion molecules in one teaspoon.
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And if we scale the same reasoning up to one mole of molecules, the estimate lands close to 10 times 10 to the 23rd.
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The exact modern value of the Avogadro constant is 6.02214076 times 10 to the 23rd particles per mole.
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That's an enormous number, and writing it all 23 zeros wouldn't make it any easier to understand.
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Scientific notation gives us a much clearer way to write it.
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A molecule might be about 10 to the negative 9th meters long.
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A mole contains about 6 times 10 to the 23rd molecules.
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Across those two experiments, we've moved from an exponent of negative 9 to an exponent of positive 23.
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That's a span of 32 powers of 10.
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This is why the exponent matters.
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It tells us the scale of the number.
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It lets us see immediately whether we're working with something extremely small or extremely large.
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There's a larger lesson here.
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People often think scientific notation is mainly a way to avoid writing zeros.
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That's useful, but it's not the real payoff.
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The real payoff is that it makes scale something you can reason with.
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If one object is about 10 to the negative fourth meters wide and another is 10 to the negative ninth meters wide, you can see a five-order of magnitude gap before reaching for a calculator.
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If a calculation that should describe a molecule gives you 10 to the positive six meters, the exponent tells you immediately that something has gone badly wrong.
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If two measurements differ only in their coefficients but share the same exponent, they live on roughly the same scale.
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This is estimation.
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It's error checking, it's scientific judgment.
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This is also why experienced problem solvers can seem so fast.
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It's not necessarily because they do the arithmetic more quickly, because they calculate, they estimate the order of the magnitude, and rule out answers that can't be right.
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Let's go back to Clapham Common.
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Franklin stands at the windward edge of a rough pond.
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He pours out less than a teaspoon of oil.
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The smooth patch begins near his feet and moves across the surface until a large section of the pond looks like glass.
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He sees the pond change.
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He records the oil in the area, but the smallest part of the experiment remains invisible.
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More than a century later, scientists recognize what those measurements contain.
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Volume divided by area gives thickness.
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Thickness gives a molecular scale.
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Scientific notation gives that scale a name.
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And once the scale has a name, you can move through it and compare it with everything around us.
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Name it, move it, compare it.
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That's the mathematical concept of this episode.
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The episode six story sheet is in study hall at storyproblem.org to practice these concepts.
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It follows the story from Franklin's teaspoon to the molecular layer and lays out the key scales so you can see how they connect.
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Then in a few days, come back for the companion office hours.
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We'll take the same numbers and work the mathematics step by step, writing scientific notation, multiplying, dividing, powers of 10, normalizing coefficients, and using exponent differences to compare scale.
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The main episode introduces the mathematics through the story.
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Office hours breaks down the problem solving steps, and study hall gives you a chance to practice them yourself.
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I'm Behi Rabani.
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This has been Story Problem.
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I'll see you next time.
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It was a teaspoon that helped Benjamin Franklin measure something he couldn't possibly see.
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And not a distant star or the depths of the ocean.
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I mean the size of a single molecule.
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Franklin was standing beside a pond outside London on a windy day.
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He poured a teaspoon of oil onto the water and watched as it spread across an enormous area of the pond.
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That gave him two measurements, the volume of the oil and the area it covered.
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And when you divide the volume by the area, you get the thickness of the oil.
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Volume divided by area equals thickness.
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This was a different way to think at the time and was something revolutionary.
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And all you had to do was divide the volume of the teaspoon of oil with the area of the pond that it covered.
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The experiment would eventually give scientists a way to estimate a distance far too small for anyone to see.
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But numbers that small create a new problem.
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Write them as ordinary decimals, and you quickly end up with a long row of zeros.
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The number becomes difficult to read, difficult to compare, and very easy to get wrong.
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So we need a better way to describe the scale.
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This is an amazing story and one that really illustrates how great narratives can help us understand math and science concepts so much better.
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So wherever you are, on a jog, in the car, doing the dishes, I hope you enjoy this episode as much as I enjoyed researching and making it.
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I'm Becky Rabani, and this is Story Problem.
