Lesson Video: Rate Graphs Chemistry

Video Transcript

In this video, we’ll become familiar with the features of graphs from rate of reaction experiments. We’ll learn how to use them to calculate the rate of reaction, and we’ll see how these graphs support collision theory. There are various ways to set up experiments for different reactions that can help us determine the rate of reaction. For instance, in this reaction where calcium carbonate in hydrochloric acid react to form calcium chloride, water, and carbon dioxide, one of our products is a gas. So we could perform this experiment by putting our reactants in our reaction vessel on a balance and then record the mass as the reactants react to form the carbon dioxide, which escapes from the vessel as the reaction proceeds.

If we were to perform this experiment, we might end up with data that looks something like this. So now that we figured out how to obtain this graph, what do we do with it? How can we use this graph to determine the rate that calcium carbonate is reacting with hydrochloric acid to produce our products? Let’s recall that, in general, a rate is a change in some variable per unit time. In the context of chemistry, we want to determine how quickly our reactants are reacting to form our products. So our rate should be defined as the change in the amount of our reactants or products per unit of time. This amount of our reactants or products might be a mass, a volume, a concentration, or even amount in moles. It just depends on how we set up our experiment.

Since we perform this experiment by measuring mass, our rate will be in terms of a change in mass per change in time. And we can, of course, make this expression look nicer by using a Δ to indicate a “change in.” If we look at a graph, we’ll notice that we have mass on the 𝑦-axis and time on the 𝑥-axis. So our rate is really a change in 𝑦 per a change in 𝑥. And now our formula to calculate the rate looks suspiciously like the formula to calculate the slope or the gradient of a line. So now let’s try calculating the rate for this data. This point on the bottom is at 420 seconds and 1.8 grams. And this point at the top is at zero seconds and three grams. So 1.8 grams minus 3.0 grams divided by 420 seconds minus zero seconds gives us a rate of negative 2.9 times 10 to the minus three grams per second.

You’ll notice that the rate that we ended up with is negative, which makes sense because what we measured here was the mass of our reactants decreasing over time as they reacted to form our products. Now let’s say we perform a different experiment. We’ll use the same reaction. But instead of measuring the mass of our reaction vessel over time, we’ll connect our reaction vessel to a gas syringe and collecting the carbon dioxide as it was produced.

This might give us data that looks something like this. We can again calculate the rate by using the data on this graph. But here our rate will be a change in volume per change in time because of how we set up this experiment. When our reaction finished at 420 seconds, we had 60 milliliters of carbon dioxide. And at the start of the reaction, at zero seconds, we had zero milliliters of carbon dioxide, which gives us a rate of 0.14 milliliters per second. This time, when we did the calculation, the rate was positive. This again makes sense because we’re measuring the rate that carbon dioxide is being formed.

So whether we have a graph of the change in our reactants over time or the change in the amount of our products over time, we can find the rate of reaction by taking the change in the amount and dividing it by the change in time. We’ll just have to keep in mind that when we calculate the rate using a graph that shows the amount of our reactants over time, the rate that we get will be negative, since the reactants are being used up. But if we use a graph that shows the amount of products over time, our rate will be positive, since we’re calculating the rate that our products are being formed.

Something to note is that generally we like to express the rate of reaction as a strictly positive number. So if we’re talking about the rate of reactants being consumed, we might just report the magnitude of the rate that we calculated. Now, unfortunately, things won’t be quite so simple when we’re looking at the results for most rate experiments. Recall that, according to collision theory, particles need to physically collide in order to react. This means that if there are fewer entities to collide with each other — that is, the concentration is lower — they won’t collide as often, which means that the rate will be slower. In other words, the concentration or amount of our reactants that we have is proportional to the rate of reaction.

So what this means for a reaction is that as the reactants collide with each other to form the products, they’re slowly being used up. So the concentration of our reactants is decreasing, which means that the rate will slow down over time, which means that we generally won’t see a graph that looks like the ones we were working with earlier, as the constant slope of this graph indicates that the rate of the reaction here is constant. Instead, we’ll often see something like this, where the graph is curved. Since the slope of this graph isn’t constant, we know that the rate here isn’t constant either. So if we used our previous method to determine the rate of reaction here, we’d actually just get the average rate of reaction over the course of the whole reaction. But since the rate is changing here over time, wouldn’t it be useful to know the rate at a specific point in time?

How do we go about doing this? Well, we might notice that if we don’t take the slope of the whole graph, but instead calculated the slope of smaller time periods around the area that we’re interested in, our results would be more accurate. If we make this time window small enough, we could instead calculate the slope of a line that’s tangent to the curve at the point in time that we’re interested in, which would allow us to calculate the rate at that point in time. If you’ve taken any calculus, you might recognize this process of calculating the slope of a graph at a specific point by using tangent lines as taking the derivative. However, you don’t need to know any calculus to use this method. All you need to know is how to calculate the slope of a line.

So let’s try calculating the rate for the three tangent lines that are drawn onto the graph. This first tangent line, labeled with the letter 𝑎, intersects the graph at 20 seconds and 0.8 molar and zero seconds and 2.0 molar. Since this tangent line is tangent to the graph at 10 seconds, we’ll be calculating the rate of reaction 10 seconds into the reaction. This calculation gives us negative 0.06 molar per second. Again, this rate is negative because we’re calculating the rate of reaction for our reactants decreasing over time.

