Kinetics first order

First-order reactions are by far the most common. They are also the simplest to study experimentally. For reactions of higher order, experimental conditions can usually be arranged so that they are first-order (see below). This simplifies the situation considerably. 

For the reaction of species A to give product B, with rate constant k 

Of course the rate of appearance of product can also be used to monitor the reaction, since 

An alternative form of equation 23-1 that sometimes is useful is 

Occasionally a first-order rate constant is obtained by experimental determination of the half-life fi/2, the time required for the reactant concentration to decrease to one-half of its original value. From equation 23-2 it follows that k = b (2)/fl/2 = 0.693/fi/2- 

The rate law for a simple first order transformation (Eq. 7.25) can be written as in Eq. 7.30. Rearranging this gives Eq. 7.31 which, upon integration, gives the integrated rate law 7.32 where [ A] is the starting concentration of A. Another common way of writing this is Eq. 7.33, which shows an exponential dependence of the concentration of A upon the rate constant. 

Therefore, if one uses a spectroscopic or chromatographic method to monitor the concentration of A at various time points f, a plotof ln[A] vs. time will give a straight line with a slope that is the negative of the rate constant. If the data can be modeled with this ec ua-tion, this suggests that the reaction is first order (see the next Connections highlight for an example). 

The half-life tm) of a reaction is the time required for 50% of the starting material to be consumed. A rule of thumb is to follow the reaction to five or more half-lives to obtain an accurate first order rate constant. For a first order reaction the half-life is t /2 = ln(2)/l = 0.693 Ik. A related term isthelifetimeofaspecies, defined as l/Ii, where Iris the rate constant for the first order disappearance of the species. For a first order reaction, the time required for reaction does not depend upon how much reactant one starts with. For higher order reactions the half-life and lifetime do depend upon the concentrations of the reactants. 

We can also work with Eq. 7.30, using d[P] /dt. However, we need to have [P] in terms of (A). Since the total concentration of A and P must always equal the initial amount of A ([A] = [A] + [P], assuming no side reactions), we derive Eq. 7.34. Now a plot of ln [A]o/ ((A] — (P])l versus f gives the rate constant as the slope. 

First Order Kinetics Delineating Between a Unimolecular and a Bimolecular Reaction of Cyclopentyne and Dienes 

SOLUTION KINETIC MODELS First-Order Kinetics 

Many important natural processes ranging from nuclear decay to uni-molecular chemical reactions are first order, or can be approximated as first order, which means that these processes depend only on the concentration to the first power of the transforming species itself. A cellular automaton model for such a system takes on an especially simple form, since rules for the movements of the ingredients are unnecessary and only transition rules for the interconverting species need to be specified. We have recently described such a general cellular automaton model for first-order kinetics and tested its ability to simulate a number of classic first-order phenomena.70 

When a reverse transition probability Pt(B,A) for the transition B — A is included, the model simulates the first-order equilibrium  

Here too, the finite size of the system causes notable fluctuations, in this case in the value of the equilibrium constant K, which fluctuates with time about the deterministic value 

The series model can be extended to longer series and to the inclusion of reversibility to illustrate a variety of fundamental kinetic phenomena in an especially simple and straightforward manner. Depending on the relative rates employed, one can demonstrate the classic kinetic phenomena of a rate-limiting step and preequilibrium,72 and one can examine the conditions needed for the validity of the steady-state approximation commonly used in chemical kinetics.70 

The simplest case to consider by far is that of first-order or linear kinetics in a constant volume batch reactor. If the rate of reaction is directly proportional to the rate of the reaction, then we call this the first order in the concentration of reactant, and the right-hand side becomes  

This is an expression for the rate of decay of the concentration of species A. (It should remind us of the expression we derived for the change in level of the draining tank for which we used a linear constitutive relationship between level and rate of flow.) The dimensions of k in this case are reciprocal time, that is, sec or min etc. The reason for this is that the rate of reaction is given in dimensions of voi n,e- Therefore to be dimensionally consistent the first-order rate constant must be in dimensions of inverse time. 

