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

(2.529) turns to

      (2.530)

      – which can be condensed to

(2.531)

      at the expense of Eq. (2.473). Departure from Eq. (2.473) applied to x and y will instead lead to

      (2.532)

      where multiplication of the first by the second factor in the right‐hand side generates

(2.533)

after lumping exponential functions in the usual fashion, Eq. (2.533) transforms to

(2.534)

Algebraic reorganization of Eq. (2.534) gives rise to

      (2.535)

      which is equivalent to

(2.536)

due to Eq. (2.473). Ordered addition of Eq. (2.531) to Eq. (2.536) unfolds

      (2.537)

      that breaks down to

(2.538)

after condensation of similar terms; while ordered subtraction of Eq. (2.536) from Eq. (2.531) yields

      (2.539)

      – where negatives may be taken of both sides to get

(2.540)

Except for the change in sign, Eq. (2.538) matches Eq. (2.325), and Eq. (2.540) likewise matches Eq. (2.321).

      Insertion of Eqs. (2.524) and (2.538) transforms Eq. (2.482) to

      (2.541)

      with division of both numerator and denominator by cosh x cosh y allowing simplification to (the more usual) form

(2.542)

      again with the aid of Eq. (2.482); whereas divison of Eq. (2.526) by Eq. (2.540) supports

      (2.543)

      with division of both numerator and denominator by cosh x and cosh y justifying reformulation to

(2.544)

The similarity of Eqs. (2.542) and (2.544) to Eqs. (2.336) and (2.333), respectively, is again noteworthy – except for the change of sign in denominator.

      A common form of Eq. (2.542) arises after setting y = x, viz.

      (2.545)

      – which breaks down to

      (2.546)

      A similar procedure, i.e. setting y = x, allows transformation of Eq. (2.538) to

      (2.547)

      or, equivalently,

(2.548)

      elimination of sinh² x via Eq. (2.496) unfolds

      (2.549)

which leads to

(2.550)

after isolation of cosh² x. Equation (2.550) is often preferable