We're going to look at doing kinetic analysis using differential analysis for batch reactor
data. And in this case, we'll be looking at a simple reaction A plus B goes to C plus
D, as shown here on the screen. The initial concentrations of A and B will be set to be
equal. In particular, they're set to their stoichiometric ratios. So it's 1 to 1 stoichiometric
ratio, and we're therefore going to run this experiment in a 1 to 1 ratio. And we've seen
in class that this can sometimes give us some useful information. Alright, if we start them
with a stoichiometric ratio, they will always be at the stoichiometric ratio, because whenever
A is consumed, B is consumed at an equal rate. And C is formed at an equal rate to which
A is consumed, so we can write an expression like the one here. And what we're interested
in is determining, using these experiments, some kind of functional form for the rate
of reaction. And here we'll postulate that we have what's called the power rate law,
in which we can raise the concentrations of the reactants to some power, in this case
alpha for A and beta for B, multiply them by an Arrhenius type rate constant, and that
will give us a good measure of the rate. But in many cases, as we'll see in particular
later in this course, we might be looking to fit a different kind of rate expression.
So, for example, Michaelis-Menten or Langmuir-Hinshelwood type rate expression. The differential analysis
we'll talk about today is kind of uniquely helpful when you are trying to use a power
rate law model. What that analysis will allow us to do is take numerical derivatives of
some data in order to determine the reaction orders and rate constants, at least in an
approximate sense. OK, the last thing we'll make use of in looking at this data are the
relationship that every mole of A we consume, we form a mole of C, and in a constant volume batch
reactor, that lets us write this expression in terms of the concentrations. OK, and then
finally what we're going to need to do in the batch differential analysis is we'll just
take the log of both sides of this expression. So we take the log of the rate, which is either
-dCA/dt or dCC/dt, and then the log of this expression. And what I'll note first, is that
since CA is always equal to CB, we can just lump the terms under CA to the alpha plus
beta power. And then finally, I'm going to call this just k CA just to the gamma power, where
gamma is defined as being alpha plus beta. And so this experiment will allow us to determine
the value of gamma. So the total reaction order. OK, and so if we take the log of this
side of the equation, then we get the log of k plus gamma times the log of concentration,
since the log of a product is equal to the sum of the logs, and we can take this power
down in front of the log. OK, so now we're ready to look at the data and try to use this
kind of relationship. What we're going to do is try to plot this versus log CA and fit,
using a linear relationship, gamma and the log of k. This is most easily done
in Excel, and I have a very simple set of data here to illustrate this process. So as
a function of time, there have been measurements conducted using the set up that have measured
the concentration of C, one of the products, as a function of time. And from that, we can
determine the concentration of A based on a mole balance, assuming we started with 0.5
moles per liter of A as we described on the previous case. And so we just have some cells
that define this. And now we want to do a differential analysis. And so the x term in
a differential analysis, when we take the log of both sides, is just CA, and so we take
the log of CA. The most difficult thing about a differential analysis is estimating a numerical
derivative. Alright, I've used the crudest possible form of a numerical derivative, which
is just a difference formula. So I'm taking delta CA between two points, divided by delta
t between two points in order to approximate the derivative. OK, this is - there are certainly
much more accurate methods that incorporate, that don't assume this linear relationship,
essentially, between the two points. So some would assume a quadratic relationship. A lot
of the most useful methods, it's useful if you do your experiment in a different way. So, for example, taking
data points at regular time intervals makes some of those methods function more easily.
And so I'm going to just use this crude method with an understand that it will give us an
initial estimate for the reaction order. And then in a subsequent screencast, we'll look
at how to get a better number for the reaction order and the rate constant. OK, so I'm just
taking this difference method, so you can see that the values of my cells just involve
dividing the difference in the concentration of A by the difference in the time. And that
gives us a dCA/dt value. And then we just take the log of these cells. OK, and then
all I've done, that we'll look at in just a minute, is a plot of the log of dCA/dt versus
the log of CA. OK, and so that plot, we see that the slope, we get a good, a fairly good
linear fit. You can see there's some curvature that's actually a consequence of the fact
that we are using a very crude estimate to our derivatives. But if we fit a straight
line to that, we get a slope of 1.93. And so that is equal to our gamma value. And we
get an intercept of -8.33, and that's equal to the log of k. And so therefore, if the
log of k is equal to -8.337, then k, we just take the exponential of that number, and get
a value of 0.00024. And that's in the units we've measured in terms of time and concentration.
OK, so we have to determine, the units of time will be in seconds, and the concentration
we have to work out by balancing the units of our rate expression. OK, so we see this
is an A plus B reaction. If it's elementary, we expect for gamma, for the total reaction
order to be 2. Here we get it to be 1.93 using a pretty crude estimate. Given the likely
error in our sampling data and the crudity of this estimate, then we would probably assume
that it's - to the best that we can determine that reaction order is 2. We'll just mention
that, in the next screencast, we'll be looking at another way of determining our constant.
Come back here, we just determined that gamma is approximately equal to 1.92. And we've
determined that k is equal to approximately 2.4 times 10 to the -4. OK, and the units
on k will need to be such that we get a total units of moles per volume per time from this
lumped term. OK, and so this would, in effect, that'd mean it's like this, where we have
liters to the 0.92 over moles to the 0.92, seconds. Then when we multiply it by a moles
per liter raised to a 1.92, then we'll get the appropriate cancellation.