As the stoichiometry for the rate of reaction of component B is the same we can show that  

Chapter 7 Reacting Systems—Kinetics and Batch Reactors 

In a similar fashion we find that the rate of appearance of component D is  

if oxygen is strongly adsorbed such that ( 02 02) 1. then the RE kinetics is simplified to first-order kinetics  

A second possibility to arrive to a similar first-order kinetics is starting from a dual site Langmuir-Hinshelwood model with oxygen adsorbed strongly in dissociative form while methane is adsorbed weakly  

Relation (16.8) for (K02P02) 1 and (/ TcHTf cH4) 1 becomes similar to (16.7) where k = k KcHi- In order to determine the kinetic parameter (k), we substitute (16.7) into Eq. (16.9) for the plug flow reactor (PER) where the 

The values of ( app) estimated in this way are given in Table 16.1, coliunn 4, and they are larger than the ( true) values estimated via the RE method and cited in column 2 of the same table. Then the values of ( ch4) can be estimated via 

The reliability of these values is supported by the fact that they reflect the amount of Ce and Sr in the catalysts as weighing factors of A02 and /Ich4. respectively. In other words, 

Denatured proteins often show irreversible behaviour at high temperature. A mathematical procedure for the analysis of heat capacity curves affected by irreversible first order unfolding was suggested by Sanchez-Ruiz et al. [12]. However, in that study the unfolding reaction was treated only as a kinetic phenomenon. Therefore no equilibrium parameters could be obtained. In a later paper [13] other treatments of the C,-curves were suggested which allowed for the extraction of equilibrium parameters from the non-equilibrium C -curve. 

Their simple model assumes that each denatured protein molecule transforms irreversibly in a first order reaction into a species from which the native form cannot be recovered. This model is called Lumry-Eyring model [55] since Lumry and Eyring were among the first to propose that proteins unfold in two steps, a reversible unfolding equilibrium of the tertiary structure followed by a first order, irreversible step involving secondary structure unfolding. 

If the magnitude of the rate constant k is small compared to the rate constants dominating the reversible folding equilibrium characterised by the equilibrium constant K, the equilibrium should be well approximated by the ratio of the fractions of unfolded ap and native molecules Qn 

The time t can be transformed to temperature T by introducing the heating rate r = dT/dt. Integration of equation 72 yields then 

Since the transition exhibits 1 1 stoichiometry, the enthalpy change is proportional to the corresponding population sizes 

In any steady state, the rate of enzyme change per unit time is zero, so that 

The rate constant for degradation is related to the half-life of an enzyme (ti/2) by the equation  

The half-life is readily obtained from a plot of time against enzyme content using semilog graph paper. 

It should be stressed that a plot of the log of enzyme content against time will only lead to a valid measurement of fi/2 or ka if the synthesis rate of the enzyme is zero. Otherwise the plot will underestimate kd-Whereas kg is near zero when synthesis is prevented by inhibitors, or 

For changes between two steady states, kg and ti/2 can be obtained if the log of the decrement ( 2 1) is plotted against time (Raining, 1971). 

---

These reactions follow first-order kinetics and proceed with racemisalion if the reaction site is an optically active centre. For alkyl halides nucleophilic substitution proceeds easily primary halides favour Sn2 mechanisms and tertiary halides favour S 1 mechanisms. Aryl halides undergo nucleophilic substitution with difficulty and sometimes involve aryne intermediates. 

Generalized first-order kinetics have been extensively reviewed in relation to teclmical chemical applications [59] and have been discussed in the context of copolymerization [53]. From a theoretical point of view, the general class of coupled kinetic equation (A3.4.138) and equation (A3.4.139) is important, because it allows for a general closed-fomi solution (in matrix fomi) [49]. Important applications include the Pauli master equation for statistical mechanical systems (in particular gas-phase statistical mechanical kinetics) [48] and the investigation of certain simple reaction systems [49, ]. It is the basis of the many-level treatment of... 

An important example for the application of general first-order kinetics in gas-phase reactions is the master equation treatment of the fall-off range of themial unimolecular reactions to describe non-equilibrium effects in the weak collision limit when activation and deactivation cross sections (equation (A3.4.125)) are to be retained in detail [ ]. 

General first-order kinetics also play an important role for the so-called local eigenvalue analysis of more complicated reaction mechanisms, which are usually described by nonlinear systems of differential equations. Linearization leads to effective general first-order kinetics whose analysis reveals infomiation on the time scales of chemical reactions, species in steady states (quasi-stationarity), or partial equilibria (quasi-equilibrium) [M, and ]. 

Horn F 1971 Generai first order kinetics Ber. Bunsenges. Phys. Chem. 75 1191-201... 

A3.13.3.2 THE MASTER EQUATION FOR COLLISIONAL AND RADIATIVE ENERGY REDISTRIBUTION UNDER CONDITIONS OF GENERALIZED FIRST-ORDER KINETICS... 

