I am going to explain the idea of reaction rate by looking at a couple of specific examples
to make clear what we mean when we get asked in this case, which reaction rate is higher.
So what I have plotted is the concentration reactant A as a function of distance down
a plug flow reactor. So isothermal reactor. It is the same reactor for these 2 conditions.
And the question is if we have a second order reaction. Where the rate is proportional to
a rate constant and a concentration of reactant raised to the second power. We want to know
which reactor has the higher rate of reaction. So these are the same size reactors, and if
we look at which reactor has the highest rate of reaction. We have to make clear that we
are not looking at which reactor has the highest conversion. So reactor 1 has the highest rate
of reaction. That is because the rate of reaction minus k(CA)^2 or if you like the rate that
we are forming the product B is k(CA)^2. So this is the rate per volume. It is normally
defined in kinetics/reactor design textbooks, and what we are interested then in the entire
reactor the rate would just be for formation of B. Would just be this integral over the
reactor over the volume or I could write this integral over the distance times the cross-sectional
area of the reactor. Since from this diagram CA1 is greater then CA2, then reactor 1 has
the highest rate of reaction. If the question is which reactor has the highest conversion,
so the highest conversion now is reactor 2. So the question might be how we obtain these
2 conditions, and if the same reactor in an isothermal.Then the molar flow rate of A entering
reactor 1 must be much greater then the molar flow rate of A entering the reactor 2, so
that a smaller fraction of the feed is converted even though the rate of reaction is higher
because the high flow rate, the percentage converted is smaller. So let's look at a similar
problem where we have diffusion and reaction within a porous catalyst pellet. The
idea being reactant diffuses in, some of it reacts, the rest continuous to diffuse
in, and we set up a steady-state concentration profile, which I have shown here for 3 catalysts.
Again we are looking at a second order reaction, and if we ask, which has the highest rate
of reaction, the answer is that catalyst pellet 1 has the higher rate of reaction, and it
is the same reasoning that the total rate of reaction within the pellet, that would
be an integral the rate of reaction per volume. Integrated over the volume. Of course we can
integrate this over radius for the appropriate values to get volume. Again because the concentration
of A in pellet 1 is greater then that in 2, greater in that than 3. Then highest concentration
means highest rate of reaction. Now if we first look at this we would say. Well the
concentration of 3 drops dramatically within this catalyst pellet. making us think that
we have reacted the reactant A faster in catalyst pellet 3. However, if we say they have the
same rate constant, and that is an important constraint, the same rate constant for each of these pellets,
then to obtain the result for catalyst 3 we must have the diffusivity of 3 as lower then
the diffusivity of 2, lower then the diffusivity of 1. For example the pores are much smaller
in catalyst 3 than in catalyst 1. So we have to be careful with what we are keeping constant,
but if they have the same rate constant then rate is proportional to concentration,
highest concentration means the highest rate of reaction.