In this video, I'm going to describe reaction rates and production rates when we have multiple
reactions, and the idea is, we want to demonstrate how to use a stoichiometry matrix to relate
the reaction rates, namely the rate of one reaction, in this case for the example here,
three reactions, r1 would be the reaction rate, to production rates.
Production rates, for example, would be the rate that C forms, or if it's negative it's
decomposing, but the rate that C or the rate that D forms.
So we have 5 rates of formation of individual reactants, and we have three reaction rates.
And then the second objective is, using again matrix calculations, calculate the rates of
reaction, calculate r1, r2, and r3, if we have measurements of the rates of A, B, C,
D and H, using a least-squares approach to account for the fact that these measurements
can have errors, and so the stoichiometry is not going to be perfectly satisfied because
of that.
So we can write the rate that species A forms, and that's minus r1, minus sign because it's
a reactant.
The rate of B, well B is involved in 2 reactions, so r1 minus r2, and likewise, C is involved
in two reactions, r2 minus r3.
D is only involved in one reaction, and then H is involved in all three reactions as a
reactant, so its stoichiometric coefficient is negative in each case.
So we're also going to define first the matrix.
I'll call this A, and I'll use in this video, the black, more intense symbols to represent
a matrix, and then this matrix is to just represent the species that are involved in
the reaction, so a 5x1 matrix, and then we want to also define a second matrix, a matrix
nu, where nu times A is going to be equal to zero.
This represents the reactions that we've written, and the easiest way is to write down what
nu is to make it clear what we're referring to.
So nu is a matrix, so this first line represents the first reaction up here, where what I'm
going to represent, this is A, B, C, D and H. So A has a stoichiometric coefficient -1,
B has a stoichiometric coefficient +1, C and D are not involved in the reaction, H also
has a stoichiometric coefficient of -1.
Using that same idea, I can write down for the second reaction, and then for the third
reaction, the stoichiometric coefficients, so this matrix nu times the matrix A equal
to zero, which is equivalent to writing these reactions with everything on the right side,
minus A minus H plus B equals zero.
So reactants have stoichiometric coefficients of minus 1, products have stoichiometric coefficients
of +1.
In this case, there's no 2 or 3 stoichiometric coefficients.
If we want to know what these production rates are, we can define a matrix R for the production
rates, and this is just these five rates that I've written the equation for.
That's the matrix for the production rates.
That's going to be equal to a matrix that's the transpose of v times a matrix that is
the rates of r1, r2, and r3.
So I've written now the rates of production in terms of the transpose R1, and so we can
write this as then that matrix for the rates of production, stoichiometric coefficient
transpose, and matrix r for the individual reaction rates.
This matrix arises, let's look at the rate of C, so it's 0 times r1, 1 times r2, -1 times
r3, and indeed that's the rate that we wrote up here for rC, and we can do it equivalently
for the others.
And what we'd like to use this for is if we measure these values and want to determine
these values, so we measure RA, RB, RC, RD and RH at some conditions, we want to know
the corresponding values for r1, r2, and r3, and of course these measurements have some
errors in them, and that means they won't satisfy the stoichiometry perfectly because
of the errors, so we want to do a least-squares fit to get to the best value.
So in the next screencast, we'll go through the calculations of how to use this matrix
notation to do this least-squares fit and get the best answers for measurements that
have errors, which of course all measurements do.