There is one special class of reaction systems in which a simplification occurs. If collisional energy redistribution of some reactant occurs by collisions with an excess of heat bath atoms or molecules that are considered kinetically structureless, and if fiirthennore the reaction is either unimolecular or occurs again with a reaction partner M having an excess concentration, dien one will have generalized first-order kinetics for populations Pj of the energy levels of the reactant, i.e. with... 

In classical kinetics, intemiolecular exchange processes are quite different from the uniniolecular, first-order kinetics associated with intramolecular exchange. However, the NMR of chemical exchange can still be treated as pseudo-first-order kinetics, and all the previous results apply. One way of rationalizing this is as... 

When the perturbation is small, the reaction system is always close to equilibrium. Therefore, the relaxation follows generalized first-order kinetics, even if bi- or trimolecular steps are involved (see chapter A3.41. Take, for example, the reversible bimolecular step... 

More generally, the relaxation follows generalized first-order kinetics with several relaxation times i., as depicted schematically in figure B2.5.2 for the case of tliree well-separated time scales. The various relaxation times detemime the tiimmg points of the product concentration on a logaritlnnic time scale. These relaxation times are obtained from the eigenvalues of the appropriate rate coefficient matrix (chapter A3.41. The time resolution of J-jump relaxation teclmiques is often limited by the rate at which the system can be heated. With typical J-jumps of several Kelvin, the time resolution lies in the microsecond range. 

B2.5.351 after multiphoton excitation via the CF stretching vibration at 1070 cm. More than 17 photons are needed to break the C-I bond, a typical value in IR laser chemistry. Contributions from direct absorption (i) are insignificant, so that the process almost exclusively follows the quasi-resonant mechanism (iii), which can be treated by generalized first-order kinetics. As an example, figure B2.5.15 illustrates the fonnation of I atoms (upper trace) during excitation with the pulse sequence of a mode-coupled CO2 laser (lower trace). In addition to the mtensity, /, the fluence, F, of radiation is a very important parameter in IR laser chemistry (and more generally in nuiltiphoton excitation) ... 

How does one monitor a chemical reaction tliat occurs on a time scale faster tlian milliseconds The two approaches introduced above, relaxation spectroscopy and flash photolysis, are typically used for fast kinetic studies. Relaxation metliods may be applied to reactions in which finite amounts of botli reactants and products are present at final equilibrium. The time course of relaxation is monitored after application of a rapid perturbation to tire equilibrium mixture. An important feature of relaxation approaches to kinetic studies is that tire changes are always observed as first order kinetics (as long as tire perturbation is relatively small). This linearization of tire observed kinetics means... 

This shows that the observed rate for this process will follow first order kinetics, even though the reaction being studied is second order. Furthennore, both and k may be detennined by observing the kinetics at different starting concentrations that vary the quantity a+b). 

However, as can be seen from Figure 8 a simple exponential expected from first-order kinetics can be fitted to the data yielding a limiting concentration of 0.005, and a rate constant of 0.0003 a.u., which translates to 1.25 x 10 s at 300 K. 

A novel technique for dating archaeological samples called ammo acid racemiza tion (AAR) IS based on the stereochemistry of ammo acids Over time the configuration at the a carbon atom of a protein s ammo acids is lost m a reaction that follows first order kinetics When the a carbon is the only chirality center this process corresponds to racemization For an ammo acid with two chirality centers changing the configuration of the a carbon from L to D gives a diastereomer In the case of isoleucme for example the diastereomer is an ammo acid not normally present m proteins called alloisoleucme... 

The integrated form of the rate law for equation 13.4, however, is still too complicated to be analytically useful. We can simplify the kinetics, however, by carefully adjusting the reaction conditions. For example, pseudo-first-order kinetics can be achieved by using a large excess of R (i.e. [R]o >> [A]o), such that its concentration remains essentially constant. Under these conditions... 

The rate of decay, or activity, for a radioactive isotope follows first-order kinetics... 

An important characteristic property of a radioactive isotope is its half-life, fj/2, which is the amount of time required for half of the radioactive atoms to disintegrate. For first-order kinetics the half-life is independent of concentration and is given as... 

Analytes A and B react with a common reagent R with first-order kinetics. If 99.9% of A must react before 0.1% of B has reacted, what is the minimum acceptable ratio for their respective rate constants ... 

