# Finding all chemical equations (Linear Algebra) [closed]

So I have been given 5 different molecules and I need to find all the possible balanced chemical equations. The molecules are:

$CO_2$, $C_7H_{10}N$, $H_2O$, $O_2$ and $NO_2$

I am confused on how to find all the possible combinations of reactions between these molecules. And there are certain things I don't know how to account for such as, you can't just have $O_2$ on one side of the equation and everything else on the other side.

If i knew how to find all the different possible reactions I technically know how to balance them individually; setting up a system of linear equations and solving that. maybe there is a more general way of balancing them all at once?

Thanks, any help would be appreciated!

• To avoid confusion you should post the exercise word by word. – callculus Apr 21 '16 at 11:16
• This is chemistry related, so you could ask in chemistry.stackexchange.com. Since I am a chemist, I can give you a hint: combustion. That should give you the only allowed chemical reaction with those molecules. – Zamu Apr 21 '16 at 16:40
• This question should not be closed, since it admits a solution that is a nice application of linear algebra. Given that in any balanced chemical reaction the number of atoms of each element must be the same in the reactants and the products, we get a homogeneous system of four equations (4 elements in the molecules) with five variables (the stoichiometric coefficient for each molecule). In this particular case, the rank of the coefficient matrix is 4, so we get infinite solutions parametrized by a free variable. Chemically, this means that there is only one (and its reverse) allowed reaction. – Zamu Apr 21 '16 at 17:10
• I think this got closed because the chemistry was not expressed as math. We can represent these molecules by vectors where the components of the vector represent the number of H, C, N and O in each. Thus, (0,1,0,2) is CO2, (10,7,1,0) is C7H10N, (2,0,0,1) is H2O, (0,0,0,2) is O2, and (0,0,1,2) is NO2. A balanced chemical equation is a linear combination of these vectors with a sum of zero. How many different balanced chemical equations exist for this set of molecules? – Karsten Theis Mar 23 '19 at 17:28
• The answer is given here: chemistry.stackexchange.com/questions/111375/… – IV_ Apr 19 '19 at 12:46