First order reactions, characteristics

FIGURE 13.9 First-order reaction characteristics (a) Decrease of reactant concentration with time (b) plot of the straight-line relationship to obtain the rate constant. The slope of the line is equal to —k. 

First-order reaction characteristics (a) Decrease of reactant concentration with time ... 

Concentration-time curves. Much of Sections 3.1 and 3.2 was devoted to mathematical techniques for describing or simulating concentration as a function of time. Experimental concentration-time curves for reactants, intermediates, and products can be compared with computed curves for reasonable kinetic schemes. Absolute concentrations are most useful, but even instrument responses (such as absorbances) are very helpful. One hopes to identify characteristic features such as the formation and decay of intermediates, approach to an equilibrium state, induction periods, an autocatalytic growth phase, or simple kinetic behavior of certain phases of the reaction. Recall, for example, that for a series first-order reaction scheme, the loss of the initial reactant is simple first-order. Approximations to simple behavior may suggest justifiable mathematical assumptions that can simplify the quantitative description. 

Non-isothermal measurements of the temperatures of dehydrations and decompositions of some 25 oxalates in oxygen or in nitrogen atmospheres have been reported by Dollimore and Griffiths [39]. Shkarin et al. [606] conclude, from the similarities they found in the kinetics of dehydration of Ni, Mn, Co, Fe, Mg, Ca and Th hydrated oxalates (first-order reactions and all values of E 100 kJ mole-1), that the mechanisms of reactions of the seven salts are probably identical. We believe, however, that this conclusion is premature when considered with reference to more recent observations for NiC204 2 H20 (see below [129]) where kinetic characteristics are shown to be sensitive to prevailing conditions. The dehydration of MnC204 2 H20 [607] has been found to obey the contracting volume... 

The half-life of a first-order reaction is characteristic of the reaction and independent of the initial concentration. A reaction with a large rate constant has a short half-life. 

FIGURE 13.14 The characteristic shapes of the time dependence of the concentration of a reactant during a second-order reaction. The larger the rate constant, k, the greater is the dependence of the rate on the concentration of the reactant. The lower gray lines are the curves for first-order reactions with the same initial rates as for the corresponding second-order reactions. Note how the concentrations for second-order reactions fall away much less rapidly at longer times than those for first-order reactions do. 

Another characteristic of first-order reactions is that the time it takes for half the reactant to disappear is the same, no matter what the concentration. This time is called the half-life ( 1/2). Applying Equation to a time interval equal to the half-life results in an equation for / i 2 When half the original concentration has been consumed,... 

One of the typical features of a (pseudo)-first order reaction is that a plot of the logarithm of the advancement of the reaction versus time (Fig. 2B) should give straight lines. However we observed deviation from linearity before the first half-life, in spite of the fact that another characteristic features of (pseudo)-first order reactions, namely that plots of the extent of reaction versus time were independant of the initial concentration (Fig. 3), was verified. We therefore investigated whether variation occured in the reaction conditions as a function of time. 

When only taking into account the concentration polarization in the pores (disregarding ohmic potential gradients), we must use an equation of the type (18.15). Solving this equation for a first-order reaction = nFhjtj leads to equations exactly like (18.18) for the distribution of the process inside the electrode, and like (18.20) for the total current. The rate of attenuation depends on the characteristic length of the diffusion process ... 

Da Second Damkohler number K l2 ID K = first-order reaction rate constant l = characteristic length D = diffusion coefficient... 

Some characteristics and applications of first-order reactions (for A - products, (—rA) = kAcA) arc noted in Chapters 2 and 3, and in Section 4.2.3. These are summarized as follows ... 

Certain general characteristics of this curve can be stated. First, the third limit portion of the curve is as one would expect from simple density considerations. Next, the first, or lower, limit reflects the wall effect and its role in chain destruction. For example, H02 radicals combine on surfaces to form H20 and 02. Note the expression developed for acnt [Eq. (3.9)] applies to the lower limit only when the wall effect is considered as a first-order reaction of... 

