I have recently discovered a chemical equation that can be balanced in infinite non-proportional ways with some restrictions:
C4H10 + aO2 = bC + cCO + 5H2O
Being the variables as follows:
2.5 < a < 4.5
b=9-2a
c=2a-5
So technically, that yields an infinite amount of solutions (fractions can be used too) for this equation. Examples:
C4H10 + 3O2 = 3C + CO + 5H2O
C4H10 + 4O2 = C + 3CO + 5H2O
C4H10 + 7/2 O2 = 2C + 2CO + 5H2O
And so on... Do any of you know more?
[Edited on 11-6-2013 by Eddygp]woelen - 11-6-2013 at 07:03
This is nothing special at all, but the underlying mathematics is fairly advanced and beyond high school level (linear algebra, concept of null-space,
kernel of matrix). Let me try to introduce some theory without using concepts of linear algebra.
Many equations in chemistry have a so-called one-dimensional solution space. This means that these equations have one degree of freedom. An example is
2H2 + O2 --> 2H2O
You can use any factor, however, so 4H2 + 2O2 --> 4H2O or 0.5 H2 + 0.25 O2 --> 0.5 H2O all are valid. The general form is 2x H2 + x O2 --> 2x
H2O, where x is a free to choose parameter.
There are reactions which can be written as a linear combination of two more fundamental reactions. Your reaction is an example of this.
There are two fundamental reaction equations:
(eq. a) 2 C4H10 + 5 O2 --> 8 C + 10 H2O
(eq. b) 2 C4H10 + 9 O2 --> 8 CO + 10 H2O
Your reactions can be any linear combination of these two equations: write x*(eq. a) + y*(eq. b) on both sides of the arrow and you see that you get
your example. A nice exercise for you would be to find x and y, such that they satisfy your equation.
Other examples of even higher dimension are:
- Combustion of C4H10 with O2, giving C, CO, CO2 and H2O. This has a 3-dimensional solution space. Try to write down the three basic reactions.
- Reaction of copper with nitric acid to give Cu(NO3)2, NO, NO2 and H2O. This has a 2-dimensional solution space.
This is a piece of software, which I have written. It identifies the dimension of the solution space and a nice description of this mathematical
phenomenon is given in the tutorial.
There are also reactions, which have a 0-dimensional solution space, although the same elements on both sides are present. An example of such a
reaction is
... PCl5 + ... H2O --> ... H3PO3 + ... HCl
If you try to balance this, then you'll find that it is not possible, except for a single solution:
0 PCl5 + 0 H2O --> 0 H3PO3 + 0 HCl
Such reactions cannot occur in reality. My program also detects that kind of reaction equations.
In more technical terms, my program derives an integer matrix equation of the form Ax = Bx, where A and B are matrices depending on the reactants and
products used. The vector x is a vector of coefficients in front of the reactants and products. The program then uses an integer decomposition method
to find the null-space of the matrix A - B, i.e. it determines the set of possible values for x, such that (A - B)x = 0. For the majority of chemical
reactions, this set is 1-dimensional. For the example you provide, this set is 2-dimensional. For combustion of butane to C, CO, CO2 and water, this
set is 3-dimensional. The mathematics behind the solving of this equation in integer space is quite advanced and is beyond this short post over here.
[Edited on 11-6-13 by woelen]bbartlog - 11-6-2013 at 07:06
Starved of oxygen
Butane burns with sooty flame
Give it some more airPickledPackratParalysis - 11-6-2013 at 07:12
This is nothing special at all, but the underlying mathematics is fairly advanced and beyond high school level
Not really advanced, but it does take a long time to write them out and do the substitution.
I use to balance these complex equations for fun. With organic oxidation, there is really not much for a point, though, because so many products are
possible.
Just an imaginary reaction for practicing equations:
Do any of you know more?
)