First-order reactions rate-concentration graphs

FIGURE 13.12 Thu ohange in concentration of the reactant in two first-order reactions plotted on the same graph When the first-order rate constant is large, the half-life of the reactant is short, because the exponential decay of the concentration of the reactant is then fast. 

The more usual pattern found experimentally is that shown by B, which is called a sigmoid curve. Here the graph is indicative of a slow initial rate of kill, followed by a faster, approximately linear rate of kill where there is some adherence to first-order reaction kinetics this is followed again by a slower rate of kill. This behaviour is compatible with the idea of a population of bacteria which contains a portion of susceptible members which die quite rapidly, an aliquot of average resistance, and a residue of more resistant members which die at a slower rate. When high concentrations of disinfectant are used, i.e. when the rate of death is rapid, a curve ofthe type shown by C is obtained here the bacteria are dying more quickly than predicted by first-order kinetics and the rate constant diminishes in value continuously during the disinfection process. 

FIGURE 13.9 The characteristic shape of the graph showing the time dependence of the concentration of a reactant in a first-order reaction is an exponential decay, as shown here. The larger the rate constant, the faster the decay from the same initial concentration. 

FIGURE 13.10 We can test for a first-order reaction by plotting the natural logarithm of the reactant concentration against the time. The graph is linear if the reation is first order. The slope of the line, which is calculated by using the points A and B, is equal to the negative of the rate constant. 

Chemical reactions are classified according to the number of reactants. In the simplest case, the first-order reaction, there is just one reactant and the rate of the reaction is proportional to the concentration of that reactant. Hence, a graph of reaction rate against reactant concentration is a straight line for a first-order chemical reaction (Fig. 7.5). 

To illustrate what Stella is and what it does, since it is somewhat less well known than equation solvers, spreadsheets, and symbolic computation tools, we present the simple case of an irreversible first-order reaction, A B. In Stella there are four icons that can be used to build models. These icons are called stocks, converters, flows, and connectors. We can use a stock to represent the concentration of A and another to represent the concentration of B. A flow then connects A to B. The flow has what appears to he a valve inserted in it. The opening of this valve can be conveniently set by means of a converter, the value of which can be thought of as the rate constant for the reaction. The connector from A to the flow allows the user to specify that the valve opening depends not only on the rate constant but also on the concentration of A. In our example helow we illustrate a first order reaction. The Diagram window for this simple reaction is shown in Figure 7. Besides the Diagram window there are three other windows available in Stella Equation Pad, Graph Pad, and... 

SECTION 14.4 Rate laws can be used to determine the concentrations of reactants or products at any time during a reaction. In a first-order reaction the rate is proportional to the concentration of a single reactant raised to the first power Rate = fc[A]. In such cases the integrated form ofthe rate law is In [A], = —kt + ln[A]o,where [A],isthe concentration of reactant A at time t, k is the rate constant, and [A] is the initial concentration of A. Thus, for a first-order reaction, a graph of In [A] versus time yields a straight line of slope —k. 

Because an Sjsjl reaction is a first-order reaction, the rate of the reaction is linearly dependent on the concentration of the reactant R-L (Eq. 14.24). For example, doubling the concentration doubles the rate. A graph of rate versus concentration thus yields a straight line whose slope is fcj. 

If an excess of L is taken (pseudo first order conditions), sketch a graph to show the variation of the observed rate constant and the concentration of L. Suggest why Lewis acids such as AlBrj accelerate this reaction. 

Figure 11.32 A graph of initial rate against concentration for a first-order reaction (decomposition of dinitrogen pentoxide) 2N20s(g) 4N02(g) + 02(g)... 

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. 

An easy test for a first-order reaction is to plot the natural logarithm of a reactant concentration versus time and see if the graph is linear. The data from Table 20.1 are plotted in Figure 20-4, and the rate constant k is derived from the slope of the line k = -slope = -(-7.30 X 10 s ) = 7.30 X 10 s . An alternative, nongraphical approach, illustrated in Practice Example 20-5B, is to substitute data points into equation (20.13) and solve for k. 

