Biomolecular rate constants and Eyring equation

[Cytotoxic]

I have always been taught in dimensional analysis that units should cancel out before you log a number. The problem I have, as stated in the title of this thread, is that units don't seem to cancel out when applying bimolecular rate constants to the Eyring equation:

ln[(kh)/(k_BT)] = -ΔH^‡/(RT) + ΔS^‡/T

where, k is the rate constant in question and the fact that lots of units are left after cancelling really bothers me. I have consulted Atkins and Google (and Wiki of course!) but to no avail. It seems that the wealth of materials available on the Internet seems to just use this equation for bimolecular reactions. This leads to a systematic error in both the enthalpy and entropy of activation (if something is indeed missing). If comparisons were to be made between enthalpies and entropies derived from rate constants with the same molecularity they would perfectly canceled out; however, if comparisons were to be made between rate constants with different molecularity then there is a problem. Am I missing something painfully obvious?

On a related note, if one were to calculate the enthalpy and entropy of activation for a reaction using pseudo-first-order rate constants, what kind of physical meaning do they have or are they meaningless by themselves (as above, this introduces a systematic difference in both the enthalpy and entropy of activation and would cancel out perfectly upon subtraction)?

Thank you very much in advance!

 

[DrDu]

If the reaction is first order, for what case the equation has been primarily derived, the units cancel.

 

[Cytotoxic]

If the reaction is first order, for what case the equation has been primarily derived, the units cancel.

That's why I specifically asked about bimolecular rate constants. :p

 

[DrDu]

I have this kinetics stuff very rusty, but thinking about it I suppose that in a bimolecular reaction, the concentrations in the reaction rate are compensated by the corresponding concentration dependence of the activation entropy.

 

[JohnRC]

The first reason why units do not cancel is because the expression should include ΔS‡/R rather than ΔS‡/T !

Is the problem because you cannot get the ΔH‡ and ΔS‡ into normal enthalpy and entropy units respectively when using the Eyring treatment? Or is there some other problem in your dimensional analysis?

 

[JohnRC]

OK, It escaped me at first: you are worried because h is in J.s and k_BT is in J.

So for a first order reaction with k in s^-1 all is well, but for a second order treatment k is in s^-1* (concentration/activity units)^-1

I suspect that the problem has to do with collision frequency, which would not ordinarily enter into an Eyring type treatment. There is a problem in trying to count microstates for a bimolecular reactant state and a unimolecular collision complex.

 

[DrDu]

That's what I meant. The activation entropy for going from two isolated molecules to a single transition state complex involves some extra concentration dependent factor \Delta S^\neq=\Delta S^\neq_0+R\ln[A]<b>/[X^\neq]</b>.

 

[JohnRC]

Right, I have it. The clue is in the value of ΔS‡. For a 2 molecule ractant state and a 1 molecule transition state, there is no especially privileged value of ΔS‡ -- it will vary by an additive term related to the log of the conversion factor between different units you might use. So if you were to consider k in s^-1 * M^-1 and forget about the units you would get one value for ΔS‡.
But if you had k in s^-1 * molecules^-1 * m³ you would arrive at quite a different value of ΔS‡, displaced by the log of the conversion factor between the two units. So for any order of reaction other than first order, you must make a selection of concentration units, and those units will determine an otherwise indeterminate scale for ΔS‡.

 

[DrDu]

That's also why one does not use usually absolute concentrations in thermodynamical or kinetic equations but dimensionless activities a referring to a standard state.
In the easiest cases, e.g. an ideal solution, a=c/c0. with c0=1 mol/l.
k is then dimensionless from the very beginning.

 

[Studiot]

Moore has a good section on transition state theory, that includes discussion of units.

Like all books on the subject his tome is entitled Physical Chemistry.

You will also find theoretical discussion/derivation and for this equation references in Denbigh (Chemical Equilibrium) p455 - 459