Happenings – 2011 Jan 8

There is a disadvantage to going so long without a diary post: it takes a while to figure out what I had been doing in the meantime, because there’s so long a time period to be searched.

I had started to put out a happenings post on Christmas morning, but I got distracted. Yes, distracted by Christmas presents… but also by the fact that I wanted to work on regression, multicollinearity in particular.

On the other hand, that may have been the last time I touched regression during these holidays. No, the Mathematica notebook shows that I put it down at noon the day after Christmas.

Whatever mathematics I’ve done since then, was done by two of my less grown-up alter egos… the kid and the grad student.

The kid decided he wanted to play with transformation geometry – what are called frieze groups and wallpaper groups – and then on New Year’s Day he decided to play with chemical reaction engineering. Meanwhile, the grad student has continued working through Dummit & Foote’s “Abstract Algebra”.

Through it all, my grown-up has been on holiday. I’ve put a fair bit of time into my annual dead-tree letter, which I’ll start mailing to friends this weekend…. and, having thoroughly enjoyed the Harry Dresden books by Jim Butcher, I thought I would try his sword and sorcery Codex Alera series. As is my habit, I bought the first volume… Barnes and Noble was still open when I finished it, so I went and bought the next four books. I finished the fourth volume last night. This is quite a distraction.

Today I’ll try to get back in the saddle and work on a technical post for Monday. Actually, this post itself has gotten me saddled up.)

Oh, speaking of Christmas… I installed Mathematica version 8 Christmas morning. I already know that it includes wavelets, and some control theory tools, but I haven’t played with them yet. Aside from verifying that I can now run either version 7 or version 8 whenever I choose, I am still running version 7.

For control theory in particular, because I already own “control system professional”, I will want to compare and contrast the capabilities of version 8 and the independent package.

For wavelets, I will want to see how much of what I’ve done out here can be done directly now… and how many more kinds of wavelets I have ready access to.

Version 6 was significantly not backward-compatible with version 5, and I have a whole lot of notebooks that will not run in version 6 or later, and I just don’t have the energy to convert them. In particular, I have version 5 notebooks about the wallpaper groups that I have never taken the time to try rewriting. I have no reason to believe that version 8 will be so destructive, but I’ll wait and see. I’m not about to risk the current generation of notebooks casually on a new version.

“Fool me once, shame on you. Fool me twice, shame on me.”

Let me say a little about chemical reaction engineering. FYI, I own 4 books: Levenspiel, “Chemical Reaction Engineering”, 2nd edition; Hill, “Chemical Engineering Kinetics and Reactor Design”; Fogler, “Elements of Chemical Reaction Engineering”; and Aris, “Elements of Chemical Reactor Analysis”.

Right now I’m interested in two specific topics: the experimental determination of rate equations, and the determination of mechanisms that account for a given rate equation.

The challenge is that there are a wide variety of rate equations. For the experimental determination of a rate equation, we need to determine the form of the equation to be fitted. This is in sharp contrast to what I’ve been doing in regression, trying to find the best variables for a linear model. For the rate equation, we know the variables but we don’t know the function. (Ultimately, of course, that is an issue in all regression analysis: should we be fitting a nonlinear model instead of a linear one?)

Then, having settled on an experimentally determined rate equation, can we break down the overall reaction into partial reactions with simpler rate equations that lead to the experimental one.

A quick example. Suppose the observed reaction is

2 A –> P + Q

and that the experimentally determined rate equation for the production of Q is

r_Q = \frac{K1\ [A]^2}{1+K2\ [A]}

where [X] denotes the concentration of X, and K1 and K2 are, now, known constants.

A possible mechanism assumes that a short-lived intermediate M (frequently a more energetic form of A) is created. That is, we hypothesize that we have three reactions… each with the simplest possible rate law for the formation or destruction of M:

Then we can show that

r_Q = \frac{k1\ k3\ [A]^2}{k3 + k2\ [A]}

and that is of the form

r_Q = \frac{K1\ [A]^2}{1+K2\ [A]}

(Yes, distinguish upper-case K1 and K2 from lower-case k1, k2, k3. To see it, start by dividing by k3.)

We have, therefore, found a possible mechanism in terms of elementary reactions which leads to the experimentally determined rate equation for the observed reaction.

(By the way, reaction mechanisms are not unique.)

This looks like fun.


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