# Magical reaction tanks

I would be interested to know people's opinion on the following matter.

Background

Often, when the far-from equilibrium behaviour of a chemical reaction system is analysed mathematically (in papers or text books), the system is placed in a gradient between a fixed non-zero concentration of "resource" species, and a fixed concentration of "waste" species (which is often zero). The other species in the reaction system are designated as "intermediates", and it is their behaviour which is of prime interest.

For example, for the oscillating Lotka-Volterra reaction system:

$R + X \rightarrow 2X$

$X + Y \rightarrow 2Y$

$Y \rightarrow W$

Species $R$ and $W$ are the fixed concentration resource and waste species respectively, and species $X$ and $Y$ are the (oscillating) intermediates.

Question

It appears as if these reaction systems are being studied in a physical scenario which could not be realised in the lab, a magical reaction tank if you like.

The two reasons most often stated as to why resource and waste are held at constant concentrations, whilst the intermediates can vary, are that:

1) The reaction takes place in a large reservoir, where resource is abundant and where any waste produced by the reaction quickly diffuses away into the vastness

or

2) The reaction has resource species quickly "pumped in" by some external agent, and likewise waste species quickly "pumped away".

My problems with these explanations are as follows:

If in explanation 1, a very large reservoir is keeping resource and waste species at constant concentration, then if follows that the intermediate species can never reach any appreciable concentration, for they too exist in the same large volume. In other words, if the waste diffuses away, then why don't the intermediates?

Conversely, in explanation 2, if a pumping process is artificially maintaining resource and waste concentrations, it is hard to see how this pumping process is being "micro selective" in completely removing the waste compounds, but leaving the intermediate compounds totally unaffected.

More physically realistic models of chemical reactions can be made by also explicitly including the reactor in the model - for example by considering the reaction as taking place in semi-permeable compartments or in bulk in a CSTR. However, additionally modelling the reactor often changes the mathematical properties for the "bare" reaction reported in the literature (for example, in a CSTR, the intermediates as well as the waste are removed).

## 1 Answer

I would say it's worth investigating. My suspicion is that the answer won't change much, but the way to find out is to try it both ways and see how the assumptions affect the answer under various conditions.

Instead of a CSTR, for this particular reaction a batch reactor might be more appropriate, since that is the easiest setup for a lab. In that case, to model the reaction all we have to do is not assume that some of the rates of change are zero, and see how that changes the reaction.

Fortunately for us, this site has already done the heavy lifting and even has a solution in a mathcad file.

I haven't looked at the solution closely enough to determine if they are making the approximations you mentioned (I don't have mathcad installed), or if they are just directly integrating all four equations. They only show graphs for the concentrations of X and Y. However, it probably wouldn't be too hard to modify in either direction to compare the results under different sets of assumptions.

• Thanks. Two points: First, from what I can see, a batch reactor reactor seems the same thing as a CSTR. Second, on the site you suggest they are holding species A constant for the LV reaction "A is continuously replaced from an external source as it is consumed in the reaction", and they are forgetting about waste B. Therefore, my question about how to ensure A and B have fixed concentrations in an experimental setting still stands. – edison1093 Jun 2 '14 at 16:22
• To add a note about the LV reaction: the paper Properties of two‐component bimolecular and trimolecular chemical reaction systems states that the LV reaction is not "structurally stable", i.e. small changes to its equations of motion (such as those introduced by modelling the reactor) generally destroy its capacity to have conservative orbits. – edison1093 Jun 2 '14 at 16:35
• The main difference between a CSTR and a batch reactor is the flow - a batch reactor has no inlet or outlet flow, a CSTR does. You are right about the solution on the site I linked assuming a fixed concentration of A - you would have to modify the mathcad file (or write your own numerical integration routine) to solve the equations without those assumptions, then compare the results. I didn't catch your question about fixed A & B in the original - I thought you were asking whether those assumptions would have an effect. My suggestion is to try it and find out. – thomij Jun 2 '14 at 17:35