Why Do We ACTUALLY Believe Atoms Exist?

This video is sponsored by Squarespace, the Albert Einstein of website builders. I want to begin with a mystery. You take some grains of pollen, suspend them in water, and then place them under a microscope.

What you observe is something truly strange and remarkable. The pollen grains jitter and move and change direction rapidly as if by some mysterious force, almost as if they are alive. But now repeat the same experiment using finely ground inorganic matter like rock dust completely inert lifeless material and you see exactly the same motion.

Now we can all agree on one thing. Rocks are not alive. So what is causing this motion?

And how can something that looks so chaotic, so irregular, and so unpredictable be explained at all? This strange behavior was first observed in the early 19th century by the botonist Robert Brown. and it deeply unsettled scientists.

You see, the motion was real. It was reproducible, but it stubbornly resisted explanation. One could imagine all sorts of possible causes from tiny currents in the fluid to temperature irregularities.

But as experiments improved, these explanations fell away one by one. But there were clues. The motion became more exaggerated for smaller suspended particles compared to bigger ones.

And the amount of agitation depended on the temperature of the fluid. higher temperature, more agitation. And it was also found that the motion was more extreme for lower viscosity fluids.

Every one of these trends pointed towards the microscopic world. An idea began to emerge that the suspended particles were being buffeted by invisible molecules of the surrounding fluid. But nobody had yet been able to show how this works.

And to make matters even worse, there was still debate as to whether atoms even existed. For more than 2,000 years, the existence of atoms had been one of the deepest unanswered questions in science. The idea itself was ancient.

Greek thinkers like Lucifus and Democrus had argued that matter must be made of indivisible units referred to as atomos, meaning uncutable. Aristotle rejected the idea. And because Aristotle's influence dominated Western thought, atomism remained controversial for nearly two millennia.

Even in the 19th century, skeptics could still argue that atoms were nothing more than clever mathematical fictions. Useful perhaps, but not real. And then, rather unexpectedly, this jittering motion of tiny particles in a liquid offered a way to settle the debate.

In 1905, a young physicist named Albert Einstein realized that brownie in motion was not a nuisance to be explained away, but a clue. In a groundbreaking paper that would change the course of history, Einstein showed that the seemingly chaotic motion of a visible brownian particle provided a window onto the invisible atomic world. If atoms were real, then a pollen grain suspended in a liquid should be constantly struck by an enormous number of invisible molecules.

Each collision would be tiny and random, far too small to track individually. But in the wondering of the pollen grain itself, perhaps there was a pattern that reflected the properties of the fluid and its microscopic constituents. But there was a problem.

If you focus on any single pollen grain under a microscope, its motion looks purely random like noise. Every trajectory is different. No two paths ever repeat.

The exact zigzag of an individual grain is unpredictable in principle. And so Einstein stopped trying to predict individual paths. Instead, he asked a different question.

How do you describe random motion quantitatively? The key insight is that randomness does not mean lawlessness. Even when individual trajectories are unpredictable, collections of trajectories can obey precise statistical laws.

And it was these statistical laws that Einstein focused his attention on. To see how this works, imagine you're standing on a straight road. Every second you flip a coin.

Heads you take one step forwards. Tails you take one step backwards. At each step there is an equal probability of moving forwards or backwards.

And you keep going. You flip then step. Flip then step.

There is no strategy, no memory, no bias, just chance. This is known as a random walk. At first glance it seems almost too simple to take seriously.

But this simple model turns out to capture the statistical essence of Brownian motion. To see why this works, consider the pollen grain again jittering randomly in two dimensions. Every tiny movement of a Brownian particle can be split into an X step and a Yep.

Those components behave independently. So when we replace the pollen grain with a tiny stickman that takes random steps left or right in the X direction, we are not changing the physics. we are isolating just one component of it.

If we can understand the statistics of this one-dimensional walk, the full brownie in motion follows automatically. That is why Einstein started here. So, let's focus on the one-dimensional walking model in the x direction and ask a seemingly simple question.

