The First-Order Rate Law

Let us consider a reaction that produces a gas, G, as the reactant, R, is transformed into the product, P. The equation can be written as 

We will represent the amount of reactant R by W, the weight of the material. If the reaction follows a first-order rate law. 

If Wo represents the amount of reactant initially present and W represents the amount at some later time, the integrated rate law is 

However, the fraction reacted, a, is given by the amount reacted (Wo — W) divided by the amount of reactant initially present. Wo- 

Because the first-order rate law applies to cases in which the initial rate is the maximum rate, the curves that show a versus time have a decreasing slope throughout. Therefore, they are deceleratory in character because the rate of the reaction decreases. This type of behavior is also exhibited by other rate laws that are based on the concept of reaction order (second, third, etc.). 

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Case 2. The particles rotate in small packets ( coherently or in phase ). Obviously, the first-order rate law no longer holds. In chapter B2.1 we shall see that this simple consideration has found a deeper meaning in some of the most recent kinetic investigations [21]. 

Several additional approaches for analyzing mixtures have been developed that do not require such a large difference in rate constants.Because both A and B react at the same time, the integrated form of the first-order rate law becomes... 

Figure 6.1 Volume of nitrogen evolved from the decomposition of AIBN at 77°C plotted according to the first-order rate law as discussed in Example 6.1. [Reprinted with permission from L. M. Arnett, /. Am. Chem. Soc. 14 2021 (1952), copyright 1952 by the American Chemical Society.]...

Develop a meehanism to aeeount for the first-order rate law for the ease where only methane is formed as a byproduet. Assume that the ehain length is large. 

Kinetic 37, 50-52, 97, 168) and stereochemical 54, 191, 192) investigations on the carbonylation of manganese alkyls and the decarbonylation of manganese acyls were already discussed in Sections III-V. The original finding (50) that the rate of CO insertion follows second-order kinetics has now been qualified 192). At higher pressures of CO (>15 atm) the first-order rate law [Eq. (23)] is obeyed. 

Students who have taken calculus will recognize that Equation results from integration of the first-order rate law. 

A sample of any unstable nuclide undergoes nuclear decay continuously as its individual nuclei undergo reaction. All nuclear decays obey the first-order rate law Rate = C. This rate law can be treated mathematically to give Equation, which relates concentration, c, to time, t, for a first-order process (Cq is the concentration present at... 

The first-order rate law is Rate = [C6H5N2Cl], which we solve for k Rate... 

If now we consider a large number of molecules N0, the fraction still in the excited state after time t would be N/N0 — e kt where N is the number unchanged at time t. This exponential law is familiar to chemists and biological scientists as the first-order rate law and by analogy fluorescence decay is a first-order process—plots of fluorescence intensity after an excitation event are exponential and each type of molecule has its own characteristic average lifetime. 

In the meantime, E. Rutherford (NLC 1908 ) studied the radioactivity discovered by Becquerel and the Curies. He determined that the emanations of radioactive materials include alpha particles (or rays) which are positively charged helium atoms, beta particles (or rays) which are negatively charged electrons, and gamma rays which are similar to x-rays. He also studied the radioactive decay process and deduced the first order rate law for the disappearance of a radioactive atom, characterized by the half-life, the time in which 50% of a given radioactive species disappears, and which is independent of the concentration of that species. 

Figure 9. Proportions of species overtime for the process of oxidation of Fe(II),( to Fe(III), followed by precipitation of ferrihydrite (Eqn. 5), as calculated using the first-order rate laws of Equations (6)-(8), and the rate constant from Figure 8. (A) Proportion of species calculated from Equations (6)-(8), assuming a kilki ratio of 10. Ferrihydrite represented as Fe(OH)3 for simplicity. (B) The residence time of intermediate species Fe(III),q, as calculated using Equation (10), for various kjh ratios.

The self-phosphorylation process catalyzed by many protein kinases as part of the regulatory mechanism for their own activation. Because true autophosphorylation is a unimolecular reaction involving enzyme both as catalyst and phosphoryl acceptor, the fraction of autophosphory-lated enzyme at any time after addition of ATP (or another phosphoryl donor) will be independent of the initial concentration of the enzyme. This criterion was first applied to the autophosphorylation of cardiac muscle cyclic AMP-stimulated protein kinase, now designated protein kinase A (PKA). At a fixed concentration of MgATP , the fraction of autophosphorylated protein will follow the first-order rate laws, [A]/[A ] where k is a first-order rate constant. 

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... 

Formal kinetic investigations (performed only with acidic ion exchange catalysts) revealed, in most cases, the first-order rate law with respect to the alkene oxide [285,310,312] or that reaction order was assumed [309,311]. Strong influence of mass transport (mainly internal diffusion in the polymer mass) was indicated in several cases [285,309, 310,312,314]. The first-order kinetics with respect to alkene oxide is in agreement with the mechanism proposed for the same reaction in homogeneous acidic medium [309,315—317], viz. 

