Half-life of reactant

On the other extreme, the rate of radical-radical reactions may be extremely fast compared to radical-molecule reactions, and then the half-life of reactant foUows the familiar law for a first-order reaction... 

Logarithm of Half-life of Reactant (Initial Pressure of 3 mm.) as Function of Rate of Light Absorption and Rate Constant ... 

Fig. 4. Half-life of reactant (CHsCHjCHO) as function of partial pressure and temperature for light source of moderately high intensity.

The equation that relates concentration and time is the integrated rate equation. We can also use it to calculate the half-life, of reactant—the time it takes for half of that reactant to he converted into product. The integrated rate equation and the half-life are different for reactions of different order. 

The functional dependence of the half-life on reactant concentration varies with the reactant order. From the integrated rate equations we obtain these results ... 

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. 

A second-order reaction has a long tail of low concentration at long reaction times. The half-life of a second-order reaction is inversely proportional to the concentration of the reactant. 

Cl 13.39 Derive an expression for the half-life of the reactant A that decays by a third-order reaction with rate constant k. 

Half-life of a reactant in a first-order reaction ... 

The half-life of a first-order reaction is independent of the initial concentration. Thus, the time required for the reactant concentration to decrease from Uq to Oo/2 is the same as the time required to decrease from Uo/2 to a jA. This is not true for reactions other than first order. 

Equation 6 would hold for a family of free radical initiators of similiar structure (for example, the frarw-symmetric bisalkyl diazenes) reacting at the same rate (at a half-life of one hour, for example) at different temperatures T. Slope M would measure the sensitivity for that particular family of reactants to changes in the pi-delocalization energies of the radicals being formed (transition state effect) at the particular constant rate of decomposition. Slope N would measure the sensitivity of that family to changes in the steric environment around the central carbon atom (reactant state effect) at the same constant rate of decomposition. 

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 concept of half-life also applies to chemical reactions. The half-life of a chemical reaction is the time it takes for the amount of one of the reactants to be reduced by half. In some reactions the reaction rate is determined by the concentration of one particular reactant as the reaction proceeds and the concentration of this reactant decreases, so does the rate of the reaction. This is the case for example, with amino acids, the components of proteins. Amino acids may occur in one of two different forms, the / and d forms (see Textbox 24). In living organisms, however, the amino acids occur only in the / form. After organisms die, the amino acids in the dead remains racemize and are gradually converted into the d form. Ultimately, the remaining amino acid, which is then known as a racemic mixture, consists of a mixture of 50% of the / form and 50% of the d form. 

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. 

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. 

Problem 1.18 A second order reaction with initial concentration of each reactant as 0.5 mol dm 3 was carried out in presence of acid as catalyst. At pH 4.0 the half-life of reaction was found to be 60 min. Calculate the observed and true rate constant for the reaction. 

Problem 1.21 The data of a chemical reaction is plotted as l/[reactant] vs time and the plot is a straight line with intercept 4.0 x 102 mol 1 dm3 and slope 4.0 mol dm3 s 1 as shown in the figure. Calculate the half-life of the reaction. 

A kinetic study requires the determination of the concentration (in mol dnr3) of at least one of the reactant or product as a function of time. In case of gaseous phase, in place of concentration, the partial pressure is determined. The method of analysis employed must be faster than the rate of reaction. The conventional methods of analysis can be applied to the reactions which have a half-life of at least a few minutes. The measurement of some physical property which is proportional to the concentration/partial pressure can also be taken for determination of the rate. In many cases of reactions in solution, it is necessary to take out aliquots from the reaction mixture at suitable intervals of time, arrest the reaction in aliquots by means of suitable means and then analyse the sample. Some conventional physical methods used to study the kinetics of slow reactions are described as follows. 

Models may also be tested by utilizing the time required for a given fraction of a reactant to disappear, since this varies with the initial concentration in a fashion characteristic of the reaction order. For example, if the half-life of a reaction is defined as the time required for one-half of the initial amount of reactant to be consumed, then Eq. (4) may be written... 

Examine Figure 6.6 to see how half-life and reactant concentration are related for the first-order decomposition of dinitrogen pentoxide. 

The half-life of any first-order reaction is always a constant, and it depends on k. In other words, any first-order reaction (that is, any reaction with a rate law equation in the form Rate = k[A]) has a half-life that is independent of the initial concentration of the reactant, A. The half-life of any first-order reaction can be calculated using the following equation. 

In this section, you learned how to relate the rate of a chemical reaction to the concentrations of the reactants using the rate law. You classified reactions based on their reaction order. You determined the rate law equation from empirical data. Then you learned about the half-life of a first-order reaction. As you worked through sections 6.1 and 6.2, you may have wondered why factors such as concentration and temperature affect the rates of chemical reactions. In the following section, you will learn about some theories that have been developed to explain the effects of these factors. 

The half-life of a reaction with a kinetic order higher than one is lengthened as the concentrations of the reactants are decreased (Sec. 1.3). Provided that there is still a sufficient change of concentration during the reaction to be accurately monitored, quite large rate constants may be measured if low concentration of reactants are used, even without recourse to the specialized techniques described in the previous section. 

Defining the half-life of the reaction, ty2, as the time needed for the concentration of reactants to drop to one-half the original value, we obtain... 

The half-life of a reactant and the mean reaction time of a reaction are two measures of the time to reach equilibrium. The half-life ty2 is the time for the reactant to decrease to half of its initial concentration, or more generally, the time for it to decrease to halfway between the initial and the final equilibrium concentration. The mean reaction time t is roughly the time it takes for the reactant concentration to change from the initial value to 1/e toward the final (equilibrium) value. The rigorous definition of the mean reaction time t is through the following equation (Equation 1-60) ... 

The time that it takes to mix reactants or to bring them to a specified temperature may be significant in comparison to the half-life of the reaction. 

An alternative method to determine the reaction order is the half-life method. The half life of a reaction (t /2) is the time it takes for 50% of the reactant(s) to be consumed. At time t /2 the concentration of A must then be [A]o/2. For a first-order reaction, Eq. 13.15 yields... 

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... 

Half-life of reaction, i.e. time for half reactant to be ... 

The half-life of a first-order reaction is a constant because it depends only on the rate constant and not on the reactant concentration. This point is worth noting because reactions that are not first order have half-lives that do depend on concentration that is, the amount of time in one half-life changes as the reactant concentration changes for a non-first-order reaction. 

The reaction A — C is first-order in the reactant A and is known to go to completion. The product C is colored and absorbs light strongly at 550 nm, while the reactant and intermediates are colorless. A solution of A was prepared, and the absorbance of C at 550 nm was measured as a function of time. (Note that the absorbance of C is directly proportional to its concentration.) Use the following data to determine the half-life of the reaction ... 

Integrated rate expressions can also be used to demonstrate that the half-life of a reaction varies systematically with [reactant], and so is diagnostic of the reaction order. 

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