# When the chemical reaction balancing fails

Not all skeletal reaction equations can be balanced. When one of the elements is present on only one side of the reaction equation, it is obvious:

S + H_{2}O → SO_{2}

but sometimes reasons are more subtle, like in:

**A**H_{2}O_{2} → **B**H_{2}O + **C**H_{2}

where no matter how we try we will be always left with excess oxygen, although there is no 'lack' of any element on either side of equation. If we try algebraic method we can set up two equations:

O: 2×A = B

H: 2×A = 2×B + 2×C

which - when solved with the help of the 'integer coefficients' rule - gives the solution:

2H_{2}O_{2} → 2H_{2}O - H_{2}

mathematically correct, but chemically unacceptable.

Sometimes we will find element balance equations are inconsistent:

**A**Na_{2}SO_{4} → **B**Na_{2}O + **C**SO_{2}

using the algebraic approach we have:

Na: 2×A = 2×B (or simply A = B)

S: A = C

O: 4×A = B + 2×C

Substituting B from the Na balance and C from S balance to the last equation, we get 4×A = 3×A - equation is inconsistent which is a signal that something is wrong. Note that this equation is not inconsistent if A=0, but that solution is not interesting for us.

Sometimes skeletal reactions can be balanced with more than one set of coefficients. That usually means that reaction is in fact sum of at least two processes. Let's take for example reaction between potassium chlorate and hydrogen chloride - KClO_{3} + HCl → KCl + H_{2}O + Cl_{2} + ClO_{2}. When balanced it may take form

2KClO_{3} + 4HCl → 2KCl + 2H_{2}O + Cl_{2} + 2ClO_{2}

or

6KClO_{3} + 28HCl → 6KCl + 14H_{2}O + 13Cl_{2} + 2ClO_{2}

(or some other). Why? This is in fact not one, but two reactions occurring simultaneously:

A: 5KClO_{3} + 6HCl → 5KCl + 3H_{2}O + 6ClO_{2}

B: KClO_{3} + 6HCl → KCl + 3H_{2}O + 3Cl_{2}

So we have two equations - and every linear combination (linear combination means we are multiplying by some numbers and adding) of these two equations, will still appear to be a valid reaction equation (although it may happen that all coefficients have to be divided by some constant).

So for example we have A+B:

6KClO_{3} + 12HCl → 6KCl + 6H_{2}O + 3Cl_{2} + 6ClO_{2}

or 2×A+3×B:

13KClO_{3} + 30HCl → 13KCl + 15H_{2}O + 9Cl_{2} + 12ClO_{2}

or 7×A+5×B:

40KClO_{3} + 72HCl → 40KCl + 36H_{2}O + 15Cl_{2} + 42ClO_{2}

and so on. We can generate infinite number of such 'different' reactions.

Algebraic method doesn't balance these equations, which is a clear signal that there is something wrong with them.

Other example of similar situation is reaction equation for TiO_{2} + C + Cl_{2} → TiCl_{4} + CO + CO_{2} process (can't be balanced by the algebraic method), which is also a sum of two, this time completely separate processes (both can be balanced by the algebraic method):

TiO_{2} + C + 2Cl_{2} → TiCl_{4} + CO_{2}

TiO_{2} + 2C + 2Cl_{2} → TiCl_{4} + 2CO

and any combination of these two looks correct:

2TiO_{2} + 3C + 4Cl_{2} → 2TiCl_{4} + 2CO + CO_{2}

3TiO_{2} + 5C + 6Cl_{2} → 3TiCl_{4} + 4CO + CO_{2}

3TiO_{2} + 4C + 6Cl_{2} → 3TiCl_{4} + 2CO + 2CO_{2}

(you may play with

- C
_{4}H_{10}+ Cl_{2}+ O_{2}→ CO_{2}+ CCl_{4}+ H_{2}O - I
_{4}O_{9}→ I_{2}O_{6}+ I_{2}+ O_{2} - HNO
_{2}+ HN_{3}→ N_{2}+ N_{2}O + H_{2}O - Zn + HNO
_{3}→ H_{2}O + N_{2}O + NO + Zn(NO_{3})_{2}

to generate other infinite sets of balanced equations).

You must be aware of the fact that all methods described here have their limitations. Let's take a look at permanganometric determination of the hydrogen peroxide. Titration is based on the reaction between permanganate and hydrogen peroxide in acidic solution. We may balance the reaction by inspection and obtain:

2KMnO_{4} + H_{2}O_{2} + 3H_{2}SO_{4} → 3O_{2} + 2MnSO_{4} + K_{2}SO_{4} + 4H_{2}O

charge and mass are conserved - yet the reaction is wrong! Proper one (and its correctness is proven by stoichiometry observed in experiment) goes like:

2KMnO_{4} + 5H_{2}O_{2} + 3H_{2}SO_{4} → 5O_{2} + 2MnSO_{4} + K_{2}SO_{4} + 8H_{2}O

Algebraic method refuses to balance the equation - there is simply not enough information to do it. The problem is, oxygen from permanganate and hydrogen peroxide is balanced together, but it takes part in two different - and separate - reactions. Permanganate oxygen atoms end in water molecules, peroxide oxygen atoms end in water and oxygen molecules. Ratio of these reactions is given by charge transfer, thus ON method, as well as half reactions method, are both capable of dealing with the case.

In general, the algebraic method has problems with balancing redox equations, where one of the reagents can disproportionate by itself:

H_{2}O_{2} → H_{2}O + O_{2}

This is already redox reaction (we start with oxygen having ON -1 and end with oxygens having ON of -2 and 0). When it is a part of larger system (like oxidation with permanganate shown above) it often poses a problem that the algebraic method is not able to overcome.

One of the main deficiencies of oxidation numbers method is the problem with assigning ON in organic compounds. Let's take a tartaric acid as an example. It has formula C_{4}H_{6}O_{6} - using standard method of ON calculation we have 6×(-2) for oxygen, 6×(+1) for hydrogen, and the molecule is neutral - so the carbon atoms must fit the equation 4×ON_{C} -12 + 6 = 0. Thing is, we get +1.5 as an answer - which just doesn't look right. Interestingly it works when used for balancing (remember that oxidation numbers are only an accounting device, but an efficient one) - but such results clearly show that something is wrong with the concept.

Half reaction method is the most sure way to balance redox reaction - but it is obviously of no use when it comes to balance non-redox reaction.