Rate-concentration graphs, determination

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. 

For example, to determine the half-life in the preceding example, the Co value and the time at which C = Co/2 were both read directly from the graph. If Fig. 3 had been a plot of In C (on a Cartesian scale) versus time, it would have been necessary to read In Co from the graph, convert it to Co, divide by 2, convert back to In (Co/2), then read the half-life off the graph. If the rate constant is determined for this example using Eq. (12), the slope must be calculated. To calculate the slope of the line it is necessary first to read two concentrations from the graph and then take the logarithm of each concentration as described in Eq. (12). 

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. 

As an initial approximation of the ideal solvent strength for HPLC, the results of TLC studies on acetylated cellulosic plates with various methanol-water mixtures and natural dye extractions were graphed (R/ vs. methanol concentration). An acetylated cellulosic TLC system is not directly comparable to a C-18 HPLC system, but TLC results were nonetheless useful as a rough estimate of solvent strength. In practice, approximately two-thirds of the solvent strength required to elute natural dyes with TLC was necessary to achieve a similar separation of major sample components with HPLC. After individual solvent concentrations were determined, samples of known dyes extracted from wool were eluted in each of the three pairs of solvents (the concentration used for each individual solvent-water system was reduced by one-half) and adjustments were made until each sample eluted with a kf value no larger than 10. Retention times and kf values are equivalent expressions of relative retention of a sample on the column if the flow rate is the same for all trials. With a flow rate of 1.4 mL/min, the maximum time... 

The CAR technique is a major alternative to the stopped-flow technique in tackling direct rate measurements on reactions with half-lives of a few milliseconds as it requires no sophisticated instrumentation and is specially suited to routine analyses. In addition, it allows mixtures of two or more analytes to be resolved. For a binary mixture, the response versus time graph typically obtained shows two consecutive linear segments of different slope from which each reaction rate can be determined and related to the concentration of each component. 

FIGURE 4.25 Indoor radon concentration as a function of the ventilation rate. Although radon-222 undergoes radioactive decay, its half-life, 3.8 days, is long compared with the ventilation rates in the graph. Therefore, radioactive decay is a relatively minor sink for indoor radon. The graph also can be applied for other indoor chemicals whose concentration is determined primarily by the balance between source strength and loss by ventilation (US EPA, 1986). 

Every rate law must be determined experimentally. A chemist may imagine a reasonable mechanism for a reaction, but that mechanism must be tested by comparing the actual rate law for the reaction with the rate law predicted by the mechanism. To determine a rate law, chemists observe how the rate of a reaction changes with concentration. The graph of the data for the NO2 decomposition reaction shown in Figure 15-6 is an example of such observations. 

H2(g) + Br2(g) 2 HBr(g) To determine the rate law for this reaction, a chemist performed two isolation experiments using different initial concentrations. Both experiments gave linear graphs of... 

Graphs such as those in Figures 8.5 and 8.6 are an ideal means of determining the rates of reaction. To obtain the rate, we plot the concentration of a reactant or product as a function of time, and measure the slope. (Strictly, since the slopes are negative for reactants, so the rate is slope x —1 .)... 

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. 

Graphs relating antipyrine concentrations and time were used to calculate clearance rates. A relationship between apparent antipyrine steady state concentrations at 120 and 240 minutes (api 2 o, ap2 o) and mussel body water and mantle cavity water was also determined (k). Mantle cavity water is that volume held between the valves when the mussels are closed, e.g., when transferred from the uptake solution (300 ml) to the elimination solution (300 ml). The initial antipyrine concentration (apo) was determined at the beginning of the experiment. Assuming no loss of antipyrine, complete mixing of the solutions, and its distribution into total mussel body water, when an apparent steady state is achieved, the following results ... 

In the following ThoughtLab, you will use experimental data to draw a graph that shows the change in concentration of the product of a reaction. Then you will use the graph to help you determine the instantaneous rate and average rate of the reaction. 

In this section, you learned how to express reaction rates and how to analyze reaction rate graphs. You also learned how to determine the average rate and instantaneous rate of a reaction, given appropriate data. Then you examined different techniques for monitoring the rate of a reaction. Finally, you carried out an investigation to review some of the factors that affect reaction rate. In the next section, you will learn how to use a rate law equation to show the quantitative relationships between reaction rate and concentration. 

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. 

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. 

Figure 3.25 Log10 plasma concentration time profile for a foreign compound after intravenous administration. The plasma half-life (fo) and the elimination rate constant (fce ) of the compound can be determined from the graph as shown.

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. 

Use the Rates of Reaction simulation (eChapter 12.2) to determine the value of the rate constant for the process at 0°C. Calculate the half-life for the process from the value of k. Select a value for initial concentration and run the reaction. Determine the half-life using the [A] vs Time graph. How does doubling the initial concentration affect the half-life ... 

Franco et al. [45] described an HPLC method for simultaneous determination of the R-( ) and (S)-(+)-enantiomers of vigabatrin in human serum after precolumn derivatization with 2,4,6-trinitrobenzene sulfonic acid (TNBSA) and detection at 340 nm. Separation was achieved on a reversed phase chiral column (Chiralcel-ODR, 25 cm x 4.6 mm) using 0.05 M potassium hexafluorophosphate (pH4.5) acetonitrile ethanol (50 40 10) as a mobile phase at a flow rate of 0.9 ml/min. The calibration graphs for each enantiomer were linear over the concentration range of 0.5-40 fig/ml with a limit of quantification of 0.5 fig/ml. No interferences were found from commonly coadministered antiepileptic drugs. 

Figure 3.5 Determination of order graphs of rate versus concentration, (concentration)2 and (concentration)1/2...

Though the reaction mechanism here is more complex than in the previous example and the kinetic equation also has non-Arrhenius parameters, it is possible to determine all the reaction rate constants. The fact is that there is a sufficient quantity of the Arrhenius complexes. In this case it appears that all "mixed complexes, i.e. complexes containing parameters of both direct and inverse reactions, are independent. Here these complexes evidently corresponding to the mixed spanning trees of the graph are coefficients for various concentration characteristics. It is this fact that permitted us to obtain the convenient eqns. (82). 

---