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Let's get to it! The story begins in 1757.
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Franklin was at sea in a fleet of 96 ships headed toward Louisburg.
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Most of the ships left the kind of broken, choppy wake you would expect, but two of them didn't.
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Behind those two ships, the water looked strangely smooth.
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Franklin asked the captain, What's going on?
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The captain gave him a very ordinary answer.
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The cooks had probably just emptied greasy water through the ship's drains.
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A little fat or oil had reached the sea and changed the surface.
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The captain said this as though everybody knew it.
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Franklin didn't.
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Why would greasy water make a ship's wake look smooth?
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Why should such a small amount of matter and could you recreate this effect yourself?
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Franklin remembered that ancient writers had described sailors pouring oil on rough water.
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He also heard stories about fishermen and divers using oil to calm small ripples so they could see more clearly through the surface.
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He kept thinking about thinking about it for years, actually.
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That part's important because Franklin didn't simply show up at a pond one day and decide to pour oil into it.
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He was following up on something he had noticed years earlier.
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By 1773, Franklin was living in London.
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One windy day he was near a large pond on Clapham Common.
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The surface was rough and he finally had the setting he wanted.
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He brought out a small container of oil and dropped a little wool onto the water.
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The first attempt didn't work.
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He had started on the leeward side, the side the wind was blowing toward.
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The wind simply pushed the oil back to shore.
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So Franklin walked around to the windward side where the ripples were beginning.
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Then he tried again.
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This time, less than a teaspoon of oil spread out with surprising speed.
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The surface immediately began to calm.
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The smooth region grew across the pond until, by Franklin's estimate, it covered nearly half an acre.
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Franklin wrote that the part of the pond covered by the oil became as smooth as a looking glass.
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Now think about how far that oil spread.
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Franklin used only about a teaspoon, but it covered roughly half an acre, about 2,000 square meters.
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That's an area larger than four basketball courts.
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The oil didn't stop the deeper motion of the pond, it didn't turn the water into glass.
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What it suppressed were the tiny surface ripples, the wrinkles that let wind keep transferring energy into the water.
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Franklin didn't have our modern explanation of surface tension and thin films.
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He was trying to understand the wave calming effect.
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But he had carefully recorded two numbers that would matter enormously later.
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How much oil he used and how much surface it covered, volume and area.
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Franklin never used the pond to calculate the size of a molecule.
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It's important to be clear about what Franklin actually did.
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He showed how far a small amount of oil could spread across the water.
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He didn't use the experiment to calculate the size of a molecule.
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That part came much later.
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At the time, scientists were still figuring out what molecules actually were.
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Franklin had performed the experiment, but it would take a later generation to realize it could be used to estimate the size of one.
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More than a century later, the physicist Lord Riley returned to the problem.
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He used carefully measured quantities of oil and measured how far the film spread.
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Then he made one crucial assumption.
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Once the oil had spread as far as it could, the film was approximately one molecule thick.
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Now the geometry becomes a lot more simple.
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Imagine the oil film as a very wide, very flat box.
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Every box obeys the same relationship.
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Volume equals area times thickness.
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So if you know the volume and the area, you can rearrange it.
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Thickness equals volume divided by area.
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And if the film is one molecule thick, the thickness of the film is an estimate of the molecule's length.
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No microscope required.
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The molecule is invisible, but its geometry is math.
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Riley's experiment put the result in the neighborhood of two billionths of a meter, depending on the oil and the experimental assumptions.
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That is the right scale for the length of an oil molecule.
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This is where the math comes in.
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To understand what happened next, we need to work with measurements so small that ordinary decimals become difficult to read and easy to get wrong.
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Let's use Franklin's rough pond numbers.
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A teaspoon is about five cubic centimeters.
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In cubic meters, that is 0.00005 cubic meters.
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Half an acre is about 2,000 square meters.
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Divide the oil's volume by the area it covered, and the estimated thickness is about 0.
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Then take 80.25 meters.
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The first non-zero digit does not appear until the ninth decimal place.
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You can write it that way.
