\[CO, \: H_2\]
and \[CH_4\]
are burnt in the presence of \[O_2\]
in a furnace. the following reactions occur.\[CO+\frac{1}{2}O_2 \rightarrow CO_2\]
\[H_2+\frac{1}{2}O_2 \rightarrow H_2O\]
\[CH_4+2O_2 \rightarrow CO_2+2H_2O\]
\[CH_4+ \frac{3}{2}O_2 \rightarrow CO+2H_2O\]
The problem is to find the minimum number of equations to completely describe the system, and find a set of independent reactions. We can write the above reactions as
\[CO+\frac{1}{2}O_2 - CO_2=0\]
\[H_2+\frac{1}{2}O_2 - H_2O=0\]
\[CH_4+2O_2 - CO_2-2H_2O=0\]
\[CH_4+ \frac{3}{2}O_2 - CO-2H_2O=0\]
Make thge assignment
\[COC) \rightarrow A_1, \: H_2 \rightarrow A_2, \: CH_4 \rightarrow A_3, \: O_2 \rightarrow H_4, \: CO_2 \rightarrow A_5, \: H_2O \rightarrow A_6\]
The sytem becomes
\[A_1+\frac{1}{2}A_4-A_5=0\]
\[A_2+\frac{1}{2}A_4-A_6=0\]
\[A_3+2A_4-A_5-2A_6=0\]
\[-A_1+A_3+\frac{3}{2}A_4-A_6=0\]
The coefficient matrix is
\[\left( \begin{array}{cccccc} 1 & 0 & 0 & 1/2 & -1 & 0 \\ 0 & 1 & 0 & 1/2 & 0 & -1 \\ 0 & 0 & 1 & 2 & -1 & -2 \\ -1 & 0 & 1 & 3/2 & 0 & -2 \end{array} \right)\]
.The non zero rows in the echelon form are linearly independent, and the number of non zero rows in the echelon form is the rank of the matrix. The number of independent reactions in this problem is equal to the rank of the matrix.
Add the first row to the fourth row.
\[\left( \begin{array}{cccccc} 1 & 0 & 0 & 1/2 & -1 & 0 \\ 0 & 1 & 0 & 1/2 & 0 & -1 \\ 0 & 0 & 1 & 2 & -1 & -2 \\ 0 & 0 & 1 & 2 & -1 & -2 \end{array} \right)\]
Add -1 times the third row to the fourth row.
\[\left( \begin{array}{cccccc} 1 & 0 & 0 & 1/2 & -1 & 0 \\ 0 & 1 & 0 & 1/2 & 0 & -1 \\ 0 & 0 & 1 & 2 & -1 & -2 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array} \right)\]
This is the echelon matrix. It has three non zero row so there are three indepemndent equations. We can take the independent equations to be the first three.
\[CO+\frac{1}{2}O_2 \rightarrow CO_2\]
\[H_2+\frac{1}{2}O_2 \rightarrow H_2O\]
\[CH_4+2O_2 \rightarrow CO_2+2H_2O\]