Array reduction for a reaction matrix

[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?

 

[maajdl]

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.

 

[Mangoes]

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!