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Scientific notation gives the number a much cleaner name, 2.5 times 10 to the negative ninth meters, or using a meter prefix, 2.5 nanometers.
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Those three forms describe the same length, 0.8025 meters, 2.5 times 10 to the negative 9th meters, or 2.5 nanometers.
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Same quantity, different name.
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That should sound familiar from our last episode on measurement and conversion, but scientific notation adds something new.
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It doesn't merely shorten the number, it makes the scale more visible to us.
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A number in scientific notation has two pieces.
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The first piece is the coefficient.
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That is the ordinary number in the front.
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In our example, it's 2.5.
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The second piece is the power of 10.
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Here it's 10 to the negative 9th.
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The coefficient tells you you are within the scale.
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The exponent tells you which scale you're actually on.
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10 to the 0 is 1.
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10 to the first is 10.
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10 to the second is 100.
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10 to the third is 1000.
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Negative exponents move in the other direction.
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10 to the negative first is 1 tenth, 10 to the negative second is 100th, 10 to the negative third is 1000th.
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And 10 to the negative 9th is 1 billionth.
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So 2.5 times 10 to the negative 9th means 2.5 billionths of a meter.
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So the first step is to rewrite the number in scientific notation.
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Don't stare at a long line of zeros and hope you counted correctly.
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Rewrite the number so the coefficient is between 1 and 10 and let the exponent tell us the scale of the number.
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Now look back at the pond calculation.
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The teaspoon is 5 times 10 to the negative 6 cubic meters.
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The pond area is 2 times 10 to the third square meters.
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We're dividing volume by area.
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The ordinary number gives us 5 divided by 2, which is 2.5.
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The power of 10 tells us how the scale moves.
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When we divide, we subtract the exponents.
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Negative 6 minus positive 3 is negative 9.
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And the units tell the same story.
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Cubic meters divided by square meters leaves meters.
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So the answer is a length, 2.5 times 10 to the negative 9th meters.
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We'll slow that math down in office hours.
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We'll work through more why exponents add when we multiply and subtract when we divide.
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For now, I want you to notice the role they play.
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We handle the 5 and the 2 just as we normally would.
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The exponents tell us how far the result moves us up or down the scale.
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That brings us to the second step, move between scales.
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Scientific notation also tells us to compare things that our eyes can't compare directly.
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Take a human hair, a typical rough value for its width is 1 10,000th of a meter.
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In scientific notation, that is 1 times 10 to the negative 4 meters.
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Our oil molecule is about 2.5 times 10 to the negative 9th meters long.
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Just look at the exponents, negative 4 and negative 9.
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They're five powers of 10 apart.
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So before doing any exact arithmetic, you already know the answer will be in the neighborhood of tens of thousands.
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Finish the division and you get about 40,000.
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Roughly 40,000 of these molecules could line up across the width of one human hair.
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That comparison is much easier to understand than a long decimal.
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The decimal tells us that the molecule is small, the ratio tells us how small it is compared to something familiar.
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That's the third move that we want to learn.
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Compare it.
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When you compare two numbers in scientific notation, start with the exponents.
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The difference between them tells you roughly how many orders of magnitude separates the numbers.
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Then compare the coefficients for the more precise answer.
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The story of the oil film did not end with Riley.
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In Germany, a young woman named Agnes Pockels had been studying the surface of water at home.
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She had wanted formal scientific training, but universities were not open to women at that time.
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She also spent much of her life caring for her family members.
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So she built a laboratory from what she had.
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Pockels made a narrow metal trough, filled it with water, and laid a movable strip across the surface.
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By sliding the strip, she could compress a film of oil or soap into a smaller and smaller area.
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With a small disc and a balance, she could measure how the surface tension changed.
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It was a simple apparatus, but it gave her control over something Franklin could only watch spread across a pond.
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When Pockles learned about Riley's work, she wrote to him.
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She explained that she had already spent years making related measurements but had no straightforward way to publish them.
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Riley recognized the quality of the work and sent a translated version of her letter to Nature.
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It appeared in 1891.
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Pockel's experiments had helped establish that these surface films could be measured as organized layers.
