## 'A Method of Defining Coefficients in the Equations of Chemical Reactions' printed from http://nrich.maths.org/

For simplicity the usage of the suggested method will be demonstrated on the following chemical reactions. Let us examine the reaction of oxidation of methyl-benzol. Instead of the unknown coefficients we will put the variables $a$, $b$, $c$, $d$:
$$a \textrm{C}_6 \textrm{H}_5 \textrm{C} \textrm{H}_3 + b \textrm{O}_2 \rightarrow c \textrm{C} \textrm{O}_2 + d \textrm{H}_2 \textrm{O}$$
$$\textrm{C}_6 \textrm{H}_5 \textrm{C} \textrm{H}_3 +9\textrm{O}_2 \rightarrow 7\textrm{C} \textrm{O}_2 + 4\textrm{H}_2 \textrm{O}$$
Let us examine one more example of using the suggested method for the reaction between potassiumpermanganate $\textrm{K Mn O}_4$ and iron (II) sulphate, $\textrm{FeSO}_4$, acidified with $\textrm{H}_2\textrm{SO}_4$.
$$a\textrm{KMnO}_4+b\textrm{FeSO}_4+c\textrm{H}_2\textrm{SO}_4\rightarrow d\textrm{MnSO}_4+e\textrm{Fe}_2(\textrm{SO}_4)_3+f\textrm{K}_2\textrm{SO}_4 +g\textrm{H}_2\textrm{O}$$
We have obtained the following system: \begin{eqnarray} a & = & 2f \\ a & = & d \\ 4a+4b+4c & = & 4d+12e+4 \\ f & = & g \\ b & = & 2e \\ b+c & = & d+3e+f \\ 2c & = & 2g \end{eqnarray} \begin{eqnarray} a & = & \frac{1}{4}g \\ b & = & \frac{5}{4}g \\ c & = & g \\ d & = & \frac{1}{4}g \\ e & = & \frac{5}{8}g \\ f & = & \frac{1}{8}g \end{eqnarray} \begin{eqnarray} a & = & 2 \\ b & = & 10 \\ c & = & 8 \\ d & = & 2 \\ e & = & 5 \\ f & = & 1 \\ g &= & 8 \end{eqnarray} $$2\textrm{KMnO}_4+10\textrm{FeSO_4}+8\textrm{H}_2\textrm{SO}_4\rightarrow 2\textrm{MnSO}_4+5\textrm{Fe}_2( \textrm{SO}_4)_3+\textrm{K}_2\textrm{SO}_4 +8\textrm{H}_2\textrm{O}$$ Thus, the suggested method is both easy-to-use and it can be used for quick definition of coefficients for complex equations of chemical reactions.