# Unit consistency in rate equations

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?

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.

• I've always wondered if I'd see a rate equation with an exponent which wasn't an integer or half-integer. Would you happen to have any reference with an explicit example? – Nicolau Saker Neto May 27 '15 at 23:15
• +1 for pointing out that it should be dimensionless. Many chemists have had similar issues because they aren't properly taught about dimensional analysis! – jvriesem May 28 '15 at 1:04
• @Nicolau Saker Neto - I just made that one up as a hypothetical example :). I searched for a while but can't find any explicit examples of real non-half-integer fractional order rate laws. I did find references to using them for complex reactions (typically involving heterogenous systems) in some of my kinetics textbooks, but there were no explicit examples there either. I know that these are most common in industrial settings, where the mechanism isn't always important. Perhaps that is why they are hard to find in the literature? – thomij May 28 '15 at 1:06
• Sorry if I wasted your time, I had no idea whether more complicated exponents were common or not. I'm not sure if this is right, but I'm under the impression that even complex reactions could in principle be discretized into several cases which cover particular steps of the reaction under sufficiently constrained conditions, and all of these small steps would have "pretty" exponents relating to the step's stoichiometry. If one doesn't care about this level of detail, it would be possible to average out some steps and try to compensate by letting the exponents vary freely as real numbers. – Nicolau Saker Neto May 28 '15 at 1:40
• @Nicolau Saker Neto - No worries - I was surprised I couldn't easily find any! You are correct, empirical rate laws with fractional exponents can be resolved to steps with whole or half integer orders. You would only see these types of fractional order laws in situations where figuring out the details costs more and provides less benefit than just using the empirical law. One of the books I read said "using them is an admission of ignorance." – thomij May 28 '15 at 12:45