# The BIG Deal

## How big are we talking about? How many molecules?

Well, a lot – think in terms of 6 x 1023 – that’s Avogadro’s number, or about how many hydrogen nuclei there are in one gram. That number is a six followed by 23 zeros, or 600 000 000 000 000 000 000 000.

It’s hard to understand just how many this is. But to give you an idea, there are about six billion, or 6x109, people on the planet. There are about 150x109 meters between Earth and the sun. In the Milky Way Galaxy, there are fewer than one trillion stars. And the gross domestic product of the world was about \$5.4x1013 in 2007. All of these numbers are not anywhere close to Avogadro’s number. If you were to count every grain of sand on the planet, you might start to get close (but by the time you finished counting, there’d be even more sand, and you’d have to start all over!).

 # hydrogen nuclei in one gram 600 000 000 000 000 000 000 000 # \$ changing hands in the world 2007 54 000 000 000 000 Distance from Earth to sun (meters) 150 000 000 000 Number of humans on Earth 6 000 000 000

The point is that these quantities – temperature, pressure, compressibility – are kind of taking an average of the behavior of this very large system.

You can find the average height of people in your classroom. If there are only two or three people in the class, this average height doesn’t mean a lot because there is probably a lot of variation about the average. On the other hand, if you have a really big class, say fifty, the average starts to be a more useful quantity. But there are always variations about the average – some people who are a little taller or shorter than average.

Same deal with pressure and temperature. For example, in a gas, particles are always whizzing around and bumping into each other, into walls, and so forth. Temperature tells us something about how fast the average particle is travelling at any given time. Actually, not all the particles are travelling the average velocity, just like not everyone in your class is of average height. But the average is still a useful number.

It turns out that, under normal circumstances, the variation about the average gets smaller as the number of molecules gets bigger. So if the number of molecules is really big (like 6 x 1023), the variation is really small.

But what if there were something that would be totally changed by even tiny fluctuations? Say, for example, you are cooking some really hard-to-make dish, like sugar candy. A small change in temperature can have a really big effect on what the finished product looks like.

What’s the analogy for critical phenomena? The response to a little variation from the average is huge at the critical point. We already saw this from the compressibility – the response of volume to a small change in pressure. So little variations about the average become really big.

## What does this have to do with critical opalescence?

Remember that little changes inside an otherwise uniform medium can scatter light? And remember that we were able to see steam bubbles in boiling liquid water because these two phases have different densities, and light bounces off the boundary between bubble and liquid?

That’s what happens at critical opalescence. There are always lots of very small variations (fluctuations) about the average quantities like temperature, pressure, and so on. These are usually too small to detect – too small to scatter light. But around the critical point, response functions like compressibility get really huge – this acts like a magnifying glass for fluctuations. So right around the critical point, really big fluctuations in density appear – big enough to scatter light.

It was Albert Einstein and Marian Smoluchowski who made this connection around 1910.

## Why’s this important? It’s just a dirty-looking gassy-liquid thing?

For a couple of reasons. First, around the time when Smoluchowski and Einstein were working on this problem, not everyone was convinced there existed such a thing as a molecule. After all, you can’t see a single air or water molecule with the human eye, can’t feel it with your hands. Materials usually don’t look like they come in little bits and pieces – they look like they fill up a space completely – that they’re continuous.

But phenomena like critical opalescence make a lot more sense in the molecular picture of the world than the continuous one. And so the study of critical opalescence helps to explain something very fundamental about the nature of matter.

It’s hard to appreciate this in the twenty-first century because we hear words like “atom” and “molecule” all the time. But this was a great (and very, very old!) debate in the early twentieth century – one of the reasons that “Einstein” is a household name is that he helped to resolve the question.

Critical opalescence is also important because it’s part of something much, much larger. It’s called “universality”, and we’ll get back to it later.

## Is this the end of the story?

Nope – the beginning. Fortunately, in science, every time you answer a question, you get a bunch more!

## Quizlet:

Under “normal” circumstances, temperature and pressure become more “exact” representations of a bunch of particles as

(a) the sensitivity of the thermometer or barometer is improved.

(b) the system approaches the critical point.

(c) the number of particles gets smaller.

(d) the number of particles gets bigger.

Critical opalescence

(a) occurs for first-order phase transitions

(b) occurs when a fluctuation gets repeated a long distance away, so that fluctuations are big enough to scatter light

(c) is an essential reduction of opal gems (essence of opals)‏

Light scatters at the critical temperature because

(a) Density variations become very large, and light travels through different densities at different speeds.

(b) The liquid water starts to boil or the gas to condense, and the steam bubbles or droplets are bigger than usual.

(c) The light photons hit each other and bounce everywhere.

(d) Einstein says so.