After taking 200 random steps, where are you? Now, if you think about it, you could be 200 steps forward or 200 steps backward, but that would be extraordinarily unlikely. Those outcomes have the same probability as throwing 200 heads in a row or 200 tails in a row.

Intuition might suggest something else. Since forward and backward steps are equally likely, perhaps you're most likely to end up right where you started. But intuition alone isn't enough.

To find out what really happens, we need to repeat the experiment many times. So instead of one walker, imagine many walkers all starting from the same point, all following the same rules, but each making their own independent random choices. Some drift to the right, some drift to the left.

We want to understand what happens to large collections of walkers on average. Perhaps if we consider such large ensembles, some sort of order may emerge from the seaman chaos. In order to do this, we're going to update our previous question and ask, if we have 500 walkers each taking 200 random steps, what does the average outcome look like?

To answer this question, we first realize that we can track each walker's displacement as a function of the number of steps taken. Each walker traces out its own displacement curve. For example, if we consider 10 walkers, we can already see a pattern emerging.

Each walker traces out a different path. Some finish with positive displacements ending to the right of where they started, and others finish with negative displacements ending to the left. So, let's now increase this to 500 walkers.

We clearly see that as the number of steps increases, the spread of displacements increases as well. So, is there a systematic way to understand this spreading? To do that, let's take a different perspective.

Instead of following individual walkers, imagine dividing the x-axis into a set of regions or bins. At any moment, we simply count how many walkers lie in each bin. Doing this gives us what's known as a histogram of displacements.

Now, let's run the simulation again, keeping a tally of how many walkers occupy each bin. Two features stand out immediately. First, the distribution is symmetric about the starting point, reflecting the fact that forward and backward steps are equally likely.

Second, as time goes on, the distribution spreads out. The histogram becomes broader. And remarkably, at any stage, it's well described by a smooth bell-shaped Gaussian curve centered on zero displacement.

This is telling us that if we were to average all of the displacements of all of the walkers, the result we would get would be zero. Mathematically we express this by saying that the expectation value of position is zero. However, we note that what does change is the width of the distribution.

If we reduce the number of steps, the spread reduces and the Gaussian becomes narrower. If we increase the number of steps, the walkers spread further and the Gaussian broadens. And it is this spreading, not the average position, that contains the real physics.

Now, although Brownie in Motion is chaotic by nature, building a website really shouldn't be. And that's where Squarespace comes in. Squarespace is an all-in-one platform for designing, building, and managing a website.

And it's what I'm currently using to build my own physics explained site. What's really stood out to me so far is how flexible the design tools are. You can start with Blueprint AI, answer a few questions about what you're building, and it generates a clean, professional layout almost instantly.

but you're never locked into it. You can tweak layouts, drop in animations, embed visuals, and structure pages around ideas, which is exactly what you want when building a website. Squarespace also makes it very easy to gate content.

You decide what's public, what's members only, and whether access is oneoff or recurring, all built directly into the site. And the built-in analytics are genuinely useful. You can see what people are engaging with, where they're coming from, and what content resonates.

If you want to try it yourself, head to squarespace. com/physicsexplained to save 10% off your first purchase of a website or domain using code physics explained. Einstein realized that if he wanted to understand how suspended particles like pollen spread through a fluid, he needed a quantity that measured not where the particles were on average, but how spread out they were.

The problem with using the mean displacement is that symmetry forces it to vanish. Forward and backward wandering cancel exactly. Our distribution in both the x and y directions are centered on zero.

Meaning that the centers of the distribution say fixed even when the particles themselves are constantly moving. To capture this spreading rather than focusing on the displacement which can be positive or negative, let's focus on the square of the displacement. Squaring removes the sign.

So leftward and rightward motion now contribute equally. For example, a displacement of x= -3 and x= + 3 would have the same square displacement of + 9. Okay, so let's square the displacements of every walker and average them.

If you do this, you find that the mean square displacement is proportional to the number of steps taken. In theory, this relationship is exact. The mean square displacement grows linearly with n.

