# Generalized reaction rate and derivation

Most textbooks give the reaction rate

$$r = k C_\ce{A}^a C_\ce{B}^b \tag{1}$$

for the elementary process $\ce{a A + b B -> P}$. However, expression (1) is not valid in the general case: e.g. it is lacking fluctuations, small-size corrections, memory effects, relativistic corrections,...

I have seen some derivations of the rate expression (1) using kinetic theory, Langevin approach, or chemical master equation, but the derivations introduced further approximations: e.g. the kinetic theory approach considered only binary collisions in diluted gases under local thermodynamic equilibrium.

Consider the process $\ce{A -> P}$, according to (1) the rate expression would be

$$r = k C_\ce{A} \tag{2}$$

However this expression not only predicts an incorrect stationary state, but it does not represent the real concentration in the system, because $C_\ce{A}$ is only an averaged concentration and not the instantaneous concentration of species $\ce{A}$ in the system. In a first approach the above rate is replaced by

$$r = k \tilde{C}_\ce{A} + \tilde{f} \tag{3}$$

where $\tilde{C}_\ce{A}$ represents the real concentration in the system and $\tilde{f}$ is a stochastic chemical 'force'. Now (3) represents the rate of the real concentration and it gives the correct stationary limit. (3) satisfies a fluctuations-dissipation theorem that relates $\langle\tilde{f}(t)\tilde{f}(t')\rangle$ to the rate constant $k$. This relation is used in experimental methods that measure chemical fluctuations to obtain the rate constant.

Equation (3) continues being an approximation, a generalization could be the introduction of memory effects, which could give something as

$$r = \int_0^t \kappa(s) \tilde{C}_\ce{A}(t-s) ds + \tilde{F} \tag{4}$$

where the kernel $\kappa$ and the generalized stochastic 'force' $\tilde{F}$ have complex expressions, but (4) continues being an approximation. Consider now the process $\ce{2A -> P}$, according to (1) the rate expression would be

$$r = k C_\ce{A}^2 \tag{5}$$

but this is not correct in general. Introduce the volume factors into $k$ to obtain

$$r = K N_\ce{A}^2 \tag{6}$$

and you can check this predicts a rate $r=K$ when $N_\ce{A}=1$, which is obviously impossible because at least two molecules are needed for the step. The rate (6) is invalid for mesoscopic systems (e.g. reactions involving small amounts of matter in living cells) and it is also incorrect for large systems in the long time limit, when $N_\ce{A} \rightarrow 0$. A corrected rate is given by

$$r = K N_\ce{A} (N_\ce{A} -1) \tag{7}$$

which cannot be derived from (1). However, (7) can be derived in combinatorial kinetics, which postulates the next rate expression for an elementary process $\ce{a A + b B -> P}$

$$r = K \frac{N_\ce{A}!}{(N_\ce{A} - a)!} \frac{N_\ce{B}!}{(N_\ce{B} - b)!} \tag{8}$$

But again (8) is an approximation.

Each rate expression I find in the literature is valid only for some specific cases, but fails for others.

What is the general expression for the reaction rate for an elementary process and how can the expression be derived from first-principles?

• What you've written is a macroscopic observable, so it is the general expression, and it is generally valid for the class of reactions for which it is designed (which is a bit circular, I realize, but that's the nature of empirical statements). All the complications of which you speak vanish in the thermodynamic limit. It sounds like what you're after is some kind of kinetic theory expression, and you're unsatisfied with dilute gas approximations, so maybe the place to start is with general formulations in the liquid state like rspa.royalsocietypublishing.org/content/188/1012/10 – Christopher Grayce May 30 '18 at 1:23
• Read my answer to this question chemistry.stackexchange.com/questions/81019/… as it largely covers the points you raise – porphyrin May 30 '18 at 7:57
• @ChristopherGrayce I didn't write a macroscopic observable ($r$ could be a mesoscopic rate) and the thermodynamic limit is an approximation used to simplify some formulas, but no real system has infinite volume and infinite number of particles. I am realy looking for a dynamical rate expression, not a kinetic one, but will look the reference. Thanks – juanrga May 30 '18 at 9:24
• @porphyrin The first part of your answer uses collision theory with the usual simplifications (including the one that I mentioned explicitly as "local thermodynamic equilibrium"). The second part mentions the approach taken by Gillespie, which is based in chemical master equation (I also mentioned this equation) which built in many simplifications including the ignoring of non-Markovian effects in the system. – juanrga May 30 '18 at 9:32
• Ok, I assume also that you know the book by Erdi & Toth (Math models of chem reactions) then in that case you should probably start by calculating reaction dynamics trajectories where you can control all parameters then average over quantum states at your choice of energy distribution. – porphyrin May 30 '18 at 11:54