Answer
\begin{array}{l}
16 M n S+13 A s_{2} C r_{1} 0 O_{3} 5+374 H_{2} S O_{4}=16 H M n O_{4}+26 A s H_{3}+130 C r S_{3} O_{1} 2+ \\
327 H_{2} O
\end{array}
Work Step by Step
\[
\begin{array}{l}
M n S=\left[\begin{array}{l}
1 \\
1 \\
0 \\
0 \\
0 \\
0
\end{array}\right] ; A s_{2} C r_{1} 0 O_{3} 5=\left[\begin{array}{l}
0 \\
0 \\
2 \\
10 \\
35 \\
0
\end{array}\right] ; H_{2} S O_{4}=\left[\begin{array}{l}
0 \\
1 \\
0 \\
0 \\
4 \\
2
\end{array}\right] ; H M n O_{4}= \\
{\left[\begin{array}{l}
1 \\
0 \\
0 \\
0 \\
4 \\
1
\end{array}\right] ; A s H_{3}=\left[\begin{array}{l}
0 \\
0 \\
0 \\
0 \\
3
\end{array}\right] ; C r S_{3} O_{1} 2=\left[\begin{array}{l}
0 \\
3 \\
1 \\
12 \\
0
\end{array}\right] ; H_{2} O=\left[\begin{array}{l}
0 \\
0 \\
0 \\
0 \\
1 \\
2
\end{array}\right]}
\end{array}
\]
A systematic method for balancing the chemical equation is to set up a vector equation that describe the number of atoms of each type present in a
reaction. since given equation involves 6 types of
atoms. Construct a vector equation for each reactant
and product
2
\[
\begin{array}{l}
x_{1}\left[\begin{array}{l}
1 \\
1 \\
0 \\
0 \\
0 \\
0
\end{array}\right]+x_{2}\left[\begin{array}{l}
0 \\
0 \\
2 \\
10 \\
35 \\
0
\end{array}\right]+x_{3}\left[\begin{array}{l}
0 \\
1 \\
0 \\
0 \\
4 \\
2
\end{array}\right]=x_{4}\left[\begin{array}{l}
1 \\
0 \\
0 \\
0 \\
4 \\
1
\end{array}\right]+x_{5}\left[\begin{array}{l}
0 \\
0 \\
1 \\
0 \\
0 \\
3
\end{array}\right]+ \\
{\left[\begin{array}{c}
0 \\
3 \\
0 \\
1 \\
12 \\
0
\end{array}\right]+x_{7}\left[\begin{array}{l}
0 \\
0 \\
0 \\
0 \\
2
\end{array}\right]}
\end{array}
\]
To balance given equation, the coefficient must satisfy
Row reduction of augmented matrix leads to these general solution.
6
\[
\begin{array}{l}
x_{1}=16 \\
x_{2}=13 \\
x_{3}=374 \\
x_{4}=16 \\
x_{5}=26 \\
x_{6}=130 \\
x_{7}=327
\end{array}
\]
To avoid fraction take $x_{7}=327$ and find other values.
7
\[
\begin{array}{l}
16 M n S+13 A s_{2} C r_{1} 0 O_{3} 5+374 H_{2} S O_{4}=16 H M n O_{4}+ \\
26 A s H_{3}+130 C r S_{3} O_{1} 2+327 H_{2} O
\end{array}
\]
Balancing equation for given chemical reaction.
