On the other hand, any vector also determines a ‘quantity’ (possibly conserved, possibly not) that assigns a number to any given choice of concentrations. It does so using the dot product: if is a choice of concentrations, we get a number

Thus, if is a vector in the stoichiometric subspace, which describes a possible *change* in concentrations, will be the corresponding change in the quantity corresponding to

If is orthogonal to the whole stoichiometric subspace, this change will always be zero, so the quantity is *conserved*.

Thus, the orthogonal complement of the stoichiometric subspace is the space of conservation laws.

It follows that the dimension of the stoichiometric subspace plus the dimension of the space of conserved quantities equals

In other words, the maximum number of linearly independent reaction vectors plus the maximum number of linearly independent conservation laws is .

In the example at hand this equation amounts to 3 + 4 = 7.

This stuff is part of reaction network theory. I think this is a pretty good, though long, introduction:

• Martin Feinberg, *Lectures on Chemical Reaction Networks*, 1979.

Any change in concentrations caused by these reactions must lie in the stoichiometric subspace: that is, the space spanned by the reaction vectors. Since these vectors obey one nontrivial relation:

[v_\alpha + v_\beta = v_\gamma + v_\delta]

the stochiometric subspace is 3-dimensional. Therefore, the space of conserved quantities must be 4-dimensional,

I see why the stochiometric subspace must be 3-dimensional, but I am not sure why the conserved quantities must be 4-dimensional. What determines the number of conserved quantities given a set of reaction vectors?

]]>(In our example from last time we didn’t keep track of conservation of hydrogen and charge, because both and ions are freely available in water… but we studied the citric acid cycle when we were younger, more energetic and less wise, so we kept careful track of hydrogen and charge, and made sure that all the reactions conserved these. So, we’ll have 7 fundamental conserved quantities.)

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