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B.  From ________, we can find the size

Molecules of gases, unlike those of solids and liquids, were thought to take up very little of the space the gas occupied, and to zoom about freely all the time except when colliding with each other or with the walls of their container.  A set of particles suspended in a fluid is very similar to this in at least two ways.

First, although the particles might be in frequent or constant contact with the fluid molecules instead of flying through almost-empty space like gas molecules, they seldom come in contact with each other, so among themselves they do behave as if they were gas molecules.  And second, if the fluid is composed of molecules in constant motion, those molecules should constantly bounce the suspended particles around, just as the fluid molecules bounce each other around.  This would put the suspended particles in constant random motion themselves, again like the molecules of a gas.

Einstein further discovered that, if the suspended particles are big enough to see with a microscope, their random motion should be visible too, as the fluid molecules, with their small invisible motions, bounce them around.  In fact, such a visible motion had long been known, and had been explored by the botanist Robert Brown.  Various causes for this motion had been supposed, including the hypothesis that molecules exist and constantly move.  When Einstein first discovered how suspended particles could make the effects of molecules visible, he didn't know enough about Brownian motion to be sure it was the same thing, but as it turns out, it is.

In proving this, Einstein also found his main clue to the size of molecules:  how far the suspended particles move should depend on the number of molecules it takes to make one "mole".  Each time a fluid molecule bounced into a suspended particle, the particle would be moved a little, so after many bounces the particle might wind up in a quite different place.  Einstein found that, if one mole equals so many molecules, the suspended particles would wander, on average, so far in one minute.  If a mole only equals one fourth as many molecules, so that each fluid molecule is four times as massive, the fluid molecules would hit hard enough for the suspended particles to wander, overall, twice as far in one minute.  The basic relation is:

(typical distance moved by a suspended particle in a given time)²
= (some other quantities that we'll look at later) ÷ (number of molecules in one mole)

Our example fits this relation: if a mole only had one fourth as many molecules, suspended particles would only move twice as far: (2)² = 1 ÷ (¼ ). The relation would also work if it took more molecules instead of fewer to make one mole.  If molecules had only one millionth of the mass they actually do, a typical suspended particle would wander only one thousandth the distance it does: (1/1,000)² = 1 ÷ (1,000,000). And if matter could be divided into infinitely small pieces-in other words, if there were no such thing as a smallest-possible unit of matter, so that a "molecule" of fluid had zero mass and it took an infinite number of them to make one mole-the suspended particles would travel zero distance: (0) ² = 1 ÷ (infinity). In that case, the suspended particles would not be bounced around at all.

To find what a molecule's mass is-i.e., the number of molecules in one mole-we need to know those "other quantities" in the above equation.  The reasoning Einstein used to determine them is interesting to follow.

Einstein showed that the suspended particles, being randomly bounced around by the fluid molecules, would diffuse throughout the fluid in accordance with an equation long known to mathematical physicists.  This equation implies a simpler one, which relates the average change in the particles' positions to the time they spend being bounced around:

average of [(change in position along any one direction)²] = 2 x (time) x D,

where D is a quantity that characterizes the diffusion rate, and depends on the nature of the fluid and the particles suspended in it.  This equation describes positional change along just one direction; the average square of the change for all the directions of three-dimensional space is three times this amount.

It turns out that D is related to the number of molecules in one mole, and to certain other characteristics of the suspension ("other quantities") that are easy to determine by experiment.  To find how they are related, Einstein had to consider four different features of the fluid-particle suspension and how they themselves are related to these "other quantities" and to each other.  These latter relationships are described by a set of equations that can be combined like pieces of a jigsaw puzzle to find the relationship of D to the size of a mole.  The result of this assembly can be expressed like this:

D x (viscosity of the fluid) x (size of a suspended particle) x x (particles in one mole)
= (absolute temperature of the fluid) x R.

Aside from the number , everything in this equation has an obvious relation to our problem except perhaps for the quantity R.  R stands for a feature of dilute gases known from experiment.  It's what you get when you multiply the volume of the gas by its pressure, and divide that by the product of its temperature and the number of moles of gas you have.  As long as the gas is dilute enough, the number is practically the same no matter what kind of gas you have, or how much there is of it, or what its pressure, volume, and temperature are.  The quantity R turns up in this equation, because one the four features of the "gas" of suspended particles-the one, in fact, that's most directly related to the number of particles in a mole-is proportional to R.

To complete our jigsaw puzzle, we only have to combine this equation with the one just before it that involves D. After a bit more algebra, we finally have the relation between how far a suspended particle should move in a given time, and the number of particles in one mole:

average of [(change in position along any one direction)²]

If we consider

average of [(change in position along any one direction)²]

as a more precise way to describe

(typical distance moved by a suspended particle in a given time)²

we see that our last equation is just a more detailed version of the equation at the beginning of this article.The more detailed equation tells us that, to find how many particles there are in one mole of any substance (and thus the mass of one molecule of a substance), we need only measure how far the suspended particles move in a given time, the viscosity and temperature of the suspending fluid, and the size of the suspended particles. If we can make those measurements, we can determine the sizes of molecules even if we can't isolate and weigh them individually. (.....continued)

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