Second-order reactions rate-concentration graphs

Compare a z) for first- and second-order reactions in a PFR. Plot the profiles on the same graph and arrange the rate constants so that the initial and final concentrations are the same for the two reactions. 

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...

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) ... 

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. 

Figure 16.9 Rate-concentration graphs for zero-order, first-order and second-order reactions... 

For a first-order reaction, the initial rate of reaction is directly proportional to the concentration of the reactant and the resulting graph is a sloping straight line (Figure 16.9). For a second-order reaction, the initial rate of reaction increases with concentration in a quadratic manner and the resulting curve is known as a parabola. 

The dependence of the rate on concentration in first-, second-, and third-order reactions is displayed graphically in Figure 4.6. The y axis is the slope taken from graphs of the sort in Figure 4.6 at various [A] values, and the X axis is [A]. For simplicity, the rate is arbitrarily set at 1 where [A] is 1 for all three reactions. In the first-order reaction, you can see that the rate drops to 1/2 when [A] is 1/2, and the relationship is linear. In the second-order reaction, the rate drops to 1/4 when [A] is 1/2 and the curve is a parabola. In the third-order reaction, the rate drops to 1/8 when [A] is 1/2 and the curve is cubic. 

This form assumes that the effect of pressure on the molar volume of the solvent, which accelerates reactions of order > 1 by increasing the concentrations when they are expressed on the molar scale, has been allowed for. This effect is usually small, ignored but in the most precise work. Equation (7-41) shows that In k will vary linearly with pressure. We shall refer to this graph as the pressure profile. The value of A V is easily calculated from its slope. The values of A V may be nearly zero, positive, or negative. In the first case, the reaction rate shows little if any pressure dependence in the second and third, the applied hydrostatic pressure will cause k to decrease or increase, respectively. A positive value of the volume of activation means that the molar volume of the transition state is larger than the combined molar volume of the reactant(s), and vice versa. 

SAQ8.16 Consider the following data concerning the reaction between triethylamine and methyl iodide at 20°C in an inert solvent of CCI4. The initial concentrations of [CH3l]o and [N(CH3)3]0 are the same. Draw a suitable graph to demonstrate that the reaction is second order, and hence determine the value of the second-order rate constant k2. 

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. 

Tangents can be drawn at various points and Table 16.7 shows the values calculated for the rate at five different concentrations of acid. These results are then plotted as a graph of rate against concentration graph. They produce a curve, indicating that the reaction is second order (Figure 16.6). 

A graph of reaction rate against concentration tells us whether a reaction is zero, first, second or third order with respect to a particular reagent (or overall). It is very rare to obtain an order with respect to a particular reagent higher than second order. Figure 22.9 shows the shapes of the graphs expected for different orders of reaction. 

Figure 22.14 A graph showing how concentration changes of hydrochloric acid or methanol affect rate of reaction. The curve shows that the reaction is likely to be second order.

But the slope of the second graph is zero The rate-determining step does not involve NaOH so adding more of it docs not speed up the reaction. The reaction shows first-order kinetics (the rate is proportional to one concentration only) and the mechanism is called S l, that is, Substitution, Nucleophilic, 1st order. 

Assuming that the reverse reaction is negligible, determine whether the reaction is first, second, or third order, and find the value of the rate constant at this temperature. Proceed by graphing ln(c), 1 /c, and 1 /c, or by making linear least-squares fits to these functions. Express the rate constant in terms of partial pressure instead of concentration. 

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