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hence the rate of reaction) are not themselves order is then the same as the molecularity.For
measurable, provided it is possible to measure a stepwise reactions there is no general connec-
chemical flux. For example, if there is a dynamic tion between stoichiometric numbers and partial
equilibrium according to the equation: orders. Such reactions may have more complex
rate laws, so that an apparent order of reaction
aA pP may vary with the concentrations of the chemi-
cal species involved and with the progress of the
and if a chemical flux is experimentally found,
reaction. In such cases, it is not useful to speak
(e.g., by NMR line-shape analysis) to be related
of orders of reaction, although apparent orders
to concentrations by the equation
of reaction may be deducible from initial rates.
α
φ−A/α = k[A] [L] λ In a stepwise reaction, orders of reaction may in
principle always be assigned to the elementary
then the corresponding reaction is of order α with steps.
respect to A,... and of total (or overall) order
n(= α + λ + ...). The proportionality factor ordering (of a set) (1) Preordering: a rela-
k above is called the (nth order) “rate coeffi- tion on a set A such that
cient.” Rate coefficients referring to (or believed (i.) ∀a ∈ Aa a;
to refer to) elementary reactions are called “rate
(ii.) ∀a, b, c ∈ Aa b, b c ⇒ a c.
constants” or, more appropriately, “microscopic”
(hypothetical, mechanistic) rate constants. (2) Partial ordering: a preordering such that
The (overall) order of a reaction cannot be
(i.) ∀a, b ∈ Aa b and b a ⇒ a = b.
deduced from measurements of a “rate of appear-
ance” or “rate of disappearance” at a single value (3) Total ordering: a partial ordering such that
of the concentration of a species whose concen-
(i.) ∀a, b ∈ Aa b or b a.
tration is constant (or effectively constant) during
the course of the reaction. If the overall rate of Examples: The inclusion is a partial ordering
reaction is, for example, given by in the power set P(X) of a set X. The relation ≥
is a total ordering in the real line R. The relation
α
ν = k[A] [B] β z w if and only if |z|≥|w| defined on the com-
plex plane C is a preordering but not a partial
but [B] stays constant, then the order of the reac- ordering.
tion (with respect to time), as observed from
the concentration change of A with time, will oregonator R.M. Noyes and R.J. Field at
be α, and the rate of disappearance of A can be the University of Oregon developed a mathemat-
expressed in the form ical model, consisting of three coupled nonlinear
ordinary differential equations, for the BZ reac-
α
ν = k obs [A] . tion (see Belousov-Zhabotinskii reaction). The
A
model was shown to have a limit cycle, hence
The proportionality factor k obs deduced from
such an experiment is called the “observed rate firmly established the theoretical basis of chemi-
cal oscillation (cf. R.M. Noyse, J. Chem. Educ.,
coefficient,” and it is related to the (α+β)th order
66, 190, 1989).
rate coefficient k by the equation:
β
k = k[B] . orientation (of an m-dimensional manifold M )
obs
A nondegenerate m-form over M.If (M, g) is a
1
For the common case when α = 1, k obs is often (pseudo)-Riemannian manifold and ds = dx ∧
m
2
referred to as a “pseudo-first-order rate coeffi- dx ∧···∧dx is the canonical local basis of m-
√
cient” (k ). forms over M, then η = gds is an orientation.
ψ
For a simple (elementary) reaction a partial The manifolds that allow orientations are
order of reaction is the same as the stoichio- called orientable, and they allow oriented
metric number of the reactant concerned and atlases, i.e., atlases with transition functions with
must therefore be a positive integer. The overall definite positive Jacobians.

© 2003 by CRC Press LLC
© 2003 by CRC Press LLC