Unit consistency in rate equations

Question

I suppose that my problem is not one of great profundity, but it is an annoying one. The problem is related to the measurement units involved in rate equations of different order. Not being a chemist myself, I have in my work encountered equations of the type: $$\frac{-\mathrm{d}[A]}{\mathrm{d}t} = k [B]^b [A]^a$$ representing the decay of some species $A$. The brackets denote concentration and $k$ is the rate constant. $B$ is some other chemical species (cooking chemical) and then there are the exponents $a$, $b$. For a first order reaction with $b = 0$ and $a = 1$ everything is still under some kind of control; if time is measured in minutes the unit of the rate constant is $\mathrm{min^{-1}}$.

However, if we have a pseudo first order reaction where the species $B$ is present albeit constant during an experimental run, problems arise. In my line of work the species $B$ is typically hydroxide ions and their concentration is traditionally given in molar, $\mathrm{mol/L}$. Even if the concentration of $B$ does not change, the initial concentration plays a role, hence the power of $[B]$ usually appears as a separate factor in the rate equation. Unfortunately, this complicates the unit balance in the equation as frequently the exponent $b$ is a non-integer number. One workaround would be to rescale the concentration to a dimensionless number, but the choice of scaling parameter would be rather arbitrary. Another, perhaps more elegant solution would be to work with molar fractions, but this would be discordant with most of the literature in the field, where the question of unit consistency mostly is disregarded.

A second order rate equation ($b = 0$, $a = 2$, or $b = 1$, $a = 1$) leads to the unit $\mathrm{L\,mol^{-1}\,min^{-1}}$ for the rate constant. I suppose this is still somehow acceptable, even if I would prefer to work with scaled concentrations (divided by the initial concentrations) with unit $\mathrm{min^{-1}}$ for the rate constant, but for fractional order rate equations there is the same problem as in the earlier described pseudo first order case.

Is there any prescribed remedy to this dilemma?

Answer 1

It is true that rate equations typically use molar concentrations for each species, and that for theoretical rate equations you can make sure the units work out by choosing the appropriate units for the rate constant.

There are two things that are important to understand about rate equations, though.

The first is that the "real" rate equations use activity, not concentration. Concentration is usually a "good enough" approximation that this doesn't matter in practice, but when you start thinking about the units of the reaction constant for things like fractional exponents in empirical rate laws, understanding that the concentrations should really be dimensionless activities makes the problem a lot easier.

The second thing to understand is that real reactions don't always follow theoretical rate equations. As a result, what you are often looking at in real life is an empirical rate law. With empirical rate laws, we don't usually worry about the physical significance of fractional exponents, and in turn, how they affect the units of the rate constant. The units of the rate constant become whatever they need to be to make the equation right. In other words, if you have

$\frac{-\mathrm{d}[A]}{\mathrm{d}t} = k [B]^{0.6}[A]^{1.3}$

as an empirically derived rate law that gives $\rm{mol_A}\rm{L}^{-1}\rm{min}^{-1}$, then $k$ would need to have units of $\rm{mol_A}^{-0.3}\rm{mol_B}^{-0.6}\rm{L}^{-1}\rm{min}^{-1}$. These physically nonsensical units are ok because the equation is entirely empirical - we are just describing what we observe on a macroscopic level, without attempting to give it physical significance beyond that.