These equations hold if an Ignition Curve test consists of measuring conversion (X) as the unique function of temperature (T). This is done by a series of short, steady-state experiments at various temperature levels. Since this is done in a tubular, isothermal reactor at very low concentration of pollutant, the first order kinetic applies. In this case, results should be listed as pairs of corresponding X and T values. (The first order approximation was not needed in the previous ethylene oxide example, because reaction rates were measured directly as the total function of temperature, whereas all other concentrations changed with the temperature.) The example is from Appendix A, in Berty (1997). In the Ignition Curve measurement a graph is made to plot the temperature needed for the conversion achieved. 

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. 

STRATEGY We need to plot the natural logarithm of the reactant concentration as a function of t. If we get a straight line, the reaction is first order and the slope of the graph is —k. We could use a spreadsheet program or the Living Graph Determination of Rate Constant (first-order rate law) on the Weh site for this book to make the plot. 

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

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. 

In zone a of Figure 2.5, the kinetics are first order with respect to [S], that is to say that the rate is limited by the availability (concentration) of substrate so if [S] doubles the rate of reaction doubles. In zone c however, we see zero order kinetics with respect to [S], that is the increasing substrate concentration no longer has an effect as the enzyme is saturated zone b is a transition zone. In practice it is difficult to demonstrate the plateau in zone c unless very high concentrations of substrate are used in the experiment. Figure 2.5 is the basis of the Michaelis-Menten graph (Figure 2.6) from which two important kinetic parameters can be approximated ... 

A kinetic study has been carried out in order to elucidate the mechanism by which the cr-complex becomes dehydrogenated to the alkyl heteroaromatic derivative for the alkylation of quinoline by decanoyl peroxide in acetic acid. The decomposition rates in the presence of increasing amounts of quinoline were determined. At low quinoline concentrations the kinetic course is shown in Fig. 1. The first-order rate constants were calculated from the initial slopes of the graphs and refer to reaction with a quinoline molecule still possessing free 2- and 4-positions. At high quinoline concentration a great increase of reaction rate occurs and both the kinetic course and the composition of the products are simplified. The decomposition rate is first order in peroxide and the nonyl radicals are almost completely trapped by quinoline. The proportion of the nonyl radicals which dimerize to octadecane falls rapidly with increase in quinoline concentration. The decomposition rate in nonprotonated quinoline is much lower than that observed in quinoline in acetic acid. 

If an amine P-NH2 is used in the aqueous solution, one obtains RCONHP instead of RCOOH. Rates of cleavage of three acyl nitrophenyl esters were followed by the appearance of p-nitrophenolate ion as reflected by increased absorbances at 400 nm. The reaction was carried out at pH 9.0, in 0.02 M tris(hydroxymethyl)aminomethane buffer, at 25°C. Rate constants were determined from measurements under pseudo-first-order conditions, with the residue molarity of primary amine present in approximately tenfold excess. First-order rate graphs were linear for at least 80% of the reaction. With nitrophenyl acetate and nitrophenyl caproate, the initial ester concentration was 6.66xlO 5M. With nitrophenyl laur-ate at this concentration, aminolysis by polymer was too fast to follow and, therefore, both substrate and amine were diluted tenfold for rate measurements. 

Since these reaction products exhibit considerable absorbance at the wave lengths utilized in the rate measurements, the calculation of rate constants required a technique incorporating this factor. Two methods of calculation were employed successfully. In some cases, limiting absorbances (A00) were determined and the rates were obtained from the slopes of graphs of log (A0—A00)/(A—A0o) vs. time. These served to demonstrate the pseudo-first-order nature of the rate constant however, the more general calculation procedure was that due to Guggenheim (11). The first-order dependence of the rate on the concentration of alkyl halide was shown by varying initial concentrations. 

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. 

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. 

Once the first-order behavior with respect to [A] has been verified, the reaction can be nm with varying concentrations of B (B still in large excess over A). A graph of /cobsd function of [B] should be linear the slope is the rate constant k. For large variations in [B], resulting in large variations in fcobsd/ h is... 

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