Method. In the steady state the first-order rate law — d In AT/dt = kg is obeyed (8, 29). Experimentally one follows the change in total film area (AT) as desorption proceeds while maintaining a constant ir. The apparatus (Figure 1) is essentially the same as previously used (8) but modified to maintain automatically a constant tt, while recording changes in At owing to desorption. 

Kinetics. Kinetic measurements using [Rh((5)-binap)(CH30H)2]C104 and geranylamine substrate indicate that (1) The initial phase of the reaction obeys the first-order rate law, but as the initial substrate concentration, [substratelo, is increased, the rate starts to deviate from the first-order plots at a relatively early stage of the reaction, implying a product inhibition. (2) The dependence of the initial rate Rq on the initial... 

It was previously normal practice to use linear forms of rate equations to simplify determination of rate constants by graphical methods. For example, the logarithmic version of the first-order rate law (Table 3.1), Equation 3.17a, allows k to be determined easily from the gradient of a graph of In Ct against time, by fitting the data to the mathematical model, y = a + bx ... 

With the development of inexpensive powerful computers and appropriate software, it has become increasingly possible to use non-linear optimisation procedures with direct experimental readings as the input. Thus, concentration-time data maybe fitted directly to the exponential version of the first-order rate law, Equation 3.17b. 

Similarly, the dose required to achieve any desired concentration change may be calculated using the first-order rate law ... 

This reaction looks very similar to the reaction of hydroxide ion with methyl chloride presented earlier, but with the negative oxygen of the acetate anion acting as the nucleophile. (The CH3C02H shown over the arrow is the solvent for the reaction.) However, investigation of this reaction in the laboratory has shown that the reaction rate depends only on the concentration of tert-butyl chloride (r-BuCl). It is totally independent of the concentration of acetate anion. The reaction follows the first-order rate law ... 

The first-order rate law is but one of many possibilities, and an interesting comparison can be made between several of the simpler rate laws using a pseudo half life. Consider the case where two measurements of pesticide residue in soil are made 1 year apart and assume, further, that one half of the added pesticide has disappeared during that year—that is, the pesticide shows a half life of 1 year for this specific concentration. In Figure 2, the path of disappearance of one concentration unit is plotted on semilog paper according to a pseudo half life of 1 year and four possible rate laws first-order, second-order, one-half-order, and zero-order—that is, whether the rate of decomposition is proportional to the concentration, the square of the concentration, the square root of the concentration, or independent of the concentration. 

One of the most significant points that we must consider in scientific studies, not limited to studies on photocatalysis, is distinction between evidence and consistency, as least as far as the author thinks. In other words, it is necessary to recognize every fact to be a necessary condition but not a sufficient condition in a strict scientific sense. For example, the fact that a reaction rate obeys the first-order rate law giving a linear relation in a plot of data as in Fig. 6 is only a necessary condition for a monomolecular reaction in homogeneous phase and also a necessary condition for heterogeneous photocatalytic reaction in diffusion-limited conditions or that in surface-reaction limited conditions with a Henry-type adsorption or a Langmuir-type adsorption in the lower-concentration region. 

Kinetics of e -h recombination may depend on its mode if one electron is excited and this is recombined with h , the recombination rate obeys the first-order rate law, while if multiple e -h+ appears at the same time within a photocatalyst particle, the rate obeys the second-order rate law. Actually, in a femtosecond pmnp-probe diffuse reflection spectroscopic analysis of tita-nia samples, photoabsorption at 620 nm by trapped electrons showed second-order decay with a component of baseline as follows ... 

It should be noted that the overall rate obeys the first-order rate law in regard to surface concentration of a substrate but not concentration in the bulk. 

In some reports on photocatalytic reaction, it has been stated that the time-course curve obeys the first-order rate law and that the rate of reaction changes following Langmuir-type adsorption behavior (so-called... 

Heterogeneous reaction (21) is currently supposed to follow the first-order rate law J=kc -kc° (21)... 

The first-order rate law for the exchange reaction can then be described by ... 

The reaction of OH with HCN in aqueous solution at 25°C has a forward rate constant kf of 3.7 X 10 L mol s. Using this information and the measured acid ionization constant of HCN (see Table 15.2), calculate the rate constant in the first-order rate law rate = r[CN ] for the transfer of hydrogen ions to CN from surrounding water molecules ... 

When the decay of intermediate B is faster than its formation, kBC > kAB, the absolute value of the pre-exponential term will become small and thus the transient concentration cB will be small at all times. For kBC 3> kAB, the appearance of product C will approach the first-order rate law, Equation 3.6 (replace the index B by C), because the transient concentration of intermediate B becomes negligible. This shows that observation of a first-order rate law for the reaction A —> C does not guarantee that there is no intermediate involved, that is, that the observed reaction A —> C is an elementary reaction. 

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