Now that we’ve spent a lot of time talking about reaction rates, it’s time to talk numbers. After all, if you’ve learned anything from this whole chemistry thing, it’s that chemistry teachers love to make you do calculations. It’s how we weed out people who don’t own calculators.

**What is a reaction rate?**

Let’s say that, for whatever reason, you’re doing a chemical reaction with the form:

A → B

The rate of the reaction can be described in the two following ways:

**rate: A measure of how fast the reagent to be used up.**According to this definition, you can measure the rate by measuring how the concentration of the reagent (in this case, A) decreases over time.**rate: A measure of how fast the product is made.**According to this definition, you can measure the rate by measuring how the concentration of the product (in this case, B) increases over time.

In chemistry, we typically measure rate according to the first definition in which the rate is a measure of how fast the reagent goes away. I suspect that this is because it’s easier to measure the original reagent than the product (i.e. you’ll have a better idea of how it can be identified), but it’s mostly just a convention we use.

If we were to graph how the concentration of the reagents in the reaction above will change over time, we might expect to see something like this:

In this graph, you can see that the concentration of the reagent decreases as time progresses (it starts at 2.500 M, and ends at 0.35 M at the end). This allows us to describe two rate-related terms, each of which will have their own section below:

**Average rate of reaction:**

The average rate of a reaction is simply the change in concentration over the period of time being measured. For example, if I were to ask you the average rate of a reaction, you’d just look at this chart and make this determination:

Initial concentration: 2.500 M, Final concentration: 0.350 M

- Change in concentration: 2.125 M

Initial time: 0 sec, Final time: 150 sec

- Change in time: 150 sec

Average rate of reaction = (change in concentration)/(time elapsed)

**= 0.014 M/sec**

If you want to find the average rate of a chemical reaction for a different time interval, just adjust your initial and ending values to find what you’d like. However…

What happens if you want to find the rate of reaction at 100 seconds? I’m not talking about the *average* rate of the reaction – I’m talking about the *instantaneous* rate of reaction at that specific time.

Using our existing equation, we find that:

- time elapsed = 0 (since we’re only using one time)
- concentration change = 0 (since the concentration doesn’t instantaneously change)
**Rate of reaction = 0/0 = [something undefinable]**

Since we chemists don’t like [something undefinable] as an answer, it’s time for us to figure out what to do in a case like this. That’s why we need to learn how to find…

**Instantaneous Reaction Rates**

If you look back at the graph above, you’ll see that the rate of a chemical reaction is always changing. In fact, I’ll just go ahead and show you the graph again:

In this updated version of the graph, what we’re going to do is figure out the instantaneous reaction rate at t = 50 seconds and t = 100 seconds. Here’s how:

- Draw a graph: That’s what we did earlier
- Draw tangent lines corresponding to the time that you’re interested in finding the rate for.
- Find the slope of the tangent line.

Before we go too much farther, it’s probably good if I tell you what a tangent is. Basically, if you’ve got a curve, the tangent to that curve at any particular point is what we estimate the slope at that point would be. In the case above, the first red line is an estimation of the slope of the blue curve at 50 seconds, and the second is our estimation of the slope of the line at 100 seconds. You should know that these slopes will really just be estimations, as it’s kind of hard to tell exactly what the slope will be. Try your best.

In any case, let’s find the slope of these lines at t = 50 and t = 100 seconds. I’ll use the points on the tangent lines below to find these slopes:

**Slope at t = 50 seconds**

- Change in concentration (between the points marked): 0.50 M
- Change in time (from the points marked): 25 seconds

*Rate at t = 50 sec = (o.500 M)/(25 sec) = 0.020 M/sec*

**Slope at t = 100 seconds**

- Change in concentration (between the points marked): 0.25 M
- Change in time (between the points marked): 40 seconds

*Rate at t = 100 seconds = (0.25 M)/(0 sec) = 0.006 M/sec*

You’ve probably guessed by now that you won’t always be given graphs to figure things out. As a result, you’ll have to learn how to find rates without them. Fortunately, I like talking about kinetics, so there’s another tutorial to check out.