In practice with a finite number of walkers, we see a line that fluctuates slightly around this ideal behavior exactly as shown in the rightmost plot for 500 walkers and 200 steps. And it was this quantity, not the individual paths, that Einstein focused his attention on. Now we make a key conceptual move.

If the random motion of a pollen grain can be thought of as a sequence of tiny uncorrelated steps, then the number of steps would simply be proportional to the amount of time the pollen had been suspended in the fluid. In other words, the number of steps n would be proportional to t. And since we've already established that the mean square displacement is proportional to the number of steps, it immediately implies that the mean square displacement grows linearly with time as well.

As a result, we can write the mean square displacement as some constant multiplied by the time t where the constant controls how quickly the spreading occurs. As we've just seen, if we plot the mean square displacement against time, we get a straight line and the constant represents the gradient of this line. Now look at the distribution itself.

As we saw earlier, the distribution along the x-axis had a classic Gaussian shape that emerges from many random walks. Now, as the Gaussian widens, the line gets steeper, signaling that the spreading occurred faster. So, what controls the rate of spreading?

Well, we can quantify the rate of spreading with a single parameter called the diffusion coefficient D. A larger D means the spreading occurs more quickly leading to a wider distribution and a smaller D means it spreads more slowly. And it turns out that the gradient and therefore constant takes a very specific form.

It is equal to 2D and that factor of two isn't a convention. It falls out of the Gaussian shape when you do the detailed analysis. So in one dimension we can write the result in its final form as the mean square displacement is equal to 2dt.

But Einstein wanted more than a descriptive law. He wanted to calculate d itself to express it in terms of ordinary measurable properties such as temperature, the viscosity of the fluid and the size of the suspended particles. The remarkable thing is that in doing this he would find a way of effectively counting the number of atoms that exist in one mole of a substance and he would provide a bridge between the macroscopic wandering of a Brownian particle and the microscopic world of atoms and molecules.

His route across that bridge was subtle and ingenious. First, as we've just outlined, he treated Brownie in motion as a diffusion process leading to the result just discussed. Next, instead of attacking Brownie in motion headon, Einstein began with a system whose behavior was already well understood, the ideal gas.

By the late 19th century, pioneers like Maxwell and Boltzman had shown that the macroscopic properties of gases could be explained if they were made of tiny molecules obeying Newton's laws. Einstein's aim was to borrow that insight and apply it to the jittering brownie in motion. The crucial step due to Yakabus Vanhoff was that a dilute solution exerts an osmotic pressure with the same mathematical form as the ideal gas law.

Einstein's bold move was to assume that this same osmotic pressure law applies to suspended Brownian particles. With that single assumption, he could treat a Brownian suspension thermodynamically like an ideal gas and connect its random wondering to measurable bulk properties. To do this, Einstein begins by considering an ideal gas in thermal equilibrium under a constant external force.

Now, it's not immediately obvious why you would introduce a force, but then not everyone is Einstein. As we'll see, the force is exactly what makes the problem solvable because equilibrium demands a specific density gradient to balance that force. Once that equilibrium condition is derived for a gas, Van Hoff's law carries it over to a dilute solution and then Einstein's bold leap is to apply it to suspended Brownian particles.

With that in mind, consider an ideal gas in thermal equilibrium at a uniform temperature T. We choose the x-axis to point to the right. Then following Einstein's strategy, we suppose that every molecule experiences the same constant external force directed to the left, i.

e. in the negative x direction. We label this force f external.

We also introduce the concept of a number density. If we consider a specific x location within the gas, we say that n of x represents the number density of the gas at location x. This is essentially a measure of the number of particles per unit volume at X.

Next, we consider a thin slab of gas between X and X + DX with cross-sectional area A. The volume of the slab can then be written as DV = A * DX where DX is the thickness of the slab. It then follows that the number of molecules contained within the slab dn will be equal to the number density multiplied by the volume.

