Why is the expression of mass action law like that???1)
I mean like->
for a reaction
aA + bB ----> cC + dD
rate of forward reaction [R] = k[A]a[ B ]b
Nomenclature is not adequate. The "[R]" means concentration of R and above the units are (concentration / time) because the 'k' have units, therefore write just R, where R means 'Rate'.2)
It is not true that
the mass-action law is for chemical equilibrium.
This is a common error of modern chemical literature. The mass-action law was introduced in 1864 by Guldberg and Waage to describe their observation about the rate of chemical reactions.
R = k rho_1 rho_2
the rhos often were not equal to real masses (or concentrations) and therefore were named 'active masses'; therein the name of the law. Today, we know why active masses are not equal to real masses; it is due to activity effects in solution and today the above law is best expressed in terms of activities in chemical kinetics textbooks.
R = k a_1 a_23)
The usual expressions in terms of concentrations are valid only as approximation and the typical discussion in terms of binary collisions really valid for phase gas kinetics. This is the reason that chemical dynamics is really developed only for gas phase kinetics systems. Next, i cite some of approximations in usual
kinetic laws of chemistry:
- Valid only for macroscopic size matter
- Valid for constant volume systems
- Valid for 'slow' chemical processes
- Valid for systems in thermal equilibrium
- Absence of long correlations (e.g. strong gravitatory fields)4)
Why has then the mass action law of chemistry that generic form?
The explanation is via canonical science. See  for a first pionnering version,  for a students level discussion of some basic stuff and the non-technical page  for some recent advances.
The basic idea is as follow. The state of any system can be represented via a vector state (we use the same Keizer original nomenclature for backward compatibility, but often the conceptual schema is different).
Any process is by definition a change in the vector state. See first equation of  or 2nd paragraph of section III of . The rate of the process is a superposition of two contributions: direct and inverse.
Take for example the direct contribution. The rate is equal to the product of its 'dinamical' rate by the posibility that process can happen. For example, imagine that rate for the chemical process A + B --> C is 45 in arbitrary units. If you have chemicals A and B spatially separated then the rate is 45·0 = 0 and there is not reaction!
The posibility for a process is directly related to the entropy of the system, i.e. to the number of favourable atomic-molecular* configurations. The section III of  is still good for some details, but modern interpretation of entropy  is more general and does not limited to macroscopic matter.
Therefore, the rate for a process is 'dinamical' · 'favourable quantum states' or
V = Omega · exp[···]
See equation (168) of . Second equation of  or equation (1) of  for the full expression (which is complicated and this board has no mathematical capabilities far from sub and sup)
The chemical reaction law R = k [A]a
[ B ]b
is a special case of canonical equation. In more general situations the canonical expression continue being valid but the kinetic law is not (for example in many processes in biological membranes).
Moreover, the Arrhenius law (and generalizations) are also derived from the canonical science, whereas in chemical kinetics textbooks simply postulated.
See section IV-B of  for a derivation
of chemical kinetic laws from canonical equation but remember that is an old approach and some concepts are outdated by new version.
The advantage of the canonical form is that is unified (explain from elementary particles to cosmology), is more general and theoretically sound that chemical kinetic law of mass action**.
 Keizer, Joel. J. Chem. Phys. 1976, 64(4), 1679.
. See equations (167) and (168).
. This is a XML page with MathML technology for mathematics, and some old browsers or nonstandard (e.g. IE) cannot render it.
This is a XML page with MathML technology for mathematics, and some old browsers or nonstandard (e.g. IE) cannot render it.
* Atomic-molecular here means microscopic structure of matter and is valid in situations where there are not molecules or atoms, for example elementary particles or Dp-branes.
** I show this in Geodome post "rate constants and chemical engineering"