Effective Concentration
I find that the concept of “effective concentration” helps to clarify my thinking about molecular processes that include catalysis, self assembly, and mechanosynthesis. The concept applies most directly to reaction rates, and it uses ordinary, solution-phase processes as a reference point.
Reactant concentration
and reaction rate
In a relevant and typical case, molecules of type A react with molecules of type B to produce molecules of type C, and the reaction rate is proportional to the concentration of both A and B:
d[C]/dt = k[A][B],
where k is a rate constant that depends on the reaction conditions (solvent, temperature, pressure…); k depends both on the mobility of the molecules and on the probability that they will react when they encounter one another. From a microscopic point of view, a single molecule of A encounters molecules of B at an average frequency that is proportional to the concentration of B: When twice as many B molecules are moving about, any single molecule of A will encounter a B twice as often. This situation is the reference point for the concept of effective concentration.
Reaction rate and effective concentration
What if a physically similar reaction occurs, not between two separate molecules, but between parts A and B of one molecule, or (equivalently) between parts of two molecules that are bound together? This mechanical constraint will affect the frequency with which A encounters B, and thereby affect the rate of the reaction. “Effective concentration” now becomes an applicable concept. The IUPAC definition is
The ratio of the first-order rate constant of an intramolecular reaction involving two functional groups within the same molecular entity to the second-order rate constant of an analogous intermolecular elementary reaction. This ratio has the dimension of concentration.
From a microscopic point of view, and again considering the point of view of a single molecule (or reactive functional group) A, the effective concentration of a neighboring bound B is the concentration of unbound Bs that would be required to have the same effect on A, that is, to result in the same reaction rate. Roughly speaking, this means the concentration that would result in the same encounter rate.
Why it matters
Higher values of effective concentration result, by definition, in higher rates of reaction; the achievable values of effective concentration are relevant both to the simplest kind of catalysis and to the simplest kind of mechanosynthesis (the line between the two is blurry, of course). The concept of effective concentration can also be helpful in thinking about the kinetics of cooperative binding in self assembly, in which partial binding of two structures can align the remaining parts, thereby increasing their mutual effective concentrations.
It’s easy to understand how the effective concentration of a bound, neighboring B, as seen by A, could be “100%”, meaning equivalent to immersion in Bs. To give a numerical example of this, the concentration of water in pure water is about 55 M (mole/liter)*.
In a follow-on post, I’ll explain why effective concentrations can easily be >55,000 M. [Here’s part 2.]
* In molecular units, about 33 nm–3.
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Mr Drexler, this leads directly to a question I have had for some time, regarding self-assembly and mechanosynthesis. What would you estimate is the best, peak, and most advanced SELF Assembly system we could build, that takes advantage of complimentary features such as charge and chemical bonding, without actually building molecular level tools to force-bond atoms? Ie, could we have a purely self-assembly method that makes say a foot of product, with atomic precision, or would that require mechanochemistry?
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