The Rate Law (Lecture 2)

We now need to introduce a certain distinction among reaction rates. Turns out, what we derived previously was the average reaction rate. Let’s look at the following graph:

For reac1, reac2, and reac3 the values of:

$$-\frac{[A{]}_2-[A{]}_1}{t_2-t_1}\text{\hspace{1em}(1)}$$

are exactly the same. However, let’s notice that reac1 achieves, let’s say, the value of $[A]=1.1[A{]}_1$ (i.e., just above $[A{]}_1$) way earlier than reac3. So there’s some difference in the rate at which molecules react, but we don’t see that difference when we calculate it using equation (1) above.

What’s really different about lines reac1, reac2, and reac3 is that only reac3 has a constant slope (it’s a line). Notice that regardless of which fixed time interval you choose (i.e. fix $\Delta t$, but choose different regions on the graph), the ratio $-\Delta [A]/\Delta t$ will be the same. The same is not true for reac2 or reac1: the value of $-\Delta [A]/\Delta t$ for reac1 is higher at earlier times. Make sure you understand why this is true!

The values of $-\Delta [A]/\Delta t$ at different positions along the curve actually represent the slope of the curve. The more steep the curve is, the larger will be the value.

So, to describe the profile of reaction rates for reac1, reac2, reac3 ideally we’d need to know the values of $-\Delta [A]/\Delta t$ at different positions along the curve. The more values you have, the better you can describe the reaction rate. Why would you want to be able to do that? Well, look at figure above again - if you want to make predictions about what happens after $t_2$, you must know what kind of reaction you have!

You might think, man, if only there was a way to know the values of $-\Delta [A]/\Delta t$ at every position without memorizing 10 or 20 different values.

Well, happily, you live in a reality in which these two fellas existed:

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On the photos above are Newton and Leibniz, both incredible scientists (one of them was also a disastrous trader) who invented calculus. Let me state two things:

1. The slope of the curve is actually the value of $-\Delta [A]/\Delta t$ when $\Delta t$ is small. Very small. Whenever math people want to say big time small, they say infinitesimally. And at that point $\Delta$ gets replaced with $d$ and the ratio becomes $d[A]/dt$. This is somewhat the motivation for calculus, which you may have taken or may decide to take in the future. What’s important for us is that once we move from $\Delta$ to $d$ we switch from average speeds to instantaneous speeds. The value of $d[A]/dt$ is the speed at any given moment of time.
2. Now that we developed the motivation for instantaneous speeds, we can get back to intuition and argue what should be the instantaneous speed. Through some derivation, which I think is partially covered in future lectures (even if not, if you’re interested, I’ll write some notes), we will end up with the formula:

$$\text{instantaneous speed}=k[A{]}^m[B{]}^n$$

which is what you have on Slide 9 and which is called the rate law.

Side note: calculus was invented in 1687, some 330 years ago. So right now, you’re learning about 300 year old tools, and if you wonder how much has happened in 300 years, I can say a lot. In the time since Newton, we created tools to study the behavior of electrons and cosmic rulers to measure the very fabric of the reality in which we exist. The human mind, channeled to the scientific quest for truth and betterment of the humanity, can reach astonishing heights. I really encourage you to keep exploring advanced science and let me know if you want to learn more about theoretical & computational chemistry and what problems does it tackle.

Reaction Orders

Consider the reaction $a\mathrm{A}+b\mathrm{B}\to c\mathrm{C}$. Its instantaneous speed is:

$$r=k[A{]}^m[B{]}^n$$

The values of $m$ and $n$ are called reaction orders with respect to reactants $A$ and $B$. The overall reaction order is $m+n$.

Is there any connection between $m,n$ and $a,b$? The slides say nah absolutely not, and that’s a good rule if you just want to solve problems correctly, but the reality is a bit more interesting. But that’s a topic for some future lecture.