Now as already mentioned Einstein's idea was to introduce an external force that acts on all the particles. It then follows that if each molecule experiences this same external force, then the total external force acting on the particles within the slab will be given by the following expression where DF external is the total force acting on the slab. Now in addition to the external force acting on the slab, the slab also experiences a force due to gas pressure.

This pressure is not caused by the external force. Rather, it arises from random thermal motion of molecules. To calculate this force along the x direction, we can focus on the force acting on each of the two faces of the slab located at x and x + dx.

The face of the slab at position x experiences a pressure force p of x * a in the positive x direction. where here P of X represents the gas pressure at location X. Whereas the face at X + DX experiences a pressure force P of X plus DX * A in the negative X direction.

The net pressure force on the slab due to thermal motion will then be given by the difference between these two forces. Next, we can factor out the cross-sectional area A. And then we note that for a sufficiently thin slab, the difference in pressure can be approximated by the gradient of the pressure at x multiplied by the width of the slab.

If we then sub this back into our slab pressure equation, we find the following result. If we then rearrange the term slightly and then refer back to our diagram, we see that the product a * dx is simply the volume of the slab dv. And so it follows that we can write df pressure is equal to minus dp by dx multiplied by dv.

And this is our approximation for the force acting on the slab due to gas pressure resulting from thermal motion of the gas molecules. And then if we recall the expression we derived for the external force acting on the slab we see that there are two forces. Now since the gas is in mechanical equilibrium and the slab does not accelerate as a whole, it then follows that the net force on the slab must be zero.

In other words, we require that df pressure plus df external is equal to zero. If we then sub in the expressions that we've derived, we find the following relation. Then since dv is not equal to zero, we can divide both sides by dv and we find that the rate of change of pressure in the x direction is equal to the number density multiplied by the external force.

This expression is essentially telling us the pressure gradient in the gas. How pressure changes with x position as a function of the number density and the external force. Now it will prove useful for later to recast this expression in terms of the mass density rather than the number density.

The key is to note that n of x represents the molecules per unit volume. Whereas row of x represents the mass per unit volume. To convert between them, we first note that the number of moles of gas per unit volume will be equal to the mass density divided by the molar mass.

And if we then multiply this by Avagadro's number na, which represents the number of molecules in a mole, then the result is the number density, namely the number of molecules per unit volume. And if we sub this expression for n of x back into our pressure gradient equation, we find the following result. Okay, so to summarize, all we've done is change variables from number density to mass density on the right hand side of the equation.

Our next task is to find a way of changing the derivative on the left hand side from pressure to density. To do that, we make use of the well-known and much loved ideal gas equation PV= NRT, where N is the number of moles within the gas and R is the molar gas constant. Rearranging for pressure, we find the following expression.

And we can then use the fact that the mass density is defined as the mass per unit volume. And this is simply equal to the number of moles multiplied by the molar mass divided by the volume. If we rearrange this, we find that the number of moles divided by the volume is equal to the mass density divided by the molar mass.

And so we can now sub this into our pressure relation and we find row / multiplied RT. So we now have the pressure P expressed in terms of the mass density row. If we then take the pressure equation that we've just derived and differentiate the pressure with respect to x, we find the following result.

And so we see that we now have a way of expressing the pressure gradient in terms of a mass density gradient. And then if we sub this result into our pressure gradient equation, we find the following expression. And after rearranging, we end up with the following result.

And this is what we were aiming for. a differential equation expressed in terms of the mass density. Now, it's important to understand what this equation represents.

It describes the mass density gradient that must exist in thermal equilibrium when a constant external force acts on a gas. And we see that if we increase the force, the concentration gradient shifts with more particles bunched on the left. If we decrease the force, the particles spread further out, but there is still a gradient.

The situation is directly analogous to the Earth's atmosphere under the influence of gravity where the force of gravity causes a density and pressure gradient in the atmosphere with higher density near the Earth's surface and lower density as the altitude increases. And this is the point at which Einstein makes the crucial switch. According to Vanto's law, a dilute solution behaves thermodynamically as if the solutes were an ideal gas.

