LLEECCTTUURREE 22:: KKIINNEETTIICCSS CCAALLCCUULLAATTIIOONNSS UUSSIINNGG TTHHEE DDIIFFFFEERREENNTTIIAALL AANNDD IINNTTEERRGGRRAATTEEDD RRAATTEE LLAAWWSS How to do the famous kinetic calculations, applied to 0th, 1st and 2nd order reactions In this lecture you will learn to do the following • Determine reaction order from units • Determine reaction order from method of initial rates • Calculate “k” from rate equation • Convert how fast (differential rate equation) into how much (integrated rate equation) using calculus • Use integrated rate law to find half lives • Use integrated rate law to find extent of reaction First we review the differential rate equation and define the pieces of the rate law: Δ [R ] = rate = [R]x A exp [-Ea/RT] = k [R ]x Δ t rate constant, k Rate From Slope rate law Note that there are four physical parameters in the rate law that determine the rate of reaction: 1. [ ] x concentration and order of reactants and products 2. Ea activation energy 3. T temperature the three factors making up k, the rate constant 4. A pre-exponential factor We will save learning to do calculations involving parameters 2-4 for Lecture 3 when we learn about kinetic theory and concentrate for now on the concentration term, [R ], and how the order of reaction, x. [R ]xOrder of Reaction In the rate equation: rate = k [R ]x , x is the order of reaction in the function that describes how concentration affects rate. Example of reaction order: 2N2O5 ----> 4NO2 + O2 rate = k[N2O5] This reaction is first order in [N2O5] and first order overall. Example of reaction order: 2NO + O2 ----> 2NO rate = k[NO][O2] This reaction is first order in NO and in O2 and second order overall. We will look at three examples of reaction order in this lecture. order rate equation effect of concentration on rate 0 rate = k[R]0 = k none 1 rate = k[R]1 rate increases linearly with concentration 2 rate = k[R]2 rate increases as the square of the concentration Example 1: 0th order x = 0 so Rate = k [R]0 = k This means that rate is independent of concentration [R] as is shown below. Note that no matter what the concentration, the slope does not change. 0th order [ ] tExample 2: 1st order x = 1 Rate = k [R]1 This means that rate varies with concentration [R] and as we saw in class, follows an exponential function—very fast initial rates and then, at longer times, a very small slope. Example 3: 2nd order x = 2 Rate = k [R]2 This means that rate varies with concentration [R] 2 and is most commonly associated with bimolecular collision reactions—what you would traditionally think of as the way a reaction happensÆ A collides with B and makes a product. A+B Æ P. This kind of reaction is actually less likely than you might think, as we will learn when we get to kinetic theory. [ ] t 1st order exponential charge [ ] t 2nd orderTIME OUT FOR CALCULUS. The rest of this lecture will follow what happens in two equations, a differential rate equation and an integrated rate equation. Just a reminder of what you should have learned in calculus as a scientist, if you learned nothing else: Differential Calculus Tells how fast something happens—it is a value found from the instantaneous slope of a function. Integral Calculus Tells us how much we have of something and is found from area under the curve made by a function. Understand that how fast (from the differential rate equation) and how much (from the integral rate expression) are really what we most want to calculate in kinetics—and we are using the simple tools we learned in calculus to do it. Can get instantaneous slopes along function Smaller slope Larger slopeWell That is pretty much it for calculus—back to kinetics. Let’s look at kinetics calculations for reactions with different reaction order. Look at x = 0, 1, 2 First, look at x = 1 = first order reaction in the simple reaction below rR Æ pP (assume r = 1) We know that the rate is found from the slope of either R or P - Δ [R] = + Δ [P] = rate = k [R]x [P]y rΔ t pΔ t and the rate law tells us how concentration matters—but how do we find out x and y? There are a variety of methods, and in fact I have a nice summary sheet posted on the many ways to find the order of a reaction. But experimentally, in lab you will use something called: Method of initial rates Run multiple kinetics experiments with different starting amounts. For each, find the slope (rate) of the reaction right at the start. Note in the data below that we are being intelligent scientists by holding one concentration constant while varying the other. This allows us to isolate how the concentration effects the rate of reaction. We want to look here at the initial data point to find rateData for a method of initial rates problem with three experimental rates determined. [R] [P] rate Experiment 1 .1M .1M 2 x 10-4 Experiment 2 .2M .1M 4 x 10-4 Experiment 3 .2M .2M 4 x 10-4 First, find order of R (the values of x) Mathematically this is the ratio of the rate of reaction to the concentration [R ] in experiments 1 and 2 where [P] is held constant. x 4 x 10-4 = 0.2 2 x 10-4 0.1 And solving: 2 = 2x so x = 1 and R is first order Now, what is the order of P (the value of y)? Do the same way using experiments 2 and 3 in which [R] is held constant. 4 x 10-4 = 0.2 y 4 x 10-4 0.1 And solving: 1 = 2y so y = 0 so P is 0th order Overall the first order rate law is: rate = Δ [R ] = k [R]1 [P]0 = k[R]1 Δ t Now can we find value of k? Stick in data from method of initial rates. Use experiment 1. Rate = 2 x 10-4 = k [0.1] 1 k = 2 x 10-3 = rate constant which is the same for all experimentsCan we also find the units of k? Simply cancel units in rate expression [R ] = k [R ] cancel and 1/t = k t This means that the units for first order rate expression are reciprocal time: like sec-1 So k = 2 x 10-3 sec-1 Integrated Rate Equation for a first order process. Recall that earlier I suggested it might be nice to know what the function was that described the relationship between
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