Today’s class presented two formalisms that organic chemists use for connecting rates, rate constants, and the shapes of PE surfaces:

• Arrhenius equation (old-school but still useful and still widely used; go here for more background on collisions & kinetics)
• transition state theory (more obvious connection to thermodynamic quantities, but web pages say this is not useful when barriers get very small)

During class I tried to show what the maximum rate constants for a unimolecular and bimolecular reaction could be by assuming that the barrier had vanished (ΔH* = 0) and the entropy change was either negligible (unimolecular) or unavoidable (bimolecular). For the latter case, I invoked ΔS* = -30 eu = -30 cal/K-mol = -126 J/K-mol. This led to a maximum bimolecular rate constant of approx 2E+6.

After class two questions were put to me that I will rephrase slightly:

• isn’t the “diffusion-controlled rate constant” the same thing as the maximum bimolecular rate constant?
• isn’t the diffusion-controlled rate constant a bit bigger than 2E+6?

My understanding of the diffusion-controlled rate constant is that it is the maximum rate that can be expected for a bimolecular process. The name, diffusion-controlled, comes from the idea that every meeting of reactants produces an immediate reaction so the only thing limiting the reaction rate is the speed at which reactants diffuse through the medium to meet each other. Wikipedia’s entry says that this rate constant is about 1E+10. I also googled for papers that mention this concept and found a recent paper that quoted a similar value for the rate constant in water:

Direct spectrophotometric observation of this reaction showed it to occur in tens of microseconds with a few micromolar •NO, consistent with a rate constant that is a factor of only 2–3 lower than the diffusion-controlled limit in water (which is ∼7E+9 M-1 s-1). (“Thiyl radicals react with nitric oxide to form S-nitrosothiols with rate constants near the diffusion-controlled limit,” E. Madej et al., DOI 10.1016/j.freeradbiomed.2008.02.01)

So why was my estimate in class 3,000-5,000 times lower than the actual value? I think much of the discrepancy can be attributed to the entropy change that I chose, ΔS* = -30 eu. This is a fairly typical value for organic bimolecular reactions, but it might not be a good choice for estimating top diffusion rates. If I change ΔS* to -15 eu (which would be unusually small for most organic bimolecular reactions), my rate constant would become 3E+9 which is similar to the diffusion-controlled constant. Another possibility is that the kT/h factor from transition state theory (TST) is inaccurate for reactions that have no barriers. I don’t know enough about TST to know if this is a reasonable criticism, but the Wikipedia entry on TST clearly says that it is not a useful theory for predicting rate constants when barriers are exceedingly small.