First-order reaction exponential decay

This equation demonstrates the exponential decay of the rate of formation of products in a first-order reaction widr time. When... 

In a first-order reaction, the concentration of reactant decays exponentially with time. To verify that a reaction is first order, plot the natural logarithm of its concentration as a function of time and expect a straight line the slope of the straight line is —k. 

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 important phenomenon of exponential decay is the prototype first-order reaction and provides an informative introduction to first-order kinetic principles. Consider an important example from nuclear physics the decay of the radioactive isotope of carbon, carbon-14 (or C). This form of carbon is unstable and decays over time to form nitrogen-14 ( N) plus an electron (e ) the reaction can be written as... 

If it is certain that the reaction is indeed an irreversible first-order reaction, one can also determine how long it takes before 50 % of the reactant has been converted into products, as for any exponential decay the half-life, ty, is related to the rate constant k as... 

Fig. 9 Exponential decay of reactant ini a first-order reaction.

That is, A decays exponentially with time determined by (kl7[B]0), as if it were a first-order reaction. Thus under these so-called pseudo-first-order conditions, a plot of ln[A] against time for a given value of [B]0 should be linear with a slope equal to ( — I7[B]0). These plots are carried out for a series of concentrations of [B](l and the values of the corresponding decays determined. Finally, the absolute rate constant of interest, kl7, is the slope of a plot of the absolute values of these decay rates against the corresponding values of [B] . Some examples are discussed below. 

Fig. 1.8. Consecutive first-order reactions with p0 = 0.1 mol dm-3, fcu = 5 x 10-3 s and k2 = 10"2s (a) the exponential decay of precursor reactant concentration, p (b) growth and decay of intermediate concentrations a(t) and b(t). Also shown in (b), as broken curves, are the pseudo-stationary-state loci, a (t) and b (t), given by eqns (1.31) and...

Figure 2. (a) Independent first-order reactions giving nonexponential decay (b) competitive first-order reactions giving exponential decay. 

If there were a first-order reaction taking place, M0 would decay exponentially, but so would Mi and M2, so that the mean and variance would be the same. If we had a number of parallel tubes communicating with each other, we would have to develop equations for the moments in each before averaging them to get the overall mean and variance. 

In this case, the sampled concentration will not represent the exponential decay of a first-order reaction at any flow rate because of the z 1/2 term. 

In the presence of noise, the signal from a first-order reaction is an exponential decay... 

Equations 13a and 13b are two forms of the integrated rate law for a first-order reaction. The variation of concentration with time predicted by Eq. 13b is shown in Fig. 13.9. This behavior is called an exponential decay. The change in concentration is initially rapid, but it changes more slowly as time goes on and the reactant is used up. 

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 12.6 Plots of (a) reactant concentration versus time and (b) natural logarithm of reactant concentration versus time for a first-order reaction. A first-order reaction exhibits an exponential decay of the reactant concentration (a) and a linear decay of the logarithm of the reactant concentration (b). The slope of the plot of ln [A] versus time gives the rate constant. 

BALLPARK Check The concentration of H2O2 (0.23 M) after 4.00 h is less than the initial concentration (0.30 M). A longer period of time (14 h) is required for the concentration to drop to 0.12 M, and a still more time (36 h) is needed for the concentration to fall to 0.030 M (10% of the original concentration). These results look reasonable. A plot of [H2O2] versus time would exhibit an exponential decay of the H2O2 concentration, as expected for a first-order reaction. 

Were it not for the coupling terms, kbAcD and k fAcA, Eq. 4.34 would have the same form as Eq. 1.55 (neglecting its constant term on the right side), with an exponential-decay solution typical of first-order reactions (Eqs. 1.56 and 4.19). Because the coupling terms are linear in the Ac, however, it is always possible to find a solution to Eq. 4.34 by postulating that a pair of time constants, r, and r2, exists such that the Ac still show an exponential time dependence ( rel axation ) 19... 

Such a chemical reaction, in which molecules are not colliding with other atoms or molecules, is called a first-order reaction because the rate at which chemical concentration changes at any instant in time is proportional to the concentration raised to the first power. Certain chemical processes, such as radioactive decay, are described by first-order kinetics. In the absence of any other sources of the chemical, first-order kinetics may lead to exponential decay or first-order decay of the chemical concentration (i.e., the concentration of the parent compound decreases exponentially with time) ... 

The direct coupling of P430 with MV was further supported by the acceleration ofthe decay rate ofthe former that is observed with increasing MV concentration. The kinetic plots showed that P430 decay was exponential at all MV concentrations examined. The decay ty, vs. MV concentration fitted a pseudo-first-order reaction, giving an estimated rate constant of 9.6-10 M - at 22 °C. ... 

