First-order reactions concentration-time graphs

Analyze and Plan To estimate a half-life of a first-order reaction from a graph, we can select a concentration and then determine the time required for the concentration to decrease by half. 

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. 

Figure 21.3 Graph of concentration c against time t for a first-order reaction...

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. 

The change in concentration over time for the first-order rearrangement of methyl isonitrile at 198.9°C is graphed in Figure 14.9 T. The first half-life is shown at 13,300 s (that is, 3.69 hr). At a time 13,300 s later, the isonitrile concentration has decreased to one-half of one-half, or one-fourth the original concentration. In a first-order reaction, the concentration of the reactant decreases by in each of a series of regularly spaced time intervals, namely, iyi- The concept of half-life is widely used in describing radioactive decay, a first-order process that we will discuss in detail in Section 21.4. 

It is important in kinetics to discover how the rate of reaction varies with concentration of the reactants. It allows chemists to deduce the order and the rate expression for the reaction (Chapter 16). One simple, but not very precise, approach is to draw a number of tangents on a concentration-time graph (Figure 6.28) and then plot a graph of the rates (the numerical value of the gradients) against concentration. Many reactants show a directly proportional relationship between concentration and rate (reactions in which this is the case are said to be first order for that reactant (Chapter 16)). 

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. 

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. 

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... 

To show that this reaction is kinetically first order, we take the logarithm of the concentration, and plot ln[A]r (as y ) against time t (as V) see Figure 8.11. That the graph in Figure 8.11 is linear with this set of axes demonstrates its first-order character. 

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. 

From the data of runs Cl to C20 and D1 to D20, calculate x, the number of moles of sucrose hydrolyzed in each time interval. If the reaction were zero order in sucrose, then we would expect that (x/0.003) = kf, where x/0.003 is the concentration of either of the product species in mol L units. Prepare a graph of the results obtained in these two series of runs, plotting x versus t, and indicate whether the data are consistent with the hypothesis that the reaction is zero order in sucrose. Note that, even if a reaction starts out being zero order in sucrose, this cannot continue indefinitely. Indeed, we expect the inversion reaction to become first order in sucrose when (S) becomes sufficiently small. 

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. 

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