# Array reduction for a reaction matrix

• Mangoes
In summary, the problem is trying to set up a reaction matrix to determine the independence of the reactions, but I'm having trouble with it. The handout I was given explains to form a matrix by the stoichiometric coefficients of the species, but I don't know how to determine the order in which I should enter the species. I was also given an equation to solve for the independent reactions, but I'm not sure if I'm doing it correctly.
Mangoes

## Homework Statement

My problem is with a portion of a much larger problem, I'm working on an exercise involving the production of synthesis gas in where there's a reactor in which it is postulated that the following reactions take place:

CH4 + CO2 ⇔ 2CO + 2H2
CO + H2O ⇔ CO2 + H2
CH4 + H2O ⇔ CO + 3H2
CH4 + 2H2O ⇔ CO2 + 4H2

One of the problems of the question asks me to write down a reaction matrix to determine the independence of the reactions.

## The Attempt at a Solution

My textbook doesn't cover this, but I was given a handout that covers the method. The handout explains to form a matrix by the stoichiometric coefficients of the species. In the example above, the matrix will be a 4x5 matrix, since there are 4 reactions and 5 total species present.

Then through elementary row operations, I am to diagonalize the matrix to determine the independence of the sets.

I'm having a problem setting up the matrix though. From what I've seen, the convention in my handout is to assign negative values to reactants and positive values to products, but I don't see how to determine the order in which I should enter the species.

To give an example of what I mean, if I were to enter the coefficients for CH4 in the first column, CO in the second column, CO2 in the third column, H2 in the fourth column, and H2O in the fifth column, the matrix would look like this:

-1 2 -1 2 0
0 -1 1 1 -1
-1 1 0 3 -1
-1 0 1 4 -2

(Apologies for the cluster, I don't know how to write matrices in here)

This can be reduced to yield the independent equations:

CO2 + 4H2 ⇔ CH4 + 2H2O
CO2 + H2 ⇔ H2O + CO

However, if I were to pick a different ordering for my entries, such as CO for my first column, water for my second column, methane for my third column, carbon dioxide for my fourth column, and hydrogen for my fifth column, I can solve that matrix to get these independent equations:

CH4 + CO2 ⇔ 2CO + 2H2
CO2 + 4H2 ⇔ CH4 + 2H2O

One of the equations in the two sets is equal, but what's the deal with the two equations that don't match? Are they independent of one another or can it be shown they are multiples of one another if I make the appropriate operations?

Last edited:
There is nothing like "the independent reactions".
There is only a possibility to pick up "a set of independent reactions".
In your case, if your calculations are correct, there would be only two independent reactions.

Observe that "CH4 + CO2 ⇔ 2CO + 2H2" in your second set is equivalent to a linear combination of your first set:

"CH4 + CO2 ⇔ 2CO + 2H2" equivalent to 2*"CO2 + H2 ⇔ H2O + CO" -1* "CO2 + H2 ⇔ H2O + CO"

and that the other equation in your second set is the same as the first equation from your first set.

You can of course chose the order of your column as you wish.
This will not change the conclusions.
It can only lead to a different choice of the independent set, but an equivalent choice.

I am not sure if the word "diagonalize" is correct, but I guess what you mean? Check.

maajdl said:
There is nothing like "the independent reactions".
There is only a possibility to pick up "a set of independent reactions".
In your case, if your calculations are correct, there would be only two independent reactions.

Observe that "CH4 + CO2 ⇔ 2CO + 2H2" in your second set is equivalent to a linear combination of your first set:

"CH4 + CO2 ⇔ 2CO + 2H2" equivalent to 2*"CO2 + H2 ⇔ H2O + CO" -1* "CO2 + H2 ⇔ H2O + CO"

and that the other equation in your second set is the same as the first equation from your first set.

You can of course chose the order of your column as you wish.
This will not change the conclusions.
It can only lead to a different choice of the independent set, but an equivalent choice.

I am not sure if the word "diagonalize" is correct, but I guess what you mean? Check.

I hadn't noted that the equation in the second set was a linear combination of my first set. I would've assumed that ordering shouldn't matter, but I didn't see it addressed and it's been a while since I've worked with matrices.

Thank you very much!

## What is array reduction for a reaction matrix?

Array reduction for a reaction matrix refers to the process of simplifying a large matrix of data in order to better understand and analyze the reactions between different molecules. This involves reducing the number of variables in the matrix and finding patterns or correlations within the data.

## Why is array reduction important for studying reactions?

Array reduction is important because it allows scientists to identify the most significant variables in a reaction and focus on them for further analysis. It also helps to reduce the amount of data that needs to be processed, making it easier to draw conclusions and make predictions.

## What methods are commonly used for array reduction?

There are several methods that can be used for array reduction, including principal component analysis, singular value decomposition, and cluster analysis. Each method has its own strengths and may be more suitable for certain types of data or research questions.

## Are there any limitations to using array reduction for a reaction matrix?

Yes, there are some limitations to using array reduction for a reaction matrix. For example, the results may be influenced by the initial selection of variables, and the reduction process may overlook important but subtle relationships between variables. It is important to carefully consider the limitations and potential biases of any array reduction method used.

## How can array reduction for a reaction matrix be applied in real-world research?

Array reduction for a reaction matrix has many practical applications, such as in drug discovery, materials science, and environmental studies. By identifying the key variables and patterns within a reaction, scientists can develop more efficient and effective processes, predict the behavior of new substances, and gain a better understanding of complex systems.

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