In this problem, we're given the concentration of a reactant inside of a porous catalyst
sphere for three different catalysts.
Same diameter, but the concentration profiles are significantly different, and the question
is, which one of these catalysts is going to react the largest amount of reactant in
a given time?
It's the same reaction, and we also require to indicate any assumptions.
So there's not really sufficient information to answer this.
We can make some assumptions, and depending on which assumptions we make, we can end up
with different answers.
So one assumption is that the rate constant is the same.
Well if we make this assumption, then it's easy to determine that catalyst 1 reacts the
most, because the amount that reacts is going to be the rate constant times the concentration
inside the catalyst, and since the concentration is changing we integrate over the volume of
the catalyst.
Each catalyst has the same volume, so if the rate constant is the same, the one with the
highest concentration is going to have the highest rate, and so that's catalyst 1.
The other extreme would be to assume that the diffusivities are the same for each of
the catalysts.
This means the rate constants are different, and now the catalyst with the highest rate
is going to be catalyst 3 because the gradient, diffusivity times the change in concentration
of A with respect to radius evaluated at the external surface, evaluated here, for how
much is diffusing into the catalyst because whatever diffuses in reacts.
If the diffusivity is the same then the one with the largest gradient, which is catalyst
3 has the largest gradient here, and therefore the most is diffusing in, and so this would
say the k3 is larger than k2, larger than k1, and therefore the most that reacts now
is in catalyst 3.
So you can see, just from these concentration gradients comparing different catalyst, we
need more information to reach a conclusion as to which is reacting the most material
per time.