[box]

Hi all,

I want to calculate the kinetics of the following system:
A -> B
          }-> BD
C -> D
          }-> DF
E -> F

In short, three parallel 1st order reactions, followed by two competing 2nd order reactions. The first part of the parallel is nailed down, the latter part also when there is no competition (either B + D -> BD or D + F -> DF). However, I fear that when there is a competition between B and F for D, the situation becomes very complicated as the reduction in the concentration of D is the sum of both reaction, which affects both rates.

I studied several Chemical Kinetics books (eg. Foundations of Chemical Kinetics by Benson, Principles of Chemical Kinetics by House, and Organic Reactions and their Mechanism by Maskill), but I would love to find a compilation of a large variety of systems, with their appropriate mathematical equations. I will work on deriving them myself, but being an organic medicinal chemist, this might take a while.

Does anyone know a place where I can find or derive the mathematical equation that describes this system? That would be very helpful.

Many thanks,

[billnotgatez]

I think someone will be interested in this enough to create a dialog.
But you might want to show some more of your thoughts  about the concepts to add to the conversation.

By the way
The forum rules are available at a link near the  center  page top.
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If you have not read them already.

[curiouscat]

Are you sure that a closed form analytical solution even exists? Why not just try a numerical solution? A numerical solution won't be hard.

Do you have values for these rate constants?

Solving for c_A, c_C & c_E seems easy but the other differential equations seem coupled.

[box]

Thanks for the comment, curiouscat.

You're right, the other differential equations are coupled, and the system can probably not be solved by a mathematical equation without making huge assumptions. I've been cracking my brains out for the last weeks, and only found solutions for the concentrations of A, C, and E; those were easy. Assuming that B, D, and F do not react, I can determine the concentration of those with [B ]t = [A]0-[A]t, etc, also easy. Then comes the impossible part, where all systems are coupled.

I'm gonna work on a numerical solution now, see how far I get; I have the rate constants of all the transformations.
Thanks again!

[curiouscat]

I'm gonna work on a numerical solution now, see how far I get; I have the rate constants of all the transformations.
Thanks again!

If you post the rate constants I can try too & we could compare our answers?

[box]

Great idea, let's see what we get.

Here are the constants:
[A]0 = [C]0 = [E]0 = 5 µM.

First order rate constants for the first three parallel steps:
k(AB) = 0.5 1/min
k(CD) = 0.05 1/min
k(EF) = 0.01 1/min

Second order rate constants for the last two parallel steps:
k(BD) = 0.01 (1/µM.min)
k(DF) = 0.02 (1/µM.min)

[Plontaj]

To solve ordinary differential system by numerical method you need to specify initial values of dependent variable (already done but not for all variables) and initial and final value of independent variable (here it is time). I've assumed that the final time is 100 min.

I have solved this system by polymath:

Code: [Select]

d(A)/d(t)=-k1*A
d(C)/d(t)=-k2*C
d(E)/d(t)=-k3*E

d(B)/d(t)=k1*A-k4*B*D
d(D)/d(t)=k2*C-k4*B*D-k5*D*F
d(F)/d(t)=k3*E-k5*D*F

k1=0.5 #1/min
k2=0.05 #1/min
k3=0.01 #1/min
k4=0.01 #1/uM*min
k5=0.02 #1/uM*min

A(0)=5 #uM
C(0)=5 #uM
E(0)=5 #uM
B(0)=0 #uM
D(0)=0 #uM
F(0)=0 #uM
t(0)=0 #min
t(f)=10 #min
A solution I give you as an image:

[curiouscat]

To solve ordinary differential system by numerical method you need to specify initial values of dependent variable (already done but not for all variables) and initial and final value of independent variable (here it is time). I've assumed that the final time is 100 min.

Great job. Maybe try a little longer times too because in this solution the F concentration hasn't maxed out yet. It should eventually.

Otherwise you solution makes sense.
 

[Plontaj]

Sure, it had to reach the max value because all reactions are irreversible and the raw materials are limited. So I forgotten about the products on the end of path reactions. So, I've added they now:

[box]

Fantastic job Plontaj! I especially like the <100 min since this shows how the concentrations vary in the initial stages of the reaction.

May I ask what the code is you used to describe the concentrations of BD and FD? I would like to see their traces in the 100 min window: they seems to ramp up slowly in the first 20(?) mins and then increase quickly, at least for FD (for BD it's not really visible).

Again, nice curves you got there ;-). I'm impressed!

Addition:
I did not know Polymath, it looks like a very nice program for these kind of things. As a Mac user I was exploring Grapher a bit, but did not get very far (will try again though). I checked the Polymath website, and it seems that the data can be exported to Excel. Does that work well?

[Plontaj]

The differential equations for final products are:

Code: [Select]

d(BD)/d(t)=k4*B*D
d(FD)/d(t)=k5*F*D
I've put bellow the BD and FD concentration (uM) and their concentration changes (in uM/min multiplied by 50 on the concentration axis) vs time:

they seems to ramp up slowly in the first 20(?) mins and then increase quickly, at least for FD

Of course. That curve is characteristic for chain reactions with limiting step as well as the rate of creating product is also characteristic (for a simple chemical reaction the maximum rate is at the beginning of the reaction). The reaction "4" and "5" is just waiting for couple min to appear a some D, B and F.

Those are a theoretical considerations or an example of a real process?

[Plontaj]

I did not know Polymath, it looks like a very nice program for these kind of things. As a Mac user I was exploring Grapher a bit, but did not get very far (will try again though). I checked the Polymath website, and it seems that the data can be exported to Excel. Does that work well?

I'm sorry I didn't notice an addition.

Yes, there is possibility to export the data to excel but without possibility to solving differential equations. If you are interested in solving of the ordinary differential equations system (ODES) you should to check out the Runge-Kutta method. For example Polymath uses RK method 4th order but not only. If you've got access to Matlab with simulink you can also solve the ODES by constructing block scheme (quite simple). I've solved the ODES in excel too, by RK 4th. I think you can get quite pretty manuals or examples in web.

[curiouscat]

+1 for Polymath.

It is an excellent program for these sort of things & very easy to use too. Too bad it's not free. I used to have a CD that had it from my Himmelblau text but misplaced it somewhere.

For stuff a bit more involved I have used Matlab & that works well too but the learning curve is a bit steeper.

[box]

Having the ability to export the data-points of the curves to Excel would already be good, at least for layout purposes. Did you try exporting the datapoints to another spreadsheet program, or to Google Sheets for example? I'm thinking about buying the program and putting it on a windows machine we have here, but that one does not have Excel (yet, gonna ask the guy to install it though).

The system was a hypothetical one for the moment, but I was thinking about some programmed synthesis kind of application. I'm more into OrgChem and MedChem, and these kind of calculations are a bit out of my comfort zone, but, as you showed now, very useful to predict the progress of reactions.

[curiouscat]

Having the ability to export the data-points of the curves to Excel would already be good, at least for layout purposes. Did you try exporting the datapoints to another spreadsheet program, or to Google Sheets for example? I'm thinking about buying the program and putting it on a windows machine we have here, but that one does not have Excel (yet, gonna ask the guy to install it though).

The system was a hypothetical one for the moment, but I was thinking about some programmed synthesis kind of application. I'm more into OrgChem and MedChem, and these kind of calculations are a bit out of my comfort zone, but, as you showed now, very useful to predict the progress of reactions.

If you plan on doing a lot of this, or more involved stuff I'd recommend Matlab / Octave over Polymath.

Polymath is great for teaching or quick calcs. but if you try to something very complex you'll hit its limitations.