The advantage of identifying the reaction order is that all reactions with the same rate law (but different characteristic rate constants) behave similarly. For example, the concentration of a reactant in a first-order reaction decays exponentially with time at a rate determined by the rate constant... 

Studies of tethered electroactive species are less sensitive to pinholes than experiments with solution reactants and blocking layers, although heterogeneity and roughness of the substrate and film defects can still play a role. The rate constant, k, in this case has units of a first-order reaction (s ). Rate constants can be determined by a voltammetric method as described earlier for electroactive monolayers (Section 14.3.3). In addition potential-step chronoamperometry can be employed, in which case the current follows a simple exponential decay (88, 90, 91) ... 

Hg. 14.11 The exponential decay of a reactant in a first-order reaction where (a) k is large (b) k is small. The larger the value of k, the faster is the decay. 

Bc2 produced by flash photolysis of seawater decays by parallel first- and second-order reactions. The environmentally important exponential decay is a pseudo first-order reaction of Br2 with the carbonate/ bicarbonate system in seawater. A chemical speciation model for the free ions and ion-pairs in seawater and in solutions at seawater ionic strength allowed us to measure the dependence of the pseudo first-order rate term, a, on individual C02-containing species. A predictive equation based on reaction of Br2 with free C03 and the hgC03°, NaC03 and CaC03 ion pairs accounts for the mean seawater a at pH 8.1 within experimental uncertainty. The reaction productfs) are unknown. 

Thermolysis of the endoperoxide in C5D5 gave lQ2( Ag) luminescence at 1270 nm which decayed exponentially as the peroxide reacted by a first-order reaction. In agreement with the photochemical work above and that reported earlier " , phthalocyanine quenched IO2 monomol emission. Weak luminescence near 700 nm was strongly enhanced by the addition of phthalocyanine. The 700 nm luminescence decay is also exponential, but the rate constant for the apparent decay is twice that observed at 1270 nm, in agreement with the photochemical work. At the same temperature and endoperoxide concentration, the decay curve of the luminescence at 700 nm is almost exactly the square of that at 1270 nm (Fig. 5). 

Equation 12.49 is the basic nondimensional equation describing the mole fraction of A in a fixed-bed reactor containing an exponentially decaying catalyst as a function of position and time in terms of two dimensionless parameters, B" and A. The performance of this reactor can be best judged by solving the equation for the reactor exit, that is, for z = 1. The solution for a first-order reaction (m = 1) is given in Table 12.7 (Sadana and Doraiswamy, 1971). It is also possible to assume various other forms of catalyst decay. Solutions are included in the table for two other forms, one of them linear. 

Equation (16.14) illustrates a characteristic of first-order reactions the concentration of the reactant decreases exponentially with time (compare dashed curve in Fig. 16.4). The paradigm for this kind of process is radioactive decay, but all monomolecular elementary reactions such as the rearrangement of cyclopropane into propane in the gas phase are included in this. There are many further decomposition reactions to be found in classical chemistry, such as decomposition of dinitrogen pentaoxide N2O5 in the gas phase according to... 

Fig. 7.3.2 Exponential decay of hydrogen atoms in a tube with catalytic walls. These data were obtained by traveling a thermocouple probe axially down a closed Pyrcx tube. At one end of the tube, a discharge generates H atoms. The atoms diffuse down the tube and recombine catalytically on the walls following a first-order reaction. The probe is covered with a very active catalyst for the exothermic recombination. As a result, its temperature is higher than that of the walls by an amount AT", which is proportional to the concentration of atoms at the location L of the probe. This dimensionless distance L is the distance down the tube divided by the tube rtidius. [See K. Tsu and M. Boudart, AcUs 2 Congres Intern. Catalyse, Vol. I, p. 593, Technip, Paris (1961).]...

A quick glance at Figures 11.6 and 11.7 shows a glaring difference between the two plots of concentration versus time one is linear and the other is clearly not. The curve shown for the first-order reaction in Figure 11.7 is an example of what is known as exponential decay. You are probably familiar with this type of curve from math classes, and you will see many other physical phenomena that show this kind of behavior as you continue your engineering studies. 

The t iRt QedioRiE es jfe i SiKg of the experiment it can be any time when we choose to start monitoring the change in the concentration of A. The exponential decay with time indicated by Equation 14.6 is a signature of first-order reactions [Eigure 14.9(a)]. 

Figure 14.9 The time dependence of reactant concentration for a first-order reaction, (a) A plot of reactant concentration versus time showing the exponential decay of Equation 14.6. (b) A log-linear plot of reactant concentration versus time gives a straight line (Equation 14.7) with a slope of —k and an intercept of ln[A]o.

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