Sunday, May 16, 2010

Le Châtelier's principle: not just for chemists

I seem to be updating my blog once a month or so. I don't intend to do that, I'm just busy. This particular entry is one I've been meaning to write for months, but just haven't gotten around to.

But before I get to it, a brief digression. My sister would like to get into a poker tournament that she needs to be voted in to. If you could take a moment and vote for Jennifer Tilly, it would be most appreciated. Thank you.

Now on to the main subject. One of my favorite principles of chemistry is Le Châtelier's principle. It has several forms, but the most general (though admittedly not perfectly accurate) is Any change in status quo prompts an opposing reaction in the responding system.

What does this mean? Well let's take a simple example. Suppose we have air in a chamber, and apply pressure to compress the chamber. From Le Châtelier's principle we expect it to push back harder. In fact it does. From the ideal gas law, applied in a simplistic way, once the volume is cut in half the pressure will double, and it will indeed be pushing back harder.

However this understates the true effect. It turns out that the act of compressing the chamber heats up the gas, increasing the temperature, and this causes it to push back even harder than it would have otherwise. This rise in temperature when you compress is called adiabatic heating. The corresponding decrease in temperature when you decompress a gas is used inside of your refrigerator or AC system to move heat from a cool place to a warmer one. Similarly when warm air rises the pressure drops, which causes it to cool down. This is why air on mountain tops is colder than air at sea level. (The drop is about 10C per km of height.)

Anyways, the fine details notwithstanding, what we find is that when you push harder on this simple system, it winds up pushing back harder. And you eventually find yourself at another equilibrium. Furthermore what works for simple systems, works for many more complicated systems.

The big question that I had as a kid in chemistry class was why. Why does this always work out? I never got a good answer from my teacher, and my dissatisfaction with "it works because it always works" answers was one of the reasons why I chose to go on in math instead.

Interestingly, many years later in an advanced math course I learned the trivially simple answer. Which requires essentially no math to understand!

Here is that answer.

A system is at equilibrium when all forces on it are balanced, and it can rest in that state indefinitely. For instance take a pencil and lay it flat on a table. It is at equilibrium there. There is also another equilibrium where it is balanced on its tip, but it is very hard to put it in that equilibrium.

Now not all equilibria are created equal. A stable equilibrium is one where any perturbation of the system will cause it to head back towards that equilibrium. An unstable equilibrium is one in which some perturbation exists that causes it to head away from that equilibrium. In the example of the pencil, laying flat on its side is a stable equilibrium, while balancing on its tip is unstable. The key fact to remember about unstable equilibria is that they have a tendency to not stick around. No real system is perfectly balanced, and the imperfection will grow over time until equilibrium disappears on its own. This is is why we run into lots of pencils lying on their sides, and none balanced on their tips.

Now what does this have to do with Le Châtelier's principle? Well if we run across a system that has settled down to the point that it has a status quo we can notice, that system is extremely likely to be at some sort of equilibrium. Based on the point I made above, we can be pretty sure that it is a stable equilibrium. But Le Châtelier's principle is just a description of what it means to be at a stable equilibrium, so Le Châtelier's principle must be true of our system!

Now I should note that there is a world of difference between a local equilibrium and a global one. The pencil that is flat on the table, would be at an equal equilibrium on a different side, as a small push demonstrates. And at a better equilibrium flat on the floor, as a strong enough push will demonstrate. A mixture of hydrogen and oxygen that is heated at little bit will heat up, which increases the pressure, which causes the container it is in to expand, which cools it down, in accord with Le Châtelier's principle. But heat the same mixture enough and a chemical reaction will begin that results in it becoming hotter still, rather than cooler. (These examples are why the overly general formulation I quoted at the top is not perfectly accurate.)

It is clear that this is a very general argument, and makes it clear that the principle has nothing really to do with chemistry per se. In fact it applies to any sort of equilibrium. In chemistry or not. On the whole I find the examples outside of chemistry to be more interesting.

A case in point appears in the classic economics paper Cars, Cholera, and Cows: The Management of Risk and Uncertainty. One of the key themes is that our risk-taking behavior as a society winds up in an equilibrium. If it is at an equilibrium, then we should expect that any change which reduces risk will cause a some sort of compensatory increase in risk. Which will undo some of the positive benefits of the change. An example from that paper is that seat belt laws increase seat belt usage. But people using seat belts feel safer, and therefore drive more aggressively, resulting in more accidents. The net result? It appears that drivers are safer, pedestrians are less safe, and benefits to society are less clear than a naive analysis would predict.

Over time I've learned that equilibria are extremely common, and therefore the "unexpected consequence" is more often something to try to anticipate than something to be surprised at. For instance when a faster variation of cheetahs is bred by evolution, it creates pressure for antelope to become some combination of faster, better at spotting predators, and willing to bolt when predators are farther away. The net result is that the more effective predators wind up about equal versus their prey. (Look up the Red Queen Hypothesis for more on this.)

But how do you anticipate the unexpected? Well there are several ways.

  1. If you know that it is common to find some sort of push-back, you know to be on the lookout for it, which will make it easier to spot. For instance suppose that you start injecting a stable compound in at a constant rate. Eventually it has to break down at the same rate, the only question is where. It was not until Sherry Rowland and Mario Molina applied this line of reasoning to CFCs that it was realized that one of the most innocuous and inert chemicals discovered by man was destroying the ozone layer.

  2. There will be a set of related push-back phenomena associated with any stable equilibrium that you can find. However equilibria are very common. So look for potential equilibria, then actively ask how they are maintained. If you find them, you frequently learn something useful. For instance suppose there is an equilibrium level of major disasters in given area of human endeavor. By what means is this level maintained? I submit that it is maintained by memories of previous disaster, and desire to not experience that again. Which means that once memory fails and people become less careful, corners will be cut until disaster happens again. However memory fails on a time scale set by human lives, which tells me that an equilibrium rate for major disasters of any particular kind is never going to be more than a small number of human generations, no matter what the engineers promise us. For example it took just over 60 years to lose the regulations that were put in place to prevent another credit crisis like the one that started the Great Depression (and about 10 more years after that to experience a credit crisis - note that before the Depression credit crises arrived on average about once every 10 years), nuclear options are now getting a boost from the fact that memories of Three Mile Island are now fading, and I am willing to bet serious money that the surprisingly good record of safety devices for offshore drilling helped result in dangerous shortcuts that were key to causing the recent BP disaster.

  3. Certain equilibria and their corresponding compensation mechanisms come up very frequently to explain otherwise puzzling events. My favorite example is that people like to maintain a positive self-impression. The result is that anything that challenges our good opinion of ourselves causes serious cognitive dissonance. People do the most amazing things to avoid this cognitive dissonance. For some of the negative results on people's ability to learn, see What you refuse to see, is your worst trap.

So, even if you're not a chemist, if you squint at the world in the right way you can see Le Châtelier's principle popping up in the most unexpected and interesting places.