What is the difference between differential yield and selectivity?

Question

I have a hard time understanding the difference between the differential yield and overall selectivity of a reaction.

Levenspiel, Chemical Reaction Engineering writes:

For convenience in evaluating product distribution we introduce two terms, p and a. First, consider the decomposition of reactant A, and let cp be the fraction of A disappearing at any instant which is transformed into desired product R. We call this the instantaneous fractional yield of R. For any particular set of reactions and rate equations cp is a function of CA, and since CA in general varies through the reactor, p will also change with position in the reactor: $ \varphi = \frac{-dC_A}{dC_S}$

and for selectivity:

The Selectivity. Another term, the selectivity, is often used in place of fractional yield. It is usually defined as follows: $$\text{selectivity} = \frac{\text{moles of desired product formed}}{\text{moles of undesired material formed}}$$

So how do they differ? How can I imagine both?

Aside from that, they also provide some nice plots:

One question I have: How can I imagine the selectivity for a MFR (gray area) crossing the selectivity curve? The formula for the integral yield for a PFR/MFR make sense, but how do I imagine that in a practical sense?

Also: What does the curve describe?

Answer 1

First, these two concepts as you may already know are useful for describing various chemical reactions taking place or multiple reactions. Why Levenspiel wants you to know these concepts? Because you are interested, typically, in one product, not all of them.

As an example, consider the following reaction scheme, where you are interested in B; and we assume power rate laws for both of them:

$$ A \rightarrow B \hspace{1 cm} r_1 = r_{1,B} = - r_{1,A} = k_1C_A^2 $$

$$ 2A \rightarrow C \hspace{1 cm} r_2 = r_{2,C} = -r_{2, A} = 2k_2C_A^3 $$

The fractional yield of $A$ and selectivity are

$$ \phi_{B/A} = \dfrac{r_{1,B}}{-(r_{1,A} + r_{2,A})} = \dfrac{k_1C_A^2}{k_1C_A^2 + 2k_2C_A^3} $$

$$ S_{B} = \dfrac{n_B}{n_C} $$

So now we can answer your first question:

1. The fractional yield gives you, as a function of the concentration of species $A$, the rate of formation of the species you want with respect to the rate of consumption (with a negative sign) of the reactant $A$. As an engineer, what do you want to do? Maximize this quantity once we give you a type of flow within a reactor. This does not mean maximizing $ B $, but will give you the best exchange between the reactant $ A $ and your valuable product $ B $.
2. The selectivity gives you, as a function of time or position, the amount in moles formed of the species you want with respect to the amount of moles of the species you do not want. As an engineer, what will they ask you? Minimize this quantity once we give you a type of flow within a reactor. As before, this does not mean that you will generate a lot of $ B $, rather, it will give you the best exchange in terms of generating $ B $ and not generating $ C $.

Levenspiel states that this definition is not useful, because it may be hard to quantify in moles all of the species that you don't want (imagine 1 valuable product and 19 not valuable products). However, a more useful mathematical definition would be the one I give you below, analogous to the yield

$$ S_{B/C} = \dfrac{r_{1,B}}{r_{2,C}} = \dfrac{k_1C_A^2}{2k_2C_A^3} $$

Lets go now to the second part. The curves you are seeing there are indeed the fractional yield vs the concentration of $A$, i.e., $\phi_{R/A}(C_A)$ vs $C_A$. They only depend on the kinetic rate laws governing the reaction scheme. However, in this case, thay are so simple that they only depend on the concentration of only one reactant. Depending on the mathematical form, they may have a maximum, or not, or have rare behaviours (imagine how the rate law may dramatically change with a catalyst, etc.). In addition, they also give you the concentration of $R$, i.e., the species you want. For a MFR, it is a rectangle of base $ C_{A0} - C_A $ and height $\phi_{R/A}(C_A)$. For a PFR, it is the area under the curve bounded between $ C_{A0} $ and $ C_A $, as you said.

The MFR can't operate passing the curve, because its type of flow is such that its operating point is the exit; such that the whole reactor "exists" as a unique point in that curve.