redox - Balancing by oxidation numbers method multiple atoms

In my Chemistry textbook, the rules for balancing a chemical equation using the oxidation-Number method are as follows:

1. Assign oxidation numbers to all the atoms in the equation.


2. Identify the atoms that are oxidized and the atoms that are reduced.


3. Determine the change in oxidation number for the atoms that are oxidized and for the atoms that are reduced.


4. Make the change in oxidation numbers equal in magnitude by adjusting coefficients in the equation.


5. If necessary, use the conventional method to balance the remainder of the equation.

I understand how to do this for simple equations, such as $$\ce{SnCl4 + Fe -> SnCl2 + FeCl3}$$

Following the steps:

1. $\ce{Sn^{4+} + 4Cl- + Fe^0 -> Sn^{2+} + 2Cl^- + Fe^{3+} + 3Cl^-}$


2. Oxidation: $\ce{Fe^0 -> Fe^{3+} (Fe)}$ reduction: $\ce{Sn^{4+} -> Sn^{+2} (Sn)}$


3. oxidation: +3; reduction -2


4. $\ce{3SnCl4 + 2Fe -> 3SnCl2 + 2FeCl3}$


5. (no change necessary, already balanced)

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The Problem: Notice how Sn and Fe (the reduced and oxidized atoms) all have subscripts of 1. However, when their subscripts differ, I'm not sure how to balance it.

For example, I can easily balance the following equations by inspection, but not so easily by this method of oxidation numbers:

$$\ce{KClO3 -> KCl + O2}$$ (O is oxidized, subscripts of 3 and 2)

$$\ce{NH3 + NO2 -> N2 + H2O}$$ (N is oxidized and reduced, oxidation numbers of -3, +2, and 0)

And this gets even more complicated when the subscripts differ and there are more than two elements whose oxidation numbers changed (are reduced or oxidized), e.g.,

$$\ce{NH4ClO4 + Al -> Al2O3 + HCl + N2 + H2O}$$ (Al and N are oxidized, Cl is reduced)

For all of these, I get stuck on the fourth step. Determining oxidation numbers and calculating the change is not the problem, but figuring out how to do it with larger or different coefficients as in the examples above is the problem.

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The five general steps in my textbook don't help me understand this method much with more complicated methods such as this. Could you give me a more thorough explanation - a better explanation or a general rule - of how to solve all of these?

Answer

In some (many?) cases you get a more parsimonious description by considering an entire reactant as an oxidant or reductant. Separating out individual atoms makes you miss the forest because you get confused by the trees.

Take the ammonium perchlorate-aluminum reaction. Since aluminum is being oxidized let us render the ammonium perchlorate as the oxidizing agent, ignoring the fact that only some of its atoms are oxidizing. You then have a reaction:

$\ce{NH_4ClO_4}\rightarrow \ce{N^0}+4\ce{H^I}+\ce{Cl^{-I}}+4\ce{O^{-II}}$

where the Roman numerals indicate oxidation states in the products. These add up to $-5$ for all the product whereas the ammonium perchlorate, as a neutral compound, started with the states adding up to zero. So five electrons must be added to the reactants:

$\ce{NH_4ClO_4}+5 e^-\rightarrow \ce{N^0}+4\ce{H^I}+\ce{Cl^{-I}}+4\ce{O^{-II}}$

Thus although some atoms might be oxidized, others reduced, and still others are just along for the ride, we identify a net oxidizing effect of the ammonium perchlorate as a whole, numerically five electrons per formula unit.

Now we know that the redox stoichiometry will be $3\ce{NH_4ClO_4}+5\ce{Al}$ and we can balance the reaction accordingly. (We will need to double the coefficients to eliminate a fraction, that's just algebra.)