A "function of proportionality" in a rate law [closed]

Question

I am currently studying rate laws and the determination of the order of a reaction. So first-order reactions remind me of linear functions $f(x)=kx$ while second-order reactions remind me of quadratic functions like $g(x)=cx^2$ in high-school algebra. Since we are dealing with reactants here, the input of the functions would be the concentrations of our substrates $[\ce{S_1}]$ and $[\ce{S_2}]$. I am wondering if there are any fancier functions that arise in "real-life" chemistry?

Answer 1

If by "fancier" you mean "uglier", then absolutely yes. Real chemistry is messy, much more than we can fully handle. The simple rate laws we learn in undergraduate physical chemistry are just approximations.

I've written elsewhere that, in general, a full description of chemical kinetics for any overall reaction requires studying an entire tree of simultaneous chemical reactions. There can be dozens, even hundreds of intermediate species involved, all mutually linked in complex ways, and with more than one final product ("by-products"). The difficulty in performing this full description has lead to the whole field of chemical reaction network theory. I hope you like matrix algebra.

To make chemical kinetics tractable in most cases, two major approximations made are:

1. Forget about the whole reaction network, and look at only the overall kinetically fastest path from reagents to products
2. Forget about all the steps and intermediates in the fastest path, and describe only the bottleneck - the slowest step

Even with these two crushing approximations, it's possible to describe many useful reactions with a fair level of accuracy. This is partly luck, and partly unavoidable (due to simplicity naturally arising in randomness). This level is what most undergraduate and even graduate work sticks to. Even when it's a bad fit, we'll sometimes stubbornly fit a system into a particular rate law, do some variational analysis, and hope the error isn't too great.

A nice example of a reaction whose kinetics is slightly more complex than the standard "nice" rate laws is the radical reaction between hydrogen and bromine.

$$\ce{H2 + Br2 -> 2HBr}$$

According to experimental data, it's possible to obtain the following rate law (which may still be approximate!):

$$\mathrm{r_{HBr}=\frac{k_1[H_2][Br_2]^{3/2}}{[HBr]+k_2[Br_2]}}$$

The deduction is nicely described in this site. As you can see, depending on the reaction parameters, the denominator can potentially be simplified down to a single term, with some degree of approximation, and a simple-ish rate law could be used. But in the general case, it's messier. Because of these reaction networks, even the concept of a "reaction order" does not exist in general. An example of a natural but extremely difficult reaction to describe kinetically in full would be the combustion of a moderately sized alkane, such as octane ($\ce{C8H18}$).

When taking full chemical networks into account, there appear cases with pretty unique kinetics. For example, if you haven't heard of oscillating reactions, they're a topic of considerable experimental and theoretical study. With a particular set of coupled reactions, it's possible for chemical systems to display the hallmarks of mathematical chaos. Even if individual steps can be described by simple kinetics, the global process cannot. There's a nice discussion of oscillating kinetics in this Chem.SE question (and possibly others).

I'm sure there are other ways to get strange rate equations. Some reactions just don't follow kinetic theory in its simplest form. This can lead to weird conclusions, such as negative activation energies. Furthermore, rate laws almost always assume macroscopic amounts of matter. When a tiny amount of reagents are involved, the discrete nature of matter calls for statistical corrections to chemical kinetics, because an integer number of molecules can't follow continuous analytical curves.