Hi Meer:
I think the key is this, the text says: "the coefficients corresponds to those without reaction, only mass transfer with reference to the whole distance x0".
First lets review some fundamentals. Lets think in a simplified case, in which diffusion described by Fick law will be applicable, you would obtain for the flux (N):
N = D*CAi/x0
with D: diffusivity, CAi: Concentration at interphase, x0: width of the liquid film, we are assuming here that the bulk concentration is zero for simplicity. Now, the text would introduce a mass transfer coefficient, k, and say:
N = k*CAi
and obviously k=D/x0, the last expression valid for the most simple case. Next the text will explain that in other circumstances k is not simply D/x0, so you could measure k, say experimentally, make correlations, etc.
Now for the case in the picture you provided, you will have (Nreacc: flux when a reaction is ocurring):
Nreacc=D*CA_i/x
Please note that I used x, the effective distance A component is travelling, now remember what the text says: "the coefficients corresponds to those without reaction, only mass transfer with reference to the whole distance x0", I want to express the flux in terms of k without reaction, I don't want this:
k_reacc = D/x (Note the "divided by x")
I want this:
k_NOreacc= D /x0 (Note the "divided by x0")
I achieve that by rearranging the terms:
Nreacc=D*CAi/x
=D*CAi*x0/(x*x0)
=(D/x0)*CAi*x0/x
=k_NOreacc*CAi*x0/x
So, long story short, the text is trying to express the flux when a reaction is happening, in terms of the mass transfer coefficient measured when NO reaction is happening, x0/x turns out to be a scale factor, same goes for the factor:
x0/(x0-x)
in the expression for the flux of B.
Best regards.