And Einstein then made the bold claim that the same should be true for dilute brownian suspensions such as pollen suspended in water. And so from this point on, row of x should be understood as the mass density of the suspended particles and m their molar mass. With that change in interpretation, we don't need to redo the algebra.

According to Einstein's reasoning, the same equilibrium relation carries over directly, provided the external force acts only on the suspended particles. In that way, we expect a similar equilibrium condition involving the suspended Brownian particles in the presence of an external force. This is the key result Einstein needs.

A constant external force does not drive steady flow in equilibrium. Instead, it produces a static concentration gradient of suspended particles. Einstein's next move is to ask an important question that will ultimately unlock everything and allow him to link microscopic and macroscopic quantities.

The question is, how could such an equilibrium be maintained dynamically? To understand what this question means and how to answer it, let's consider a simulated experiment. Imagine that we have a system of brownian suspended particles undergoing random walk motion.

And to be clear, I've massively sped up the simulation so that we can observe change easily. Then imagine that a constant magnitude external force is suddenly applied just as before. We see that as a result of the external force, there is a flow of Brownian particles in the direction of the force and we see that this leads to a concentration gradient.

Then if you switch off the force completely, we see that the flow of particles is now in the opposite direction in the form of a diffusion current flowing from high to low concentration. Now in the steady equilibrium state, these two effects act simultaneously. Namely, we have a drift current caused by the external force which we label J drift and we have a diffusion current from high mass density to low mass density which we label J diff.

And here is the crucial point. In equilibrium, these two currents must cancel. Otherwise, the concentration profile would not remain stationary.

In other words, we require that J drift plus J diff is equal to zero. This is Einstein's dynamical equilibrium condition. So, let's now work out the consequences of this condition.

So let's again consider a suspended particle where here I've represented the surrounding fluid with a light blue coloration. Now imagine switching on an external force. The suspended particle initially begins to accelerate.

But because it's moving through a viscous fluid, a drag force immediately acts in the opposite direction. This drag force grows with speed and almost immediately balances the applied external force. So the particle reaches a constant drift velocity with Brownian jitter superposed on top.

This fact allows us to calculate the drift velocity directly using Stokes's law. Consider a spherical particle of radius A suspended in a fluid of viscosity eater. Then imagine that an external force is applied to the particle.

It quickly speeds up until the drag force exactly balances the external force. The equation for the drag force is provided by Stoke's law which says that the drag force is equal to 6 pi EA ava is the fluid viscosity, a is the size of the particle and v is the speed of the particle. Now at terminal drift speed the applied external force is exactly balanced by the viscous drag.

And so we can write that the external force is equal to the drag force. And if we then sub in Stoke's law and rearrange, we find an expression for the drift velocity in terms of the external force. We can then use this drift velocity equation to estimate the drift current of the suspended particles.

The drift current is defined as the number of particles crossing a unit area per unit time in the direction of the applied force. It is simply the product of the number density and the drift velocity. And if we then use the expression for the number density in terms of the mass density that we derived earlier together with the drift velocity equation and sub these in, we end up with the following compact expression for the drift current due to the external force.

This expression represents the steady flow of suspended particles driven by the external force. more specifically the number of particles that cross a unit area each second as a result of the applied force. And this equation makes intuitive sense.

First, we see that the drift current is proportional to the applied external force. If you double the force, you double the drift current. Second, we see that reducing the viscosity or reducing the size of the particle increases the drift current as one might expect.

At the same time, as we've already seen, a mass concentration gradient drives diffusion in the opposite direction. Random thermal motion causes particles to spread out. And this diffusive flow is described by fix law, which can be written as J diff= minus d * dn by dx where n is the number density and d is the very same diffusion coefficient that we encountered earlier.

This equation is simply saying that the diffusion current is proportional to the number density gradient. How steeply the concentration of particles changes as you move along the x-axis. The minus sign indicates that diffusion always acts to transport particles from regions of higher concentration to lower concentration.

Now just as before we can write the number density in terms of the mass density using the fact that n= row * n a / m. And then if we take the derivative of n with respect to x, we get the following expression relating the number density gradient to the mass density gradient. And if we then sub this into the diffusion current equation, we find the following result.

