2015 AP Chemistry free response 5a: Finding order of reaction

- [Voiceover] Blue food coloring can be oxidized by household bleach, which contains hypochlorite ion, or OCI-, to form colorless products, as represented by the equation above. So we have this equation where we have blue food coloring, which has this chemical formula, and it's reacting with bleach, or hypochlorite, and it's making colorless products. So we're getting this color change from blue to colorless. A student used a spectrophotometer set at a wavelength of 625 nanometers to study the absorbance of the food coloring over time during the bleaching process. In the study, bleach is present in large excess, and I will underline that 'cause that's important. It's present in large excess so that the concentration of hypochlorite is essentially constant throughout the reaction. The student used data from the study to generate the graphs below. So we'd like to answer, in part a, based on the graphs above, what is the order of the reaction with respect to the blue food coloring? So we have a lot of information here. We have these three graphs. If we write absorbance as A, we're looking at the concentration of A over time. We're looking at the ln of A over time, the natural log, and we are also, lastly, looking at one over the concentration A over time. So, we would like to know, what is the order of the reaction with respect to our blue? My first instinct when I see this problem, is actually to be a little bit panicky because, hey we have a lot of information here, so how do we deal with it? We also have a little bit more information for this problem. We also have the equation sheet. So this is the part of the equation sheet that's relevant for this problem. So it's all of the equations that have to do with kinetics, which is looking at reaction rates, so that is what we're looking at here. So we have three possible things that might be helpful for this problem. Maybe not, right? So we have this first equation which gives us the first order integrated rate law. We have the second equation, which gives us the second order integrated rate law, and we have this third equation which is actually an equation for the half-life. So it's saying half-life is .693 divided by k, and that is the half-life for a first order reaction. And the important thing to remember here, which will come up again in a second, is that this is a constant, because k is a constant, so our half-life is also a constant for a first order reaction. So that's the information we have. Let's look back at our graphs. We're gonna start by looking at just the first graph. It turns out we can get a lot of information just by looking at the concentration of A over time. So if we look at this first graph, and we look at the concentration of our blue stuff going down over time, we can see immediately that this is not linear. So the fact that it's not linear immediately tells us it's not zero order. If it was zero order, we would expect to see a straight line, which I will draw in pink, with a dotted line because that's not actually what we're seeing. So it's not zero order. We can also get some more information by looking at this one graph. We know it's not zero order. Can we tell from this first graph if it's first order? And the answer is, we actually can. If you remember from looking at the equations sheet, we said that first order reactions, so for first order reactions, we know, and I'd say this is maybe one of the most important things about first order reactions, is that half-life is constant. So that means that the time it takes for the concentration to go down to half of what it was, is the same no matter what your initial concentration is, no matter what point in your reaction you're at. So that's something we can actually investigate just by looking at this first graph. We can see how much time it takes to get to half the concentration, and see if that's something that changes over time. So for example, if we start with .8 right here, half of .8 is .4, so we can see how much time it takes to get from .8 to .4, which is right here, and if we go down we can see it takes 20 seconds to have the concentration starting at .8 absorbance units. We can look at the half-life now to go from .4 to .2. So if we go from .4 to .2, and I will change colors to make this look more obvious. We'll do this in orange. So going from .4 to .2 takes another 20 seconds. And so we can do this again and again. We could go from .2 to .1. We'll just see the continuation of a pattern here, which is that, we can see that it always takes 20 seconds to halve the concentration. And so the fact that half-life is constant based on Graph I, immediately tells us that our reaction is first order because our half-life is constant. We don't even have to look at any of the other graphs. There is another way you could do this problem. You might remember, based on the equations that we have, or based on just equations you know off the top of your head. So one reason we know, and one reason we could say this is first order, is because t 1/2 is constant. The other reason why we know this is first order is, when a reaction is first order in your reactant, you know that graphing ln of your reaction versus time, gives you a straight line, and we can see from Graph II that this is the case. So we have these two pieces of evidence, from Graph I and Graph II, that tell us that this particular reaction is actually first order with respect to the blue food coloring.