Reaction-order half-life

Sucrose (Ci2H22On) hydrolyzes into glucose and fructose. The hydrolysis is a first-order reaction. The half-life for die hydrolysis of sucrose is 64.2 min at 25°C. How many grams of sucrose in 1.25 L of a 0.389 Af solution are hydrolyzed in 1.73 hours ... 

The concentration of the reactant does not appear in Eq. 7 for a first-order reaction, the half-life is independent of the initial concentration of the reactant. That is, it is constant regardless of the initial concentration of reactant, half the reactant will have been consumed in the time given by Eq. 7. It follows that we can take the initial concentration of A to be its concentration at any stage of the reaction if at some stage the concentration of A happens to be A], then after a further time tv2, the concentration of A will have fallen to 2[AJ, after a further tU2 it will have fallen to [A], and so on (Fig. 13.13). In general, the concentration remaining after n half-lives is equal to (t)" A 0. For example, in Example 13.6, because 30 days corresponds to 5 half-lives, after that interval [A ( = (j)5 A]0, or [A]0/32, which evaluates to 3%, the same as the result obtained in the example. 

FIGURE 13.13 For first-order reactions, the half-life is the same whatever the concentration at the start of the chosen period. Therefore, it takes one half-life to fall to half the initial concentration, two half-lives to fall to one-tourth the initial concentration, three half-lives to fall to one-eighth, and so on. The boxes portray the composition ot the reaction mixture at the end of each half-life the red squares represent Inc reactant A and the yellow squares represent the product. 

This is the integrated rate equation for a first-order reaction. When dealing with first-order reactions it is customary to use not only the rate constant, k for the reaction but also the related quantity half-life of the reaction. The half-life of a reaction refers to the time required for the concentration of the reactant to decrease to half of its initial value. For the first-order reaction under consideration, the relation between the rate constant k and the half life t0 5 can be obtained as follows ... 

The purified tetraethyl pyrophosphate is a colorless, odorless, water-soluble, hygroscopic liquid (24, 4 )- It possesses a very high acute toxicity (28), exceeding that of parathion, and is rapidly absorbed through the skin. There is no spray-residue problem, however, for tetraethyl pyrophosphate hydrolyzes even in the absence of alkali to nontoxic diethyl phosphoric acid. Hall and Jacobson (24) and Toy (47) have measured its rate of hydrolysis, which is a first-order reaction. Its half-life at 25° C. is 6.8 hours and at 38° C. is 3.3 hours. Coates (10) determined the over-all velocity constant at 25° C. k = 160 [OH-] + 1.6 X 10 3 min.-1 Toy (47) has described an elegant method for preparing this ester as well as other tetraalkyl pyrophosphates, based upon the controlled hydrolysis of 2 moles of dialkyl chlorophosphate ... 

The addition of 2,3-dimethyl-1,3-butadiene to 17 gives only one structural isomer, 21. The pseudo-first order half-life of this Diels-Alder addition reaction is 47.5 minutes at 22.2°C in neat diene solution. The Ea of this reaction is estimated to be 9.2 kcal/mol. This reaction rate is 50 times faster than the rate of addition of this diene to methyl methacrylate (33). 

The half-life of a reactant is the time required for half of that reactant to be converted into products. For a first order reaction, the half-life is independent of concentration so that the same time is required to consume half of any starting amount or concentration of the reactant. On the other hand, the half-life of a second-order reaction does depend on the starting amount of the reactant. 

The half-life of the reaction depends on the concentration of A and, thus, this reaction cannot be first-order. For a second-order reaction, the half-life varies inversely with the... 

A zero-order reaction has a half life that varies proportionally to [A]0, therefore, increasing [A]0 increases the half-life for the reaction. A second-order reaction s half-life varies inversely proportional to [A]0, that is, as [A]0 increases, the half-life decreases. The reason for the difference is that a zero-order reaction has a constant rate of reaction (independent of [A]0). The larger the value of [A]0, the longer it will take to react. In a second-order reaction, the rate of reaction increases as the square of the [A]0, hence, for high [A]0, the rate of reaction is large and for very low [A]0, the rate of reaction is very slow. If we consider a bimolecular elementary reaction, we can easily see that a reaction will not take place unless two molecules of reactants collide. This is more likely when the [A]0 is large than when it is small. 

More generally, for an nth-order reaction, the half-life is given (from equation 3.4-9) by... 

A second-order reaction may typically involve one reactant (A -> products, ( -rA) = kAc ) or two reactants ( pa A + vb B - products, ( rA) = kAcAcB). For one reactant, the integrated form for constant density, applicable to a BR or a PFR, is contained in equation 3.4-9, with n = 2. In contrast to a first-order reaction, the half-life of a reactant, f1/2 from equation 3.4-16, is proportional to cA (if there are two reactants, both ty2 and fractional conversion refer to the limiting reactant). For two reactants, the integrated form for constant density, applicable to a BR and a PFR, is given by equation 3.4-13 (see Example 3-5). In this case, the reaction stoichiometry must be taken into account in relating concentrations, or in switching rate or rate constant from one reactant to the other. 

The reaction half-life, tV2, is the amount of time that it takes for a reactant concentration to decrease to one-half its initial concentration. For a first-order reaction, the half-life is a constant, independent of reactant concentration and has the following relationship ... 

