In this video, I will demonstrate how, using matrix calculations, we can determine reaction
rates from measurements of production rates using a least-squares approach.
This is explained in more detail in the previous video on this topic.
We're looking at three reactions where small r indicates the rate of that individual reaction,
then we use the matrix notation, where the intense black notation indicates a matrix,
and we can write these three reactions now in terms of the matrix notation as a stoichiometric
coefficient times the matrix A, where the stoichiometric coefficient has the reactants
negative and the products positive, then the rates of reaction for the individual species
is the transpose of that stoichiometric coefficient times the rate of the individual reaction.
For example, let's suppose that we were able to know r1 was 1.2, and we won't worry about
units here, it could be moles per liter per minute or per second, so r1, r2, and r3 are
known at some conditions.
We can then calculate the R matrix by substituting r1, r2 and r3 here, and we would calculate
the R matrix as being equal to -1.2, 0.4, 0.3, 0.5, and -2.5.
This is the rate of A, the rate of B, C and rate of H, so A is disappearing, H is disappearing
net, B, C and D are increasing.
They start out, more is being made than is used up for these species.
In reality, these are the things we would measure in the laboratory.
We would measure concentrations as a function of time for individual species and calculate
rates from that data, and question is, if we know these rates, can we go back and calculate
what the r value is.
To do that, let's assume we have the measurements of the rates of individual species, but we're
also going to assume that there's some error, as there always is in these measurements,
so I'm going to use these values, values where I've included some error, and the value is
not identical to values that we know if we were to accurately be able to measure these,
and as a result, the stoichiometry of these species, which are interrelated, are not going
to be exactly satisfied.
So if we want to determine r, the least squares approach is to take that stoichiometric coefficient,
take its transpose, multiply those together, take their inverse, multiply by the stoichiometric
coefficient, and multiply this matrix R of the measured values that we have here.
And we've done this calculation using and Excel spreadsheet.
So I've written down the stoichiometric matrix and its transpose, multiplied them together,
and then taken the inverse.
This then is that inverse.
This is a 3x5, this is a 5x3 matrix, so we end up with a 3x3 matrix.
So once we have this inverse, we multiply it again by the stoichiometric coefficients,
and then the measured rate matrix.
So here's the inverse again, stoichiometric coefficient, this is a 3x3, this is a 3x5,
we multiply, and then this is a 5x1, so we end up with this 3x1 matrix, where these are
the calculated rates.
So this is calculated rate for r1, r2 and r3 from our measurements, which had some errors,
and originally to generate the measurements that we then added errors to, we started with
rates of 1.2, 0.8 and 0.5, so you can see the errors in those measurements introduce
some errors into our calculated rates, but this is the best fit of the data using a least-squares
method.
These matrices were all generated, the inverse, the transpose, the multiplication was all
done in Excel using the matrix notations there.