Second-order reactions concentration-time graphs

Figure 8.9 Kinetics of a second-order reaction the racemization of glucose in aqueous mineral acid at 17 °C (a) graph of concentration (as y ) against time (as V) (b) graph drawn according to the linear form of the integrated second-order rate equation, obtained by plotting 1 / A, (as V) against time (as V). The gradient of trace (b) equals the second-order rate constant k2, and has a value of 6.00 x 10-4 dm3mol 1s 1...

Figure 23.6 Graph of reciprocal of concentration 1/c against time t for a second-order reaction...

Expressions similar to equation (17) may easily be derived for various second-, third-, and higher-order reactions. These expressions are readily integrated for all second-order reactions and for many third- and higher-order reactions, yielding (in many cases) relations analogous to equation (18), which define useful concentration-time graphs. The dimensions of the rate constant k for an nth order reaction are (concentration) (time) ... 

FIGURE 18.7 For a second-order reaction such as 2 NO2 —> 2 NO + O2, a graph of the reciprocal of the concentration against time is a straight line with slope 2k. 

A second-order reaction is one for which the overall reaction order is 2. If a second-order rate law depends on the concentration of only one reactant, then rate = k[A], and the time dependence of [A] is given by the integrated form of the rate law 1/[A], = 1/[A]q + kt. In this case a graph of 1/[A] t versus time yields a straight line. A zero-order reaction is one for which the overall reaction order is 0. Rate = fc if the reaction is zero order. 

In the case of first-order (Figure 16.2) and second-order reactions (Figure 16.3), the graph obtained is a curve. If the reaction is second order then a deeper curve is obtained for the graph of concentration against time. The first-order curve is an exponential curve and the second-order curve is a quadratic curve. It can be hard to distinguish between these two types of kinetic behaviour using experimental data with random uncertainties. 

Sketch, without carrying out the calculation, the variation of concentration with time for the approach to equilibrium when both forward and reverse reactions are second order. How does your graph differ from that in Fig. 7.2 ... 

Figure 22.12 The half-life of zero-, first- and second-order reactions can be determined from graphs of concentration against time.

In any case what is usually obtained is a graph showing how a concentration varies with time. This must be interpreted46 to obtain a rate law and a value of k. If a reaction obeys simple first- or second-order kinetics, the interpretation is generally not difficult. For example, if the concentration at the start is A0, the first-order rate law... 

To determine whether the reaction is first order or second order, calculate values of In [NO2] and 1/[N02], and then graph these values versus time. The rate constant can be obtained from the slope of the straight-line plot, and concentrations and half-lives can be calculated from the appropriate equation in Table 12.4. 

The reaction begins without delay. The rate decreases as the degree of conversion increases. A graph showing vinyl concentration according to second order kinetics plotted against time gives a linear relationship (Fig. 5). 

If we were to change the kinetics so that the first reaction was second order in A and the second reaction was first order in B, then we would see largely the same picture emerging in the graphs of dimensionless concentration versus time. There would of course be differences, but not large departures in the trends from what we have observed for this all first-order case. But what if the reactions have rate expressions that are not so readily integrable What if we have widely differing, mixed-order concentration dependencies In some cases one can develop fully analytical (closed-form) solutions like the ones we have derived for the first-order case, but in other cases this is not possible. We must instead turn to numerical methods for efficient solution. 

The available data are the reactant concentration as a function of time for a single experiment, so we will need to use graphical techniques to determine the order of the reaction. There are three possibilities we can explore using the integrated rate laws we ve examined. The reaction could be zero order, first order, or second order with respect to NO2. We will need to manipulate and plot the data in various ways to determine whether there is a good fit with any of these models. (Other orders are also possible, so we should be aware that all three tests could conceivably fail.) With a spreadsheet or a graphing calculator, such manipulation of data is easy. For this example, first we will calculate all of the data needed for all three plots and then make the appropriate graphs to find the linear relationship and determine the rate law. 

Interpolation is a technique where a graph is used to determine data points between those at which you have taken measurements. Figure 11.34 is a graph of concentration of hydrogen peroxide against time. It is an exponential graph and the dotted construction lines are interpolation lines to prove that it is a first-order reaction (Chapter 16). The half-life of the reaction is approximately 25 seconds. 

The accompanying graph shows the concentration of a reactant as a function of time for two different reactions. One of the reactions is first order and the other is second order. Which of the two reactions is first order Second order How would you change each plot to make it hnear ... 

Kinetics. Madinaveitia and Quibell (112) and McClean and Hale (106) stated that the reaction time R to the half-viscosity is independent of the substrate concentration and inversely proportional to the enzjrme concentration. Actually, a graph of the data (112) shows a linear relationship between R and enzyme concentration only for reaction times up to 20 minutes. Similar observations were made by Haas (57) and by Dorfman (26). It is obvious that the enzyme should be assayed at concentrations high enough to give reaction times R < 20 minutes in order to avoid serious errors. Dalgaard (23), for the same reason, used a reaction time of 300 seconds. 

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