Tackling Chemical Equations with Multiple Balancing Solutions

In rare cases the reader may find chemical equations that can be balanced in multiple ways, providing several possibilities for the coefficients, without considering multiples of other solutions.

For instance, take the following equation.

$$a*CrC{l}_3+b*KI+c*C{l}_2+d*KOH\to e*K_{2}Cr{O}_4+f*KI{O}_4+g*KC{l}_3+h*H_{2}O$$

With some iterations, the following solutions might be found.

$$2*CrC{l}_3+2*KI+11*C{l}_2+32*KOH\to 2*K_{2}Cr{O}_4+2*KI{O}_4+28*KC{l}_3+16*H_{2}O$$ $$8*CrC{l}_3+3*KI+24*C{l}_2+88*KOH\to 8*K_{2}Cr{O}_4+3*KI{O}_4+72*KC{l}_3+44*H_{2}O$$

Both are mathematically correct. Are they chemically correct? First, we will explore why there are two possible answers. By transforming the chemical equation into a linear system, one equation is obtained per each chemical element and the coefficients a ,b...h  defined as variables.

$$$$$$\mathrm{Cr:\; a}=e$$
$$\mathrm{Cl:\; 3}a+2c=g$$
$$\mathrm{K:\; b}+d=2e+f+g$$
$$\mathrm{I:\; b}=f$$
$$\mathrm{O:\; d}=4e+4f+h$$
$$\mathrm{H:\; d}=2h$$
$$\mathrm{<br>}$$

Be noticed that the amount  of equations is directly linked to the number of chemical elements, while the variables are linked to the quantity of compounds. Therefore, the system is underdetermined, thus that either there are no solutions or there are infinite ones. In the previous case we already mentioned possible solutions hence there are infinite solutions that balance the chemical equation.

What would make the reaction proceed with different proportions in real life conditions? A given solution would be favored by the physicochemical conditions in which the reaction takes place. In more complex situations, some solutions will be "more correct" because they may describe correctly the true reaction by taking into account its mechanism.

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To avoid these chaotic situations, many chemical processes shan't be combined into a single equation. They must hold their independence to avoid ambiguities. Such simplifications may lead to quantitative errors. Similar expositions on the topic may be seen here or there. Not every chemical equation follows stoichiometry in real life reactions, be referenced to Cotton's Advanced Organic Chemistry where it talks about non molecular solids.