Half first order reaction

II [Anisole] = 2 x lo mol i" first-order reactions. For the experiment using pure nitric acid the half-life was about i min, but for that using fuming nitric acid reaction was complete in < 30 s. 

Even when there is a transport disguise, the reaction order remains one for a first-order reaction. But for reactions that are not intrinsically first order, the transport disguise changes the observed reaction order for an intrinsically zero-order reaction, the observed order becomes 1/2 and for an intrinsically second-order reaction it becomes 3/2 when 0 10. For all reaction orders the apparent activation energy is approximately half the intrinsic... 

The half-life tvi is defined to be the time required for the reactant concentration to decay to one-half its initial value. To find tvi for a first-order reaction we use Eq. (2-6) with the substitutions Ca = c°/2 and t = finding... 

Evidently the measurement of should be accomplished with at least the same level of accuracy as the measurement of the A, values, so the question arises When does t = 00 That is, when is the reaction essentially complete For a first-order reaction, we calculate, with Eq. (2-10), that reaction is 99.9% complete after the lapse of 10 half-lives. This would ordinarily be considered an acceptable time for the measurement of... 

Find a relationship between the half-life tn2 and the lifetime t of a first-order reaction. 

Suppose that Cy = 0, Cz = 0, as is often the case. Then the final product concentrations are found by setting f = < in Eqs. (3-12) and (3-13) we obtain Cy = ( ki/k and c = c /Ji lk. The half-life for the production of Y is then given by Eq. (3-12), setting Cy = Cyl2 when t = t i. We find ha = In Hk, and the same result is obtained for product Z. Thus, the products are generated in first-order reactions with the same half-life, even though they have different rate constants. 

A more serious problem is that we lose all kinetic information about the system until the data collection begins, and ultimately this limits the rates that can be studied. For first-order reactions we may be able to sacrifice the data contained in the first one, two, or three half-lives, provided the system amplitude is adequate that is, the remaining extent of reaction must be quantitatively detectable. However, this practice of basing kinetic analyses on the last few percentage of reaction is subject to error from unknown side reactions or analytical difficulties. 

FIGURE 14.4 Plot of the course of a first-order reaction. The half-time, <1/9, is the time for one-half of the starting amonnt of A to disappear. 

The analysis of Example 11.3c reveals an important feature of a first-order reaction The time required for one half of a reactant to decompose via a first-order reaction has a fixed value, independent of concentration. This quantity, called the half-life, is given by the expression... 

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 ... 

At high temperatures, the decomposition of cyclobutane is a first-order reaction. Its activation energy is 262kJ/mol. At 477°C, its half-life is 5.00 min. What is its half-life (in seconds) at 527°C ... 

The logarithmic plot is not linear, of course, since this is not a first-order reaction. Note, however, that even In [A], is linear in time to about 50 percent reaction. One cannot use these procedures to establish the kinetic order without data taken to at least two half-times, and preferably longer. 

Consider the interconversion of two chiral molecules to yield ultimately the racemic mixture. This is simply the situation of opposing first-order reactions of A and P, treated in Chapter 3, for the special case of an equilibrium constant of unity. Recall that for such an equilibrating system ke = kf + kr because of that, knc is one-half the experimental rate constant. 

In a thin flat platelet, the mass transfer process is symmetrical about the centre-plane, and it is necessary to consider only one half of the particle. Furthermore, again from considerations of symmetry, the concentration gradient, and mass transfer rate, at the centre-plane will be zero. The governing equation for the steady-state process involving a first-order reaction is obtained by substituting De for D in equation 10.172 ... 

We already know that the higher the value of k, the more rapid the consumption of a reactant. Therefore, we should be able to deduce a relation for a first-order reaction that shows that, the greater the rate constant, the shorter the half-life. 

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. 

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. 

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 13.13 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. 

Suppose that a pollutant is entering the environment at a steady rate R and that, once there, its concentration decays by a first-order reaction. Derive an expression for (a) the concentration of the pollutant at equilibrium in terms of R and (b) the half-life of the pollutant species when R = 0... 

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

Therefore, if a plot of In [A] against t is linear, the reaction is first order and k can be obtained from the slope. For first-order reactions, it is customary to express the rate not only by the rate constant k but also by the half-life, which is the time required for half of any given quantity of a reactant to be used up. Since the half-life ti/2 is the time required for [A] to reach Aq/2, we may say that... 

For the same value of K, first-order reactions proceed much more rapidly than second-order reactions. The reaction rate for a hrst-order reaction will decrease to half its original value when the concentration has decreased to half the original concentration. For a second-order reaction, the reaction rate will decrease to a quarter the original rate when the concentration has decreased to half the original concentration compare Equations (1.16) and (1.17). 

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,... 

Equation does not contain the concentration of A, so the half-life of a first-order reaction is a constant that is independent of how much A is present. The decomposition reactions of radioactive isotopes provide excellent examples of first-order processes, as Example illustrates. We describe the use of radioactive isotopes and their half-lives to determine the age of an object in more detail in Chapter 22. 

Recall also from Chapter 15 that for first-order reactions, the time required for exactly half of the substance to react is independent of how much material is present. This constant time interval is the half-life, Equation... 

If it is certain that the reaction is indeed an irreversible first-order reaction, one can also determine how long it takes before 50 % of the reactant has been converted into products, as for any exponential decay the half-life, ty, is related to the rate constant k as... 

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. 

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 above equation implies that the half-life of a first-order reaction is independent of concentration. This result in only true for a first-order reaction. 

The half-life is thus seen to depend on the initial concentration for the second order reaction considered. This is in contrast to first-order reaction where the half-life is independent of concentration. For this reason half-life is not a convenient way of expressing the rate constant of second-order reactions. 

The equation involving t for the general case of a reaction of the nth order as shown above applies to any value of n except n = 1, for this case the treatment leading to exponential equation shown in first-order reaction (In a/(a- x) = kt) must be employed. The equation is applicable for n = 2. Other cases, including those of nonintegral orders, can easily be worked out. The half-life, t0 5, is seen to be inversely proportional to k in all cases, and inversely proportional to the (n - 1) power of the concentration. 

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 ... 

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