Lecture 22: The Arrhenius Equation and reaction mechanisms. As we wrap up kinetics we will: • Briefly summarize the differential and integrated rate law equations for 0, 1 and 2 order reaction • Learn how scientists turn model functions like the integrated rate laws into straight lines from which useful information can be found in the slope and y-intercept • Learn about the three factors found in the rate constant: A, Ea and T are related in the Arrhenius equation • Learn how the activation energy can be extracted from concentration time data using the combined Arrhenius equation • Learn about two theories developed to explain kinetics: collision and transition state theory • Learn about how the rate law for a reaction is created from the reaction mechanism • Look at some famous catalysts First, a summary of the differential and integrated rates laws from the first kinetics lectures: In addition to the equations we have derived, note the comparison curves for first and second order integrated equations are also provided. In plot (a) note that in the first t1/2 of 1.73 s, the concentration of A falls from 1.0M to 0.5M. It falls again by half from 0.5M to 0.25M in the next 1.73 s. And on and on. Contrast this with a second order reaction in (b) where during the first 2.5 s t1/2, the concentration falls from 1.0M to 0.5M. However the second t1/2 takes 5 s for the concentration to be cut in half; the third t1/2 takes 10 s. Note the concentration dependence of t1/2 for all reactions that are not order one.23Curve Fitting: Graphing the integrated rate equation. Something scientists do a lot is to try to find a best fit of a theoretical equation to experimental data. Let’s see if there is any value in doing it for the integrated rate equations that were and are shown in the table above. Remember these are the equations that let us find out how much stuff we have after a reaction has been going on awhile. In trying to fit experimental data, it is important to know what kind of function will fit the data. Scientists hope to find simple relationships like straight lines or parabolas because they aren’t crazy about doing hard math either. Well the good news for the integrated rate equations is that each of them can be arranged to fit a straight line. Remember that a straight line has the form: y = mx + b with two constants, a slope, m, and a y-intercept, b. The dependent y-variable is plotted as a function of the independent x-variable.4One of the reasons scientists try to fit their data is that parameters like the slope or y-intercept actually correspond to important scientific information. For example, in doing kinetics, a couple important pieces of information are the rate constant, k, and the amount of starting material in a reaction, [Ao]. Wouldn’t it be great if we could extract that kind of information from a kinetics plot? Let’s try to fit the first order integrated rate equation we derived. to a straight line plot. [Ao] ln ⎯⎯⎯ = a k t [A] Using a bit of knowledge about the properties of log functions, we can rewrite the natural log term as a subtraction. a ln ⎯ = ln(a) - ln(b) ln [Ao] - ln[A] = akt b Now let’s rearrange: ln[A] = -akt + ln[Ao] y = mx + b This ought to look an awful lot like a straight line function and in fact that is what we see in the graph above where a plot of ln[A] versus time yields useful information.5We can do the same for the second order integrated rate equation: 1 1 ⎯⎯ = akt + ⎯⎯ [A] [Ao] and for zero-order integrated rate equation: [A] = -akt + [Ao] This idea of fitting data to a known function for the purpose of extracting useful scientific information is incredibly important in science. It is presented as yet another example of an effort to make math real to budding scientists like you. More than understanding its application here to kinetics, I’d rather you recognize the general concept of fitting functions to experimental data because you will do it a lot during your science courses at UT. For example, you will probably have to do it again before the end of the kinetics chapter for an entirely different application. Example. To get you started on this with some real data, look at the plot of concentration time data on the next page—it is taken from the answer key of your worksheet. Note that for the data I recalculated the data by taking a natural log in one column and a reciprocal in another column. I can now look at the results—only one of the columns will have data that has a straight line (the slope doesn’t change. Which one is ? Answer: The ln [C] plot which corresponds to the first order integrated rate law equation. So it must be true that this is the data for a first order reaction and I can now go on and collect more kinetics data—like Ea with the combined Arrhenius equation.6 Reaction Mechanisms. If life was really simple, then a reaction mechanism for a reaction like: A + B Æ AB would simply involve a bimolecular reaction between A and B. The rate expression for the reaction would be rate = k [A][B] and the reaction would be first order in [A] and [B]. Sadly, it is rarely this simple. Instead reactions often go through a series of simple steps which taken together make up the reaction pathway.7reaction mechanism: The step-by-step process by which a reaction occurs. It usually involves a series of smaller unimolecular or bimolecular reaction steps. Example. Reaction of NO2 + CO -----> NO + CO2 In a simple world the mechanism for the reaction would be rate = k[NO2][CO] but what if I told you the actual rate law was rate = k[NO2]2 This result should tell you right off that experimental kinetics is hard. The result above is indeed the experimentally verified mechanism for the reaction, found by the same methods we used in method of initial rates. So what is happening in this reaction? No one really knows for sure, but several mechanisms have been postulated including NO2 +NO2 -----> NO3 +NO NO3 + CO ----> NO2 + CO2 _____________________________________________ NO2 + CO -----> NO + CO2 NO3 is referred to here as a reaction intermediate. It is neither a
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