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She could estimate how much surface area an individual molecule occupied.
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Now we can estimate another dimension of the molecule.
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Franklin's pond gives us a rough estimate of the oil layer's thickness.
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Pockel's trough helps us estimate how much surface area each molecule occupies.
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Later, Irving Langmuir built a broader theory of molecules arranged in single layers at surfaces.
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His work in surface chemistry earned the Nobel Prize in Chemistry in 1932.
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So the scientific journey was more of a relay race than one heroic sprint.
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Sailors notice greasy water, Franklin turns the observation into an experiment, Riley turns the experiment into a molecular estimate, Pockles turns the surface into something that can be controlled and measured, and Langmere builds a theory of the molecular layer.
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The same mathematical idea connects all these experiments.
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Start with quantities you can measure and use them to estimate something too small to see.
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Now let's move in the opposite direction.
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A single oil molecule is about a few billionths of a meter long, but a teaspoon of oil contains an enormous number of them.
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Using a simplified molecular shape and the measurements from the thin film, we can estimate the volume of one molecule.
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Then we can divide the total volume of oil by the volume of one molecule.
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For Franklin's teaspoon, the estimate lands around 10 to the 22nd molecules.
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That's about 10 sixtillion molecules in one teaspoon.
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And if we scale the same reasoning up to one mole of molecules, the estimate lands close to 10 times 10 to the 23rd.
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The exact modern value of the Avogadro constant is 6.02214076 times 10 to the 23rd particles per mole.
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That's an enormous number, and writing it all 23 zeros wouldn't make it any easier to understand.
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Scientific notation gives us a much clearer way to write it.
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A molecule might be about 10 to the negative 9th meters long.
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A mole contains about 6 times 10 to the 23rd molecules.
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Across those two experiments, we've moved from an exponent of negative 9 to an exponent of positive 23.
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That's a span of 32 powers of 10.
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This is why the exponent matters.
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It tells us the scale of the number.
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It lets us see immediately whether we're working with something extremely small or extremely large.
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There's a larger lesson here.
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People often think scientific notation is mainly a way to avoid writing zeros.
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That's useful, but it's not the real payoff.
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The real payoff is that it makes scale something you can reason with.
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If one object is about 10 to the negative fourth meters wide and another is 10 to the negative ninth meters wide, you can see a five-order of magnitude gap before reaching for a calculator.
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If a calculation that should describe a molecule gives you 10 to the positive six meters, the exponent tells you immediately that something has gone badly wrong.
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If two measurements differ only in their coefficients but share the same exponent, they live on roughly the same scale.
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This is estimation.
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It's error checking, it's scientific judgment.
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This is also why experienced problem solvers can seem so fast.
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It's not necessarily because they do the arithmetic more quickly, because they calculate, they estimate the order of the magnitude, and rule out answers that can't be right.
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Let's go back to Clapham Common.
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Franklin stands at the windward edge of a rough pond.
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He pours out less than a teaspoon of oil.
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The smooth patch begins near his feet and moves across the surface until a large section of the pond looks like glass.
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He sees the pond change.
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He records the oil in the area, but the smallest part of the experiment remains invisible.
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More than a century later, scientists recognize what those measurements contain.
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Volume divided by area gives thickness.
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Thickness gives a molecular scale.
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Scientific notation gives that scale a name.
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And once the scale has a name, you can move through it and compare it with everything around us.
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Name it, move it, compare it.
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That's the mathematical concept of this episode.
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The episode six story sheet is in study hall at storyproblem.org to practice these concepts.
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It follows the story from Franklin's teaspoon to the molecular layer and lays out the key scales so you can see how they connect.
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Then in a few days, come back for the companion office hours.
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We'll take the same numbers and work the mathematics step by step, writing scientific notation, multiplying, dividing, powers of 10, normalizing coefficients, and using exponent differences to compare scale.
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The main episode introduces the mathematics through the story.
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Office hours breaks down the problem solving steps, and study hall gives you a chance to practice them yourself.
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I'm Behi Rabani.
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This has been Story Problem.
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I'll see you next time.
