Answer: Balance the Chemical Equation
Saturday, 16 of February , 2008 at 11:05 am
Question: Balance the chemical equation using Linear Algebra.
Answer: In order to balance the chemical equations, we need to find the five unknowns (ab,c,d,e).
Since the number of chemical elements are conserved, we know that:
we need to solve the following linear system.
This system of equations can easily be solved with Maple.
> with(Student[LinearAlgebra]):
> A := Matrix([[1, 1, 0, -1, 0, 0],
[1, 2, 0, 0, -1, 0],
[0, 0, 2, -4, -2, 0]]):
> ReducedRowEchelonForm(A)
According to maple, we can reduce the linear system to:
According to the above equation, we don’t have a unique solution. However, we know that the linear system will have a solution 
where
Since we know that 
(where 
), we can add additional constraints to our linear system. If we look at the above equations, it is easy to see that the following constraint must hold:
Using the linear system and the above constraint, we find the following solutions…
…
…
As you can see, there is still an infinite amount of solutions for this chemical equations. Some can be reduced, however majority of the equations can’t be reduced
Question: Balance the chemical equation using Linear Algebra.
Answer: In order to balance the chemical equations, we need to find the five unknowns (ab,c,d,e).
Since the number of chemical elements are conserved, we know that:
we need to solve the following linear system.
This system of equations can easily be solved with Maple.
> with(Student[LinearAlgebra]): > A := Matrix([[1, 1, 0, -1, 0, 0], [1, 2, 0, 0, -1, 0], [0, 0, 2, -4, -2, 0]]): > ReducedRowEchelonForm(A)
According to maple, we can reduce the linear system to:
According to the above equation, we don’t have a unique solution. However, we know that the linear system will have a solution where
Since we know that (where ), we can add additional constraints to our linear system. If we look at the above equations, it is easy to see that the following constraint must hold:
Using the linear system and the above constraint, we find the following solutions…
…
…
As you can see, there is still an infinite amount of solutions for this chemical equations. Some can be reduced, however majority of the equations can’t be reduced
Category: Answers, Linear Algebra
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