Thank you very much, that textbook reference was exactly what I was looking for. I had no doubt the derivation existed, but for some bizarre reasons those books on kinetics that are on my shelf right now only treat the irreversible case.

Thanks for the 1984 paper, too. I do have a few papers by Elizabeth Boeker from that era in my collection, but this one somehow slipped out.   Incidentally attached here is a more (typographically) compact form of the same integrated rate equation.

Thanks again...

To my opinion, as your equation it is not found directly in textbooks, you still may write a pertinent contribution to the Journal of Chemical Education provided that you include illustrative examples of application.

Dr. Van der Zwan,

I think I can use a tiny bit more help from you, if you please don't mind.  You stated that "the equation [I have derived] is given in the book by J.E. House, Principles of Chemical Kinetics, p59ff."   So today I purchased the book (as an "Amazon Kindle" electronic edition) and I started looking for what would be the equivalent of page 59, where (according to your email) "the equation" is supposed to be listed.

But unfortunately I cannot find it. I am currently looking at the book's second edition, 2007.  Here is what I see:

Chapter 1, "Fundamental Concepts"

Section 1.2 "Dependence of Rates on Concentrations"

• 1.2.1 First order
• 1.2.2 Second order ... this shows "A + A ---> B" only: d[A]/dt = -k[A]2
• 1.2.3 Zero order
• 1.2.4 Nth order

In any case Chapter 1 discusses only irreversible reactions (whereas the situation I interested in discussing is the reversible case).

 So then we move to ...

Chapter 2, "Kinetics of More Complex Systems"

• 2.1 Second order: "A + B ---> C", irreversible
• 2.2 Third order (not relevant)
• 2.3 Parallel reactions (not relevant)
• ...
• 2.6 Reversible reactions
• 2.7 Auto-catalysis
• ...

In section 2.6 "Reversible reactions", which sounds promising, the only situation that is discussed is the uni-molecular reversible mechanism, "A B".

Am I looking in the wrong place?  Is the second edition different from the first edition?  Is the Kindle edition different from the print edition?  Or was my original question posed so confusingly that it was not at all clear I am discussing the reversible, bi-molecular case "A + B C"?

Thanks in advance for any clarification you may be able to offer, Dr. Van der Zwarn.

I am still hunting for that textbook reference to the integrated rate equation for reversible bi-molecular kinetics.  Just found a 1981 paper from a Hungarian journal (see attached) referring to a 1960 book by Benson:

• S. W. Benson: The Foundations of Chemical Kinetics. McGraw Hill, New York, 1960.

Unfortunately the book is out of print and the nearest research library available to me no longer has paper books, only electronic.  Anybody happens to have the Benson book easily available, and if so could you post the equation with a page reference?

Thanks!

Dear Gert,

thank you for the explanation, I figured something like that happened. 

Incidentally I do not regret getting the House book at all (Kindle edition on sale for USD $23, so not too bad) . At least I now have additional "evidence" documenting that the classical, essentially algebraic models for chemical and biochemical kinetics are grossly deficient (as a body of knowledge and a general approach, not the rate equations considered individually, which are of course perfectly fine).   The principal disadvantage of the algebraic approach is that only very few special cases can be handled by closed-form (integrated) rate equations.

In contrast, the differential equation formalism (requiring a numerical, iterative solution, of course) can be used successfully to handle any and all reaction mechanisms, using the same general approach based solely on (a) the mass action law and (b) the mass conservation law.   With these two theoretical and well understood principles in mind, it is truly trivial to compose a system of differential equations for any conceivable reaction scheme, no matter how simple or how complicated.

In fact, the compilation of those differential equations is so trivial that it can be entirely automated and performed by the computer.  This is what happens for example in the software program "DynaFit" I have been developing for almost a couple of decades now.  The program is available free of charge to all academic researchers (students, faculty, staff, post-docs, anybody with a .EDU email or international equivalent):

• http://www.biokin.com/dynafit/

In DynaFit, we would describe the reversible bimolecular equilibrium as follows:

[mechanism]

   A + B C    :    k(on)     k(off)

And the machine will do the rest.  Now someone might be asking why even bother to get algebraic (integrated) rate equations since it is so easy to devise a "ODE compiler" based on symbolic input, like what I am showing above.  Well, in my particular case it is probably "just because": I like the challenge of doing some mathematics by hand, from time to time, like a crossword puzzle.   But also, it is true that the algebraic models (in the few special cases where they do exist) can be set up in any software system, or even just calculated "by hand".

In any case, I invite everyone to take a peek at DynaFit.  By now it has been cited more than 900 times in the biochemical and chemical kinetics literature, so you'd be in good company.

Thanks again,

Petr

Gert, thanks very much for the offer to copy pages from the book, but in the meantime I got a copy of it from a used-book vendor.   The bottom line is the Nelson book does not show the equation I had to derive, either.  However, I did find something reasonably close to "my" equation:

• Pladziewicz, J. R., Lesniak, J. S., and Abraham, A. J., Treatment of kinetic data for opposing second-order and mixed first- and second-order reactions, J. Chem. Educ. 63 (1986) 850-851.

It's not the same thing as I have, but it's a good jumping off point for a possible paper presenting the newly derived formula.

Thanks again.

Have you seen, then, that very interesting papers are published in the J. Chem. Ed.?

Dear Dr. Kuzmic,

Your formula is absolutely accurate and can be very useful. However, I consider the more general case, when the initial concentration of C is not equal to zero. Maybe my attachment will be interesting for you.

Dear Victor,

Thanks for the post.  Your newly derived formula (assuming that the initial complex concentration is nonzero) can also be very useful, for example in the "concentration jump" experiment.  More generally, your formula could be useful in trying to understand what happens over time upon high dilution of a complex preformed in solution.

In some types of experiments in enzymology or biophysics, we pre-incubate  the enzyme and inhibitor to obtain a fully equilibrated mixture with concenrtrations E_eq, I_eq, and C_eq.  Those equilibrium concentrations are given by a simple formula. Then this mixture is highly diluted, e.g. 10-fold, and we can watch over time how the the composition gradually shifts to a new equilibrium.

So if the dilution ratio were 1:10, to use your formula we would simply set E_0 = E_eq/10, I_0 = E_eq/10, and C_0 - C_eq/10 and go from there.  It's a "complementary" experiment (dissociation) relative to the one were we consider the complex formation over time (association).  This shows that your new formula indeed has physical relevance, so thanks again for posting it.

Sorry, a small correction (see in attach).

Has anyone found a literature reference?

If not... what's the method of solution? My guess is that the first step would be to make a change of variable to obtain a Bernoulli equation.

William, I did eventually found a reference:

Benson, S. W. (1982) The Foundations of Chemical Kinetics, Krieger Publishing, Malabar, FL, p. 29-30.

This is an "updated and corrected" reprint of an original 1960 edition by McGraw-Hill, New York.

 Thank you Petr and Viktor. I'm attaching my notes on this matter, which includes a derivation of the solution.