F. Xavier Malcata - Mathematics for Enzyme Reaction Kinetics and Reactor Performance

The question still remains as how to calculate the Α ’s in Eq. (2.205); to do so, one should start by multiplying both sides by (or , for that matter), thus generating

(2.216)

After splitting the outer summation, Eq. (2.216)becomes

(2.217)

one may further write

(2.218)

if the extended products are, in turn, splitted as and . Equation (2.218)may be rewritten as

(2.219)

after lumping powers in the argument of the first summation, or else

(2.220)

following explicitation of the independent term in the first summation; since Eq. (2.220)is universally valid, it should hold in particular when x = r 1– in which case one obtains

(2.221)

that breaks down to

(2.222)

due to r 1− r 1being nil, as well as any (significant) power thereof. Isolation of Α 1,1in Eq. (2.222)finally unfolds

(2.223)

A similar reasoning may be followed with regard to any other root r k– provided that x − r kis singled out in Eq. (2.216)instead of x − r 1as done in Eq. (2.217), and eventually setting x = r k; one accordingly finds

(2.224)

In attempts to determine Α 1,2(should s 1≥ 2), one may to advantage differentiate both sides of Eq. (2.220)with regard to x so as to obtain an independent relationship, viz.

(2.225)

– by resorting to the rule of differentiation of a product; the said rule, coupled with the rule of differentiation of a sum, allows subsequent transformation of Eq. (2.225)to

(2.226)

The rule of differentiation of a power may now be invoked to transform Eq. (2.226)to

(2.227)

after setting x = r 1again, Eq. (2.227)becomes

(2.228)

which dramatically simplifies to

(2.229)

due to the nil value of r 1− r 1and any significant powers thereof. After splitting as the ratio of to r 1− r r, and lumping the former to the extended product, Eq. (2.229)will take the form

(2.230)

– where may, in turn, be taken off the summation to yield

(2.231)

for being independent of r as counting variable; A 1,1may now be eliminated via Eq. (2.223), viz.

(2.232)

where cancelation of identical extended products between numerator and denominator gives rise to

(2.233)

Isolation of Α 1,2from Eq. (2.233)is finally in order, according to

(2.234)

generalization then ensues as

(2.235)