Reaction Rates and Production Rates: Part 1

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 5×1 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