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

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Название Mathematics for Enzyme Reaction Kinetics and Reactor Performance
Автор произведения F. Xavier Malcata
Жанр Химия
Серия
Издательство Химия
Год выпуска 0
isbn 9781119490333



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

      (2.234)equation

      generalization then ensues as

      (2.235)equation

      Although seldom of relevance, higher order constants, say Al,3, Al,4, …, may be calculated in a similar fashion – by resorting to higher order derivatives of Eq. (2.220), but accordingly requiring more cumbersome algebraic manipulations.

      2.2.5 Power

      According to Newton’s theorem, it is possible to expand the nth power (with n integer) of a sum of (real) terms, x and y, via a sum of a finite number of products of integer powers of x and y, with exponents adding up to n in every case; more specifically,

      The simplest case pertains indeed to n = 2, and accordingly reads

Image described by caption.

      – also known as another notable case of multiplication; however, a simple geometrical interpretation is no longer possible (due to the negative term). A similar proof may be constructed in three dimensions, corresponding to the volume (x + y)3 of a cube of side x + y; it may indeed be decomposed as

      (2.239)equation

      The binomial coefficients in Eq. (2.236), of the form images, count in how many ways one can pick up a subset with k elements out from a set of n elements in total; in mathematical terms, this is equivalent to writing

      (2.241)equation

      where factoring n! coupled with elimination of inner parenthesis give rise to

      (2.242)equation

      the factorials in denominator may, in turn, be rewritten as

      (2.243)equation

      based on their definition, thus allowing further factoring out of (k − 1)! and (nk)! as

      (2.245)equation

      that degenerates to

      (2.246)equation

      the outstanding factors may, in turn, be lumped with the existing factorials to yield