A mixture of two analytes, A and B, is analyzed simultaneously by monitoring their combined concentration, C = [A] + [B], as a function of time when they react with a common reagent. Both A and B are known to follow first-order kinetics with the reagent, and A is known to react faster than B. Given the data in the following table. 

The relationship of this type of model to a tme differential analysis has been discussed for the case of linear equiHbrium and first-order kinetics (74,75). A minor extension of this work leads to the foUowing relations for a bed section in which dow rates of soHd and Hquid are constant. For the number of theoretical trays,... 

If first-order kinetics are assumed, k /is the linoleic selectivity ratio and k l is the selectivity ratio for reduction of linoleic acid to stearic acid. Figure 2 shows a typical course of hydrogenation for soybean oil the rate constants are = 0.367, = 0.159, and k = 0.013. With a selective nickel catalyst,... 

First-order kinetics (ie, n = 1) is frequently assumed and seems adequate to describe the kinetics of most flotation processes. However, highly hydrophobic particles float faster and very fine particles or coarse ones outside the optimal flotation size range (see Fig. 1) take longer to coUect in the froth layer. ExceUent reviews of the subject are available in the Hterature (27). 

A cis-elimination mechanism has been postulated for this decomposition which foUows first-order kinetics (120). The rate is accelerated by addition of lithium j iZ-butoxide [4111-46-0] and other bases, and by an increase in temperature (120). Pyrolysis of j iZ-butyUithium in the presence of added alkoxide is one-half order in alkyUithium and first order in alkoxide (120). Thermal decomposition of j iZ-butyUithium at 0.18% alkoxide at 25, 40, 50, and 60°C is 0.1%, 0.6%, 2.0%, and 6.8%/d, respectively (121). 

Decomposition of diphenoylperoxide [6109-04-2] (40) in the presence of a fluorescer such as perylene in methylene chloride at 24°C produces chemiluminescence matching the fluorescence spectmm of the fluorescer with perylene was reported to be 10 5% (135). The reaction follows pseudo-first-order kinetics with the observed rate constant increasing with fluorescer concentration according to = k [flr]. Thus the fluorescer acts as a catalyst for peroxide decomposition, with catalytic decomposition competing with spontaneous thermal decomposition. An electron-transfer mechanism has been proposed (135). 

Hydrated amorphous silica dissolves more rapidly than does the anhydrous amorphous silica. The solubility in neutral dilute aqueous salt solutions is only slighdy less than in pure water. The presence of dissolved salts increases the rate of dissolution in neutral solution. Trace amounts of impurities, especially aluminum or iron (24,25), cause a decrease in solubility. Acid cleaning of impure silica to remove metal ions increases its solubility. The dissolution of amorphous silica is significantly accelerated by hydroxyl ion at high pH values and by hydrofluoric acid at low pH values (1). Dissolution follows first-order kinetic behavior and is dependent on the equilibria shown in equations 2 and 3. Below a pH value of 9, the solubility of amorphous silica is independent of pH. Above pH 9, the solubility of amorphous silica increases because of increased ionization of monosilicic acid. 

The first mechanistic studies of silanol polycondensation on the monomer level were performed in the 1950s (73—75). The condensation of dimethyl sil oxanediol in dioxane exhibits second-order kinetics with respect to diol and first-order kinetics with respect to acid. The proposed mechanism involves the protonation of the silanol group and subsequent nucleophilic substitution at the siHcone (eqs. 10 and 11). 

Kinetics of Pesticide Biodegradation. Rates of pesticide biodegradation are important because they dictate the potential for carryover between growing seasons, contamination of surface and groundwaters, bio accumulation in macrobiota, and losses of efficacy. Pesticides are typically considered to be biodegraded via first-order kinetics, where the rate is proportional to the concentration. Figure 2 shows a typical first-order dissipation curve. 

For those pesticides that are cometabolized, ie, not utilized as a growth substrate, the assumption of first-order kinetics is appropriate. The more accurate kinetic expression is actually pseudo-first-order kinetics, where the rate is dependent on both the pesticide concentration and the numbers of pesticide-degrading microorganisms. However, because of the difficulties in enumerating pesticide-transforming microorganisms, first-order rate constants, or half-hves, are typically reported. Based on kinetic constants, it is possible to rank the relative persistence of pesticides. Pesticides with half-hves of <10 days are considered to be relatively nonpersistent pesticides with half-hves of >100 days are considered to be relatively persistent. 

---