Mathematical Formulation of Reaction Rate (Lecture 2)

In previous lecture, we argued that:

we intuitively associate rate with the speed at which the number of particles changes

Let’s now start making mathematical formulations of reaction rate for a reaction $A\to B$. Translating the text above into math:

$$r=\frac{N_B(t_2)-N_B(t_1)}{t_2-t_1}=\frac{\Delta N_B}{\Delta t}$$

where $r$ is the reaction rate, $N_B$ is the number of particles of B, and $t$ is time. Recall that when we explored how we can increase speed of the reaction, we noticed that we care not just about number of particles, but about the number of particles per unit of space/volume, which is concentration. Because it matters a lot for the number of collisions whether a $100$ particles are within some tiny box or in a large container, it may be intuitive to conclude that the reaction rate should also consider that. So, a better formulation would be in terms of concentration, rather than absolute numbers.

$$r_B=\frac{[B{]}_2-[B_1]}{t_2-t_1}=\frac{\Delta [B]}{\Delta t}$$

we use square brackets to denote concentration, subscripts to denote time.

Teachers like to sneakily use words such as obvious, clearly, evidently, intuitively as I did above. Sometimes I may even do that subconsciously, just because it feels good™, but please whenever it’s not clear or intuitive, let me know! I once was helping a student with a problem, and at some point I said “Obviously, this means X is true”. The student argued “nah, not true, here’s why”, and it turns out I was wrong. Now, when I look back, I have no idea how could I ever think X was true.

Remember last time we noticed that for a reaction $A\to B$:

we might infer the rate judging both by appearance of B and by disappearance of A.

Thus, we could also write:

$$r_A=\frac{\Delta [A]}{\Delta t}$$

But we should be careful and notice that $\Delta [B]$ is positive, while $\Delta [A]$ is negative. However, a) it doesn’t make a lot of sense for a rate to be negative b) we really don’t care about the distinction between whether we count disappearance of A or appearance of A. We could say let’s define reaction rate to be the absolute value. $r=|r_A|=|r_B|$. The problem is, modulus as a mathematical operation is nasty and inconvenient, so instead we say: $r=r_B=-r_A$. That’s why you’ll see:

$$r=-\frac{\Delta [A]}{\Delta t}=\frac{\Delta [B]}{\Delta t}$$

Moving to more complex reactions

Let’s look at a more complicated reaction $\mathrm{H}{}_2+\mathrm{I}{}_2\to 2\mathrm{HI}$. We need to make the following observations:

1. whenever a particle of $\mathrm{H}{}_2$ disappears (reacts), so must $\mathrm{I}{}_2$. Therefore, $\Delta [\mathrm{H}{}_2]=\Delta [\mathrm{I}{}_2]$ (strictly speaking, what is directly implied is $\Delta N(\mathrm{H}{}_2)=\Delta N(\mathrm{I}{}_2)$, which we can convert to concentrations if the volume is fixed.)
2. whenever a particle of $\mathrm{H}{}_2$ disappears (reacts), two molecules of $\mathrm{HI}$ appear. Thus, at a given time period, $\Delta [\mathrm{HI}]=2\Delta [\mathrm{H}{}_2]=2\Delta [\mathrm{I}{}_2]$.

From (1), we can write the reaction rate to be:

$$r=-\frac{\mathrm{\Delta }[\mathrm{H}{}_2]}{\Delta t}=-\frac{\mathrm{\Delta }[\mathrm{H}{}_2]}{\Delta t}$$

which is the application of concepts we learned before, nothing new. Now, when we try to write:

$$r_{\mathrm{HI}}=\frac{\Delta [\mathrm{HI}]}{\Delta t}$$

Given (2), we can observe that $r_{\mathrm{HI}}=2r$. Now we’re faced with a problem: do we want to have two different formulations of reaction rate? I can’t see a lot of reasons for, but a big reason against: it makes it harder to communicate:

Rick: Morty, how fast is this reaction going? Give me the numbers! Morty: Uh, it’s 3 mol per minute, Rick. Rick: 3 mols of what, Morty? A, B, or C? Be specific! Morty: Oh, uh, it’s B, Rick. 3 mol of B. Rick: So, that means we’re dealing with 3 mol of A per minute, right? Morty: No, no, Rick! It’s actually 6 mol of A. The coefficient is 2, remember? Rick: 🤬

How can we fix this? Yeah well, if we want to have a unified formulation through $r$, from $r_{\mathrm{HI}}=2r$ it follows that it’s suffice to divide $r_{\mathrm{HI}}$ by 2!

$$r=-\frac{\mathrm{\Delta }[\mathrm{H}{}_2]}{\Delta t}=-\frac{\mathrm{\Delta }[\mathrm{H}{}_2]}{\Delta t}=\frac{1}{2}\frac{\mathrm{\Delta }[\mathrm{HI}]}{\Delta t}$$

Now, we can write the formulation for a general reaction $a\mathrm{A}+b\mathrm{B}\to c\mathrm{C}+d\mathrm{D}$:

$$r=-\frac{1}{a}\frac{\mathrm{\Delta }[\mathrm{A}]}{\Delta t}=-\frac{1}{b}\frac{\mathrm{\Delta }[\mathrm{B}]}{\Delta t}=\frac{1}{c}\frac{\mathrm{\Delta }[\mathrm{C}]}{\Delta t}=\frac{1}{d}\frac{\mathrm{\Delta }[\mathrm{D}]}{\Delta t}$$