This problem deals with the analysis of some batch kinetic data for the reaction that involves
A+2B irreversibly going to some products. The initial concentration of B is twice that
of A, and a measurement of the concentration of A is made with respect to time. When the
reciprocal of that concentration is plotted versus time, we get this straight line relationship
and based on that, we have these competing claims on what that means in terms of the
reaction order with respect to component A. And that is equal to the rate at which A accumulates
or the negative of the rate with which it's consumed. So that's the rate constant times the concentration
of A to some power alpha that we're going to try to evaluate and the concentration of
B to some power beta. The trick in this problem, if you can call it that, is that the concentration
of B initially is equal to double the concentration of A and in fact, at all in times in the reactor,
it's going to be double because this is a stoichiometric reaction and so every time
we consume a mole of A, we'll consume 2 moles of B so that we are always, we can write this
relationship as being generally true. Alright and so because of that, we can actually replace
the concentration of B with 2 times the concentration of A. We then can collect terms in that expression,
and I'll write that down here, that now our differential equation reads that dCA/dt is
equal to -k*2^beta*CA^(alpha+beta). And so we see that we have this alpha+beta term.
You can analyze this a few different ways but anticipating what the answer is, we're
going to assume that alpha+beta is equal to 2. So this represents a way of doing the integral
analysis with batch kinetic data, where we're going to pick a reaction order and then see
if we were correct basically in terms of the experimental data we observe. Okay, so if
alpha+beta is equal to 2, and then we can separate out these two sides of the differential
equation, and on this side we get dCA/CA^2 and on the other side, we'll get -k*2^beta, again times time
because we'll have the dt over there but these two terms do not depend on time, they're just
constants with time, and also constants with respect to concentration, and so we just get
an integral of dt which gives us time. On this side, we're going to evaluate from CA0
to CA. And if we do that evaluation, then we'll simply get a 1/CA-1/CA0. Okay, and so
we see that if we plot the concentration of A, or the reciprocal of the concentration
of A, versus time, then we should get a straight line relationship where this would be the
slope and our intercept would be determined by the reciprocal initial concentration. But
this doesn't mean that we have an order with respect to A of 2, all it means is that we
have our total order is 2 and we actually can't tell whether the order in A is 1 or
2 or zero or some other number and so we actually have to do another experiment where the feed,
the starting material is not in stoichiometric concentration ratio, for example with one
of the components in great excess, in order to determine what the reaction order is with
respect to the individual components. Here, we can only determine what the total reaction
order is.