Now let’s try calculating the rate for the tangent line labeled with the letter 𝑏. This one intersects the graph at 30 seconds, so the rate we’ll be calculating will be the rate 30 seconds into the reaction. This tangent line intersects the graph at the points 45 seconds, 0.2 molar, and 15 seconds and 1.0 molar. Performing this calculation gives us negative 0.0266 repeating molars per second. Our final tangent line is tangent to the curve at 80 seconds. So we’ll be calculating the rate 80 seconds into the reaction. This line intersects the graph at 90 seconds and zero molar and 70 seconds and zero molar, which gives us a rate of zero molar per second.

So now we’ve calculated the rate for three different points in time for our reaction. Looking at the magnitude of the rates that we calculated, we can see that the rate decreases over time as the reactants are used up, just as we expected. We can also compare our calculations to visually inspecting the graph. We can see that at the beginning of the reaction, our rate graph has a steep slope, which corresponded to a higher rate. As the reaction proceeded, the slope of the graph became shallower, which corresponded to a lower value for the rate, meaning that the reaction was slowing down. And then at the end of the reaction, when all of the reactants were used up, the graph had a slope of zero, which corresponded to the rate of the reaction also being zero.

So as we can see, we can get a lot of information about the rate of the reaction from just visually examining a rate graph. We don’t even have to do any math to tell when the rate of reaction is faster and when it’s slower. Before we move on to looking at some more graphs to get some more practice with this, there’s something else that’s interesting about the math that we just performed. If we plot the rates that we calculated against the concentration of our reactants at that point in time, we end up with a straight line, which tells us that the rate is proportional to the concentration, which is exactly what collision theory tells us.

When the concentration of our reactants is high, we have more collisions and therefore a higher rate of reaction. It’s always nice to know that the model that we’re working with to understand a phenomena, in this case, understanding the rate of reaction through energies colliding with each other, isn’t some abstract thing but something that can be supported through our experimental data. In other words, this shows that science is working.

Now let’s go back to looking at some rate graphs. We just looked at different parts of the same graph and learned how to visually determine where the rate was fastest. We can also use this skill to determine which experiment will have a faster rate by comparing two different curves. For instance, we can perform the reaction with calcium carbonate with calcium carbonate either in large chunks or ground up into a fine power. As we know, the surface area is related to the rate of reaction. If our reactants have a larger surface area, the rate of reaction will be higher. When you grind calcium carbonate into a fine powder, it increases the surface area.

So let’s look at these two curves on this graph that has the volume of carbon dioxide produced over the reaction. And let’s see if we can figure out which curve belongs to the large chunks of calcium carbonate and which one belongs to the fine powder of calcium carbonate. If we compare the slope of these two curves at around the same time near the beginning of the reaction, we can see that the curve labeled 𝑎 is much steeper than the one that’s labeled 𝑏. And as we know, a steeper slope on our curve corresponds to a faster rate of reaction. Of course, this curve levels off more quickly as well, indicating that the reactants are being used up more quickly.

So when we’re making these comparisons, we want to look at the rate near the beginning of the reaction. We can also see that these two curves end at the same volume of carbon dioxide produced. It just takes curve 𝑏 longer to get there, which makes sense, since we’re starting with the same mass of calcium carbonate in both cases. But as we know, a steeper slope corresponds to a faster rate, and curve 𝑎 has the steeper slope initially, which means that it has a faster rate. And the fine powder of calcium carbonate should have a faster rate of reaction than the calcium carbonate in large chunks. So curve 𝑎 must correspond to the fine powder of calcium carbonate, and curve 𝑏 must correspond to the calcium carbonate in large chunks. We didn’t even have to do any math to tell these two curves apart.

Let’s look at one last example. This graph shows the concentration of the products of the reaction where ammonia decomposes to form nitrogen and hydrogen. Let’s see if we can figure out which curve belongs to the concentration of nitrogen over time and which one corresponds to the concentration of hydrogen over time. Looking at our reaction, we can see that the stoichiometric coefficients in front of our products aren’t the same. The stoichiometric coefficient in front of nitrogen is one, and the stoichiometric coefficient in front of hydrogen is three.

This means that for every mole of ammonia that decomposes, we’re going to get more hydrogen than we do nitrogen, which means that hydrogen is being produced faster, which means that the rate for hydrogen should be higher. So let’s compare the slope of these two graphs. Whichever one has the steeper slope would have the higher rate and therefore that one should correspond to the rate of hydrogen being produced. Again, we wanted to compare the initial rates of reaction as opposed to the rate of reaction towards the end when the reaction is nearing completion and both curves are leveling out.

We can see that the slope of curve 𝑎 near the beginning of the reaction is much steeper than the slope for the curve 𝑏 near the beginning of the reaction. So the curve that corresponds to hydrogen being produced is curve 𝑎, and the curve that corresponds to nitrogen being produced is curve 𝑏. Looking at this graph, we can also see that these curves don’t end in the same place, which makes sense because this reaction is going to produce more hydrogen than nitrogen.

So now we’ve learned how to calculate the rate of reaction and how to visually examine rate graphs. So let’s summarize everything we have learned with the key points for this video. We can calculate the average rate of our reaction by taking the change in the amount of our reactants or products and dividing it by the change in time. And this amount might be a change in mass, a change in volume, a change in concentration, or something else, depending on how we set up our experiment.

But if we’re interested in calculating the rate at a specific time, we can do this by calculating the slope of a line that’s tangent to the graph at that point. The rate of reaction changes over the course of a reaction, which corresponds to the reactants being used up as the reaction proceeds. By visually looking at a graph, we can comparatively determine the rate because a large rate corresponds to a steeper slope on our graph and a small rate corresponds to a shallower slope on our graph.