OK, so this problem is a follow-up on a problem that we considered for a batch reactor using
differential kinetic analysis in order to determine the overall reaction order, gamma,
and the reaction rate constant, k. And in this screencast, instead of using differential
analysis, or purely differential analysis, we're going to take the information that we
got from the differential analysis, which was that gamma was approximately equal to
1.93, I think, and k was equal to 2.4 times 10 to the -4, and use those as initial guesses
in a nonlinear regression in order to solve for the rate. And so we'll see how to do this
fairly simply in Excel. If you have a very complicated kinetic expression then Excel may not
be the best way to go and you may want to use a more robust modelling package. But Excel
should work fine for this simple power rate law type expression. So let's go over to Excel.
What I've done is I've taken the same data that we looked at last time. So we had as
a function of time, various concentrations of C that we had measured. By a mole balance,
we determined the concentration of A, and I've just pasted that over here. OK, and then
what I've done is I've put in a predicted value for CA. And so this is one that's calculated
based purely on a model. And so I have formula in here that predicts CA for a particular
model. So let's take look at this, OK, I've taking this expression that we have - and
here, and just for simplicity I've raised it to the n instead of the gamma - I have
dCA/dt is equal to negative kCA to some power. That's the power I want to solve for. And
we can actually write a general formula for this value of the concentration of A in terms of k and
n, which are the two things we're trying to solve for. And so what I can do is take this
formula and put it straight into the column here, alright, which is what I for the predicted
value of CA, and then reference that wherever I need k and n to values that we're going
to change using the Solver routine in Excel. OK, and so the values that I've initially
set are the values that we obtained from out differential analysis of 0.00024 and 1.93
for k and n, respectively. And then what I do, is I compute the predicted concentration
of A at those values, I take the difference between what's predicted from the equation
and what was obtained from the experimental data, and square that difference. And I do
that for every data point, and then I sum the squared errors. So this gives me the squared
error for each data point, I sum those, and this gives me something that we want to try to
minimize using Solver. OK, so we can see it's already pretty low because we've made a good
initial guess based on our previous solution to this problem, but this should give us a
better model fit since we aren't having to make any calculations of numerical derivatives.
So let's go ahead and use the Solver routine, and what we're going to do is we're going
to set the target cell, the sum of squared errors, equal to a minimum in order to get
our best fit. And we'll do that by changing the two cells that control our model fit,
which are k and n. Alright, so those will be changed. And then we're going to set this
equal to the value of the target cell. And then, subject to the constraints that k, for
example, should be a positive number, and we can probably think of some other constraints
that we put in here - and this becomes a lot more important if you don't have a very good
initial guess or if you have a more complicated model - but for this two parameter model we
should be OK. OK, and Solver has converged on a solution with all constraints satisfied.
So we'll keep that solution. We see that we got a little bit of a difference in values.
Now n is even closer to 2, so 2.02, and k decreasing a little bit. That results in a
very low sum of squared errors. And so now we know that we have a good fit, a good second order fit
for the reaction rate. OK, now, if you recall, - OK, one thing we won't talk about in this
screencast but we'll definitely talk about later on in another screencast, alright, is
we've just estimated gamma and k, alright, but we can't, we have to do another experiment in order
to actually estimate alpha and beta. Alright, we can't do that from this single experiment.
Alright, so one kind of experiment that we typically would run in addition to this one
would be to run with one component in great excess of this as constant. And that will
allow us to estimate the reaction order of the other component, and once we know the
total reaction order and the reaction order of one component, then we can determine the
others by this addition. OK, so we'll explore this later, but I just wanted to point out
that we will need additional experimental data that we'll analyze using various similar techniques
so that we can analyze with differential analysis or with this minimization of the sum of squared
errors that we've just talked about in order to determine the individual values of alpha
and beta.