Question #306965

Balance the given chemical equation using linear equations (gussis Jordan elimination method).





Na₂CO3 + C + N₂ ---> NaCN + CO

1
Expert's answer
2022-03-07T17:25:05-0500

Let's write the equation in the form

a Na2CO3 + b C + c N2 ---> d NaCN + e CO

where a, b, c, d, and e are unknown numbers.


Equate the left and right sides of the equation:

2 a = d for Na

a + b = d + e for C

3 a = e for O

2 c = d for N


or in matrix form



[20010110113000100210][abcde]=[00000]\begin{bmatrix} 2 & 0 & 0 & -1 & 0 \\ 1 & 1 & 0 & -1 & -1 \\ 3 & 0 & 0 & 0 & -1 \\ 0 & 0 & 2 & -1 & 0 \end{bmatrix} \cdot \begin{bmatrix} a \\ b \\ c \\ d \\ e \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}

Then using Gauss-Jordan method

[20010110113000100210][1001/20110113000100210][1001/200101/210003/2100210]\begin{bmatrix} 2 & 0 & 0 & -1 & 0 \\ 1 & 1 & 0 & -1 & -1 \\ 3 & 0 & 0 & 0 & -1 \\ 0 & 0 & 2 & -1 & 0 \end{bmatrix} \rarr \begin{bmatrix} 1 & 0 & 0 & -1/2 & 0 \\ 1 & 1 & 0 & -1 & -1 \\ 3 & 0 & 0 & 0 & -1 \\ 0 & 0 & 2 & -1 & 0 \end{bmatrix} \rarr \begin{bmatrix} 1 & 0 & 0 & -1/2 & 0 \\ 0 & 1 & 0 & -1/2 & -1 \\ 0 & 0 & 0 & 3/2 & -1 \\ 0 & 0 & 2 & -1 & 0 \end{bmatrix} \rarr


[1001/200101/210003/2100210][1001/200101/21002100003/21][1001/200101/210011/200003/21]\begin{bmatrix} 1 & 0 & 0 & -1/2 & 0 \\ 0 & 1 & 0 & -1/2 & -1 \\ 0 & 0 & 0 & 3/2 & -1 \\ 0 & 0 & 2 & -1 & 0 \end{bmatrix} \rarr \begin{bmatrix} 1 & 0 & 0 & -1/2 & 0 \\ 0 & 1 & 0 & -1/2 & -1 \\ 0 & 0 & 2 & -1 & 0 \\ 0 & 0 & 0 & 3/2 & -1 \end{bmatrix} \rarr \begin{bmatrix} 1 & 0 & 0 & -1/2 & 0 \\ 0 & 1 & 0 & -1/2 & -1 \\ 0 & 0 & 1 & -1/2 & 0 \\ 0 & 0 & 0 & 3/2 & -1 \end{bmatrix} \rarr


[1001/200101/210011/2000012/3][10001/301004/300101/300012/3]\begin{bmatrix} 1 & 0 & 0 & -1/2 & 0 \\ 0 & 1 & 0 & -1/2 & -1 \\ 0 & 0 & 1 & -1/2 & 0 \\ 0 & 0 & 0 & 1 & -2/3 \end{bmatrix} \rarr \begin{bmatrix} 1 & 0 & 0 & 0 & -1/3 \\ 0 & 1 & 0 & 0 & -4/3 \\ 0 & 0 & 1 & 0 & -1/3 \\ 0 & 0 & 0 & 1 & -2/3 \end{bmatrix}


Therefore if e = 3 then a = 1, b = 4, c =1, d = 2 and the equation is

Na2CO3 + 4C + N2 ---> 2NaCN + 3CO




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