For second-order reactions, the half-life does depend on the reactant concentration. We calculate it using the following formula ... 

Therefore, in case of third order reactions the half-life period is inversely proportional to the square of initial concentration. 

A simple way to characterize the rate of a reaction is the time it takes for the concentration to change from the initial value to halfway between the initial and final (equilibrium). This time is called the half-life of the reaction. The half-life is often denoted as ti/z. The longer the half-life, the slower the reaction. The half-life is best applied to a first-order reaction (especially radioactive decay), for which the half-life is independent of the initial concentration. For example, using the decay of " Sm as an example, [ Sm] = [ Sm]o exp( kt) (derived above). Now, by definition,... 

Half-Life Method For a zero-order reaction the half-life (tll2) is proportional to the initial concentration. The half-life for a first-order reaction is independent of the initial concentration while a second-order reaction is proportional to 1/initial concentration. 

FIGURE 13.12 For first-order reactions, the half-life is the same whatever the concentration at the start of the chosen period. 

FIGURE 12.7 Concentration of a reactant A as a function of time for a first-order reaction. The concentration falls from its initial value, [Alo, to [A]q/2 after one half-life, to [A]0/4 after a second half-life, to [A]0/8 after a third half-life, and so on. For a first-order reaction, each half-life represents an equal amount of time. 

Reaction A half-life increases as concentration decreases, second order. 

Reaction B half-life decreases as concentration decreases, zero order. 

Reaction C half-life remains constant, first order. 

Equation 3-39 shows that in the first order reactions, the half-life is independent of the concentration of the reactant. This basis can be used to test whether a reaction obeys first order kinetics by measuring half-lives of the reaction at various initial concentrations of the reactant. 

Important characteristics of zero-order reactions are that (1) a constant amount of drug is eliminated per unit time since the system is saturated (maximized) and (2) the half-life is not constant for zero-order reactions but depends on the concentration. The higher the concentration, the longer the half-life. Therefore, the term zero-order half-life has little practical significance since it can change and (3) zero-order kinetics is also known as nonlinear or dose-dependent. For example, if the body can metabolize ethanol at a rate of 10 ml per hour, then if one consumes 60 ml, it will take 3 hours to metabolize half of it (the half-life under these circumstances). However, if 80 ml is consumed the half-life will now become 4 hours. This is particularly significant regarding ethanol toxicity. 

A plot of Aj versus would produce a straight line with slope -k in units of mass per unit time (e.g., mol min-1) (Fig. 7.4). In the case of a zero-order reaction, its half-life is l/2ff, where tf represents the total time needed to decompose the original quantity of compound A (A0). Another way to express such reactions is shown in Figure 7.5. The data show A1 release from y-Al203 at different pH values. The data clearly show that the reaction is zero-order with k dependent on pH. 

This is the general equation for the half-life of a first-order reaction. Equation (15.3) can be used to calculate tV2 if k is known or k if t1/2 is known. Note that for a first-order reaction the half-life does not depend on concentration. 

Time course of drug concentration Drug passage across cell membranes Order of reaction Plasma half-life and steady-state concentration Therapeutic drug monitoring... 

Unlike t for the first-order reactions, the half-life of the second-order reaction is dependent on the initial concentration of reactants. It is not possible to derive a simple expression for the half-life of a second-order reaction with unequal initial concentrations. 

Thus for first-order reactions, the half-life is constant and independent of the initial reactant concentration and can be used directly to calculate the rate constant k. For non-first-order reactions, Eq. (7-171) can be linearized as follows ... 

If we assume that the second-order reaction given in equation 1 can be expressed as an equilibrium reaction, then K = kf/kr (K is the equilibrium constant, kf is the forward rate constant of the reaction, and kr is the reverse rate constant of the reaction.) If ° = -1.49 V, then K = 6.2 X 10 26. For the reverse reaction, which is spontaneous, we can assume an upper diffusion-controlled limit for kT of 1 X 1010 M"1 s"1 (17). Thus, k = 6.2 X 10"16, and we calculated a second-order half-life of about 200 billion years at 02 concentrations of 250 xM in the photic zone of the ocean. These rates and the... 

This relates the half-life of a reactant in a first-order reaction and its rate constant, k. In such reactions, the half-life does not depend on the initial concentration of A. This is not true for reactions having overall orders other than first order. 

This is a first-order reaction. The half-life of benzoyl peroxide at 100°C is 19.8 min. (a) Calculate the rate constant (in min ) of the reaction, (b) If the half-life of benzoyl peroxide is 7.30 h, or 438 min, at 70°C, what is the activation energy (in kJ/mol) for the decomposition of benzoyl peroxide (c) Write the rate laws for the elementary steps in the above polymerization process, and identify the reactant, product, and intermediates, (d) What condition would favor the growth of long, high-molar-mass polyethylenes ... 

Determining the rate constant and order of a reaction is tedious and time-consuming. For many studies, this detail is unwarranted and the half-life is measured instead. The half-life is the time required for half of the original concentration of reactant to disappear. For the particular case of a first-order reaction, the half-life ( i /2) is directly related to the reaction rate constant k by... 

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