The differential (rate) forms are (1.16), (1.18) and (1.20), and the corresponding integrated forms are (1.17), (1.19) (or (1.19a)) and (1.21). The designations [A]q and [A], represent the concentrations of A at zero time and time /. Linear plots of [A], In [A], or [A], vs time therefore indicate zero-, first, or second order dependence on the concentration of A. The important characteristics of these order reactions are shown in Fig. 1.1. Notwithstanding the appearance of the plots in 1.1 (b) and 1.1 (c), it is not always easy to differentiate between first-and second-order kinetics.Sometimes a second-order plot of kinetic data might be mistaken for successive first-order reactions (Sec. 1.6.2) with similar rate constants. 

Important quantities characteristic of a first-order reaction are the half-life of the reaction, which is the value of t when [A], = [A](,/2, and t, the relaxation time, or mean lifetime, defined as k. ... 

Solution of the equations for successive reversible reactions is quite formidable even for the first-order case thus, we illustrate only the general characteristics for a few typical cases. Consider the reversible first-order reactions... 

As mentioned in Section 4, the analysis of rate data resulting from unimolecular reactions is considerably easier than the analysis of such data for bimolecular reactions, and the same is true for pseudounimolecular reactions. Kinetic probes currently used to study the micellar pseudophase showing first-order reaction kinetics are almost exclusively compounds undergoing hydrolysis reactions showing in fact pseudofirst-order kinetics. In these cases, water is the second reactant and it is therefore anticipated that these kinetic probes report at least the reduced water concentration (or better water activity in the micellar pseudophase. As for solvatochromic probes, the sensitivity to different aspects of the micellar pseudophase can be different for different hydrolytic probes and as a result, different probes may report different characteristics. Hence, as for solvatochromic probes, the use of a series of hydrolytic probes may provide additional insight. 

One particular point of interest is the expression for the half-life of a reaction f,/2 this is the time required for one half of the reactant in question to disappear. A first order reaction is unique in that the half-life is independent of the initial concentration of the reactant. This characteristic is sometimes used as a test of whether a... 

Equation (5.59) gives the solution in the limiting case of a first-order reaction, where Thiele modulus L is the characteristic length, which is half the width of a flat plate, r/2 of a long... 

In this equation, the conversion term for a first-order reaction (1-X) is expressed as a function of the characteristic temperature levels of the scenario. First-order is a conservative approximation, since for higher reaction orders the reactant depletion is even higher. Zero-order would even be more conservative, but is generally unrealistic. 

There is, however, one difficulty with this interpretation. One of the unusual features of nitration in acetic anhydride is that the rate of formation of the electrophile cannot be made rate-determining even when the concentration of the aromatic substrate is as large as 0.5 mol dm 3 and when the substrate (mesitylene) appears to react at the limiting rate (Marziano et al., 1974). This implies that the rate coefficient for the back reaction of the electrophile (considered as a first-order reaction) must be > 1010 s I for otherwise the rate of the back reaction would not markedly exceed the rate of encounter of the electrophile with this concentration of the substrate. It follows therefore that the half-life of the electrophile must be below 10-10 s. This is the condition that leads to predominant reaction via pre-association (p. 11). Since the characteristics of the reaction accord with nitration by the nitronium ion, the simplest interpretation is to assume that this ion is formed within an encounter pair consisting of the aromatic substrate and an inactive precursor of the nitronium ion (e.g., CH3CO. ON02H+). 

The half-life, t /2, is independent of the initial concentration of A. This independence of [A]o is characteristic only of first-order reactions. Also, note that t /2 and k are independent of the units in which A is expressed (although molarity is the most common unit, as indicated by [A]). 

Problem 4 What is a first order reaction Derive the rate expression for it and discuss its characteristics. Give some examples of first order reaction. 

Determination of clearance rate (for first-order reactions) also allows us to calculate another very important drug characteristic, namely its half-life (t112), according to the following formula ... 

---