Thread by @dasingleton

Order, chaos, bah. This is chemistry, in fact it is my entropy and free energy lecture. We teach both so badly that most chemists, for God’s sake, understand neither. I use analogies like this, or a simple marble shaker, to show the ideas. Let’s go.

https://twitter.com/page_eco/status/1234047673100976128

The shaking rate is important, let’s name it T.
The average height of the mass of the nails seems important, let’s give it a name, I dunno, H. The H started high and went lower, but with greater T or lighter nails or less gravity it could have gone up.

But something a little more subtle is also important, and that is the number of ways the nails can be arranged. A low H limits the positions for the nails, so there are fewer ways they can be arranged. At a high H there are more. Let’s call the number of ways, maybe, omega.

Now to figure out what the nails are going to do we need to allow for the H and the T and the omega. Ignoring omega for now, gravity favors a lower H, but with a little shaking we can get a higher H, disfavored by some factor that increases with H. If we go up in H again, gravity

again disfavors it by the same factor. The repeat of factor vs H makes the relationship an exponential, so we can suggest a relationship between the ratio at high versus low H, call it k, and the shaking T that looks like k=exp(-H/RT) where I have thrown in a constant and called

it R (I might have called it k also, but that would be confusing). Think about this equation and you see it works perfectly. Lower shaking, k gets smaller. Double the H and the k (less than 1) should square.

Now I have to allow for the omega. If a given H has twice the omega,

then that H will be favored by a factor of 2. So I might change the equation to k = exp(-H/RT) times omega(high H)/omega(low H), but that is a bit unwieldy. I really want an equation where I can work with a single number that is conveniently additive and that go into a simple

exponential equation. No problem, I just have to define some new terms based on the ones I have, but that allow a simplified equation. To get my ratios of omegas into an exponential, I just need to take the log, and it won’t matter if I throw in an extra constant, so I’ll define

something as R ln omega and call it, oh, S. Now I can write my equation as k = exp(-H/RT) times exp(S/R) and it is just the same - the R is thrown in just to cancel out the previous R, and the exp just reversed the ln. I have really just allowed for the count of configurations,

the omega, in a fancy way.

So far so good, but we can make the equation simpler if we define a single term that combines the H and the S. Let’s use G for the new term, and define it as G = H - T S. The T and the minus are thrown in just to be canceled out later, to let us

write k = exp(-G/RT).

Of course this only works if the lowest H is assigned as 0, but we can compare any two using a deltaG.

Some notes: every time a student learns that entropy is “a measure of the disorder in a system” the world gets collectively dumber. I hate that useless

definition. It is an intuition killer. Entropy is a count of the possibilities, done in a fancy way that lets you work with the count of possibilities and allow for it easily.

Also, people will says that the video is backwards. I don’t know but it is irrelevant. At very high shaking you would get a high H and one of the very many disorders possibilities. At low shaking and a long time, you get a low H and one of the fewer ordered possibilities.

The equilibrium can be taken either way, whether it is nails or ice crystals, by changing the T (as shaking or temperature).