And so we now have expressions for both the diffusion current that we've just arrived and the drift current caused by the external force. And in equilibrium, the sum of these two currents must be zero. If we then sub in the expressions for the currents, we find the following condition.

Next, note that we can factor out n a / m. And so we see that in order for there to be no net current, the term inside the bracket must vanish. And so we end up with the following equation.

And if we rearrange slightly, we see that we now have a second mass density differential equation. So let's pause a moment and take stock. We have just derived a mass density differential equation by balancing the drift current due to the external force and the drag current.

If you recall, we earlier derived another mass density differential equation by considering the forces acting on a slab of gas and applying Vanto's law. So we now have two equations describing how the mass density changes as a function of position in the x direction but each contains a different set of variables. So if we now set these two equations equal to each other we see that the external force that we introduced and that lay at the heart of Einstein's entire approach as if by magic cancels out.

And if we then rearrange and simplify we find a truly remarkable and historic result. first obtained by Albert Einstein himself in 1905. We have managed to derive an expression for the diffusion coefficient expressed purely in terms of measurable quantities such as temperature, viscosity, and the size of the suspended particle.

And the genius of Einstein's approach was that he only introduced the external force as a clever trick. It was never meant to be measured. Rather, it helped carry the argument until it was no longer needed.

and it gracefully bowed out. And so we can now finally bring all of the pieces of the puzzle together. If you recall, we found earlier that the mean square displacement of our brownian particle was equal to 2dt where d was precisely the same diffusion coefficient that we've just been discussing.

And so if we now combine our two results, we find the following beautiful equation. And this is the wow moment because every symbol in this equation is something that can be measured with ordinary laboratory equipment. For example, you can track a particle under a microscope and plot the mean square displacement against time.

You can measure the temperature T. You can measure the viscosity EA and you can measure the particle radius A. And the molar gas constant is already known from macroscopic thermodynamics.

And this means there is only one unknown left. Na. And Na is not some adjustable parameter.

It's the number of atoms or molecules in one mole of a substance. The conversion factor between the human scale world and the hidden microscopic world of atoms and molecules. What Einstein had done was extraordinary.

He had effectively found a way to count the invisible by watching something visible. The decisive experimental confirmation came a few years later through the work of Jean Baptist Perin in an article titled Brownian Movement and Molecular Reality. Perne's key achievement was to take Einstein's prediction out of the realm of plausibility and turn it into a precision measurement of Avagadro's constant.

Beginning in 1908, he worked with carefully prepared dilute suspensions of nearly spherical resin particles in liquids whose viscosity was known. First, he tracked the translational brownie in motion itself. He would record a particle's position, wait a fixed time interval, and then record it again and repeat this many times across many particles.

He then calculated the mean square displacement and showed that it grew linearly with time just as Einstein had predicted. Using his results and Einstein's brownie in motion equation he was able to estimate Avagadro's constant obtaining values of around 6 * 10 23 exceptionally close to the current day measured value. In concluding his 1909 article, Perin wrote, "I think it will henceforth be difficult to defend by rational arguments a hostile attitude to the molecular hypothesis.

" Perin would go on to win the Nobel Prize in physics in 1926 for his work on the discontinuous structure of matter. Einstein, on the other hand, was just getting started. Einstein's work on Brownie in motion was just one of four groundbreaking papers that he published in 1905 that would change the course of history.

In the next video in this series, we will look at the often misrepresented work that ultimately led Einstein to being awarded the Nobel Prize, the idea that light is in fact comprised of particles. I think it only appropriate to end with the wise words of Einstein himself. There are two ways to live.

You can live as if nothing is a miracle. You can live as if everything is a miracle. So, here's to living as if everything is a miracle.

And on that note, thank you for listening. And until next time, goodbye. And as always, a massive thank you to all my patrons.

And a special shout out to the following who have been incredibly generous with their support. Thank you so much. I couldn't do it without you.