[Big-Daddy]

I tend to balance equations algebraically if it's not immediately obvious how to balance them from first glance.

But this one seems to resist being solved directly:

MnO₄^- + H⁺ + H₂O₂    Mn²⁺ + H₂O + O₂

An immediate inspection means we can write the following equations to try and solve:
[Mn] a=d;

• 4a+2c=e+2f; [H] b+2c=2e; [e^-] -a+b=2d
  
  4 equations and 6 variables, there's no reason we should be able to force a solution out of this and I do not know how to. But an answer is still given. How would I go about solving such equations? (Not this specific one, but general cases where I am balancing and I end up with less equations than I have variables.)

[Borek]

Permanganate and hydrogen peroxide is a tricky problem, one that is in general difficult to balance. Problem is, hydrogen peroxide decomposes on its own, and details of the final equation depend on the mechanism details. Even after balancing the equation:

2MnO₄^- + 5H₂O₂ + 6H⁺  2Mn²⁺ + 5O₂ + 8H₂O

you can always add to it any number of

2H₂O₂  2H₂O + O₂

and still get a balanced reaction, for example:

2MnO₄^- + 19H₂O₂ + 6H⁺  2Mn²⁺ + 12O₂ + 22H₂O

In general, there are always less equations that unknowns, but the difference should be not larger than 1, if it is - there is no way of dealing with the problem (it is a signal something is wrong with the way system is set up).

[Big-Daddy]

Permanganate and hydrogen peroxide is a tricky problem, one that is in general difficult to balance. Problem is, hydrogen peroxide decomposes on its own, and details of the final equation depend on the mechanism details. Even after balancing the equation:

2MnO₄^- + 5H₂O₂ + 6H⁺  2Mn²⁺ + 5O₂ + 8H₂O

you can always add to it any number of

2H₂O₂  2H₂O + O₂

and still get a balanced reaction, for example:

2MnO₄^- + 19H₂O₂ + 6H⁺  2Mn²⁺ + 12O₂ + 22H₂O

In general, there are always less equations that unknowns, but the difference should be not larger than 1, if it is - there is no way of dealing with the problem (it is a signal something is wrong with the way system is set up).

Wow, I am alarmed to find such a large problem in a question that came from a past paper of mine!

How did you arrive at the "correct" answer, by the way? Your original answer is that given by my mark scheme and is the one redox equations seem to come to, but it is true there are infinite solutions to this problem. How then did you go about finding the correct answer?

And is it then safe to say that if there aren't (at most) 1 less equations than there are variables, there is no unique solution?

[Borek]

How did you arrive at the "correct" answer, by the way? Your original answer is that given by my mark scheme and is the one redox equations seem to come to, but it is true there are infinite solutions to this problem. How then did you go about finding the correct answer?

You can get the correct answer assuming oxygen is produced only from the hydrogen peroxide, and not from the permanganate. In other words - oxygen atoms are not equivalent here.

While this approach gives the correct answer here (read: it reflects the stoichiometry observed in glass), I am not convinced it is a correct approach in general. Don't ask further questions - I simply don't know.

And is it then safe to say that if there isn't 1 less equations than there are variables, there is no unique solution?

Even having n equations is not guaranteed to give solution, as they have to be independent.

Generally speaking when you have n unknowns and less than n independent equations, there is no unique solution (this is math, not chemistry, such system is called underdetermined). In the case of balancing we have additional condition that all unknowns should be integers and the smallest possible ones, which is in a way equivalent to adding n^th equation.

[Big-Daddy]

Even having n equations is not guaranteed to give solution, as they have to be independent.

Generally speaking when you have n unknowns and less than n independent equations, there is no unique solution (this is math, not chemistry, such system is called underdetermined). In the case of balancing we have additional condition that all unknowns should be integers and the smallest possible ones, which is in a way equivalent to adding n^th equation.

OK, and is it ever possible to solve when you have the n variables and n-2 equations? Or do you always need n-1 for it to be feasible? (And then they must be independent, which in the case of balancing they should be, or else the solution be trivial.)

[Borek]

OK, and is it ever possible to solve when you have the n variables and n-2 equations?

You mean "are there systems that give n-2 independent equations, yet we know solution exists"?

Yes. This is aspirin prepared from salicylic acid and acetic anhydride:

C₇H₆O₃ + C₄H₆O₃ C₉H₈O₄ + HC₂H₃O₂

Equation is already correctly balanced, but algebraic method fails with just two independent equations and four coefficients. This is purely accidental.