so that takes concentration out We have looked at first order kinetics and the integrated rate law for first order kinetics. We are ready now to look at second order kinetics. We are going to have a very similar process. We are going to derive the equations that we will use. The integrated rate law for second order reactions, we are going to see how you come up with the half-life equation. So my goal or you is to be able to use the second order integrated rate law to calculate the interdependence between concentration and time, including the half-life calculations. We will finish this lesson with a brief discussion of zeroth order kinetics and also tie the three together with some graphical representations some plot of the three. So lets look here at the same basic reaction. A reactant of some sort going to So lets look here at the same basic reaction. A reactant of some sort going to products, but this time it is a second order reaction. We know from the rate expression, that we have the express for rate. We also know, since it is second order, that this is the rate law for the reaction. Once again if I set these two portions here equal to each other this portion equal to portion, and I employ an integration I will come up with this expression. Which relates concentration and time. This equation is one to commit to memory. It is a second order reaction, integrated rate law. We are not going to, in the lesson time, use this equation to do calculations, we you will have an opportunity to view some of those within my talk hand lessons. Lets do a little graphical manipulation, and get it into the slope intercept form of a line again. Lets do a little graphical manipulation, and get it into the slope intercept form of a line again. All I have done is a rearrangement. When you rearrange it you suddenly see you have it in the form of a line, a linear relationship, the slop intercept form, that one can plot one over a [1/A] on the y-axis and plot t along the x-axis. When I do that, if it is second order we will get a straight line. Sometimes that is a way to obtain the order of the reaction is to plot, as time goes by, a relationship of 1 over concentration verse time, and see if gives a straight line, and if it does it is a first order reaction. Now when, you have that linear relationship it is a first order reaction. Now when, you have that linear relationship there and y = mx + b what is the slope going to be equal to? You think about it and choose your answer. Did you say 'k', certainly that is correct. That is the relationship of what m is and that is 'k' So a plot of 1 over A [1/A] along the y-axis time [t] along the x-axis will give a straight line, but the straight line will go off in this direction, as a positive slope. Lets work on the half-life information here. We know that half-life is represented with t 1/2, so we are going to substitute that in for t in the equation. We are going to be at half the original amount so I am going to replace the A here with this portion. With those substitutions, we have this and once again if I combine the A naught terms and solver for half-life this will give me T 1/2 is equal to 1 over 'k' [1/k] time the initial concentration. Let look at this relationship here this is very different then our half-life relationship for a first order reaction. In a first order reaction we saw that the amount of time it takes to get to half your start, whatever that time was will be the same time it will be to drop in half again. This is not true for second order kinetics. Second order kinetics, when we drop it in half and then we drop it in half again, that second half-life is going to be longer because the concentration is part of that expression. Concentration is in the denominator so as we drop it in half and now we have less amount, we are putting a smaller number in that denominator that is going to stretch out the half-life so it is going to take even longer to drop it in half a second time, and so forth. That is the second order half-life equation. We want to look at zeroth order equations next. What do we know about zeroth order reactions they are rated to the zeroth power of the equation, rate is just a constant. So if rate is a constant a plot of concentration verses time is going to give you a nice straight line. What would that straight line look like? Well, if it is concentration and time plotted like this concentration is certainly going to drop off but it is going to drop off in a linear fashion. So, it is not a common reaction type but it is going to drop off in a linear fashion. So, it is not a common reaction type but it is out there. What about a plot of rate verses time? If instead of concentration I put rate as time goes by. rate is a constant, it does not change so I would get a nice straight line. That is a bit of the graphical representation of zeroth order reactions. I would like to finish with just a summary of those three. Sometimes you are given in a problem a list of concentrations and time. So they give you some data points. I will use these line to represent the data points. They will say, with this information, tell me what is the order of the reaction. So the reaction is A going to products and you are trying to figure out what is the order. I suggest that you start with what is the order. I suggest that you start with first order, because that is the most common. If you were to plot natural log [ln] of A verse time and you get a straight line then you know it is first order. So you would just take all these A's and determine the concentration the natural log [ln] of those values so you get new values here plot these tow portions on our graph and see if it is a linear relationship. If it is, it is first order, on our graph and see if it is a linear relationship. If it is, it is first order, but if it is not, if it is not a nice straight line, then you want to go to the next option. if it is not a nice straight line, then you want to go to the next option. Which is typically to plot one over A [1/A] verse time. So you come back up to your data points here and you figure each of those out in terms of one over the concentration [1/concentration]. Now if you plot those one over the concentration [1/concentration]. Now if you plot those and that gives you a straight line then you know it is second order. If that does not give you a straight line, you might want to finish with the zeroth order. You can do these in any order you want. But if you were to plot these and get a straight line, then it is a zeroth order. If you were to plot all three of those and not get a straight line you do not know the order but you do know what it is not. You know it is not zeroth, you know it is not first, you know it is not second order. Now, also from each of these you could get the value of the rate constant. If you run into a situation where time and concentration are given and they want to know the order then you want to do it and they want to know the order then you want to do it using these linear plots. Now we have looked at the second order kinetics. We have looked at its integrated rate law we have looked at half-life for second order. We have done a brief mention of zeroth order and we have given a summary of first, second, and zeroth order kinetics, of you were to plot concentration information verse time. And this completes this lesson.
