Reaction rate half-time

A further factor which must also be taken into consideration from the point of view of the analytical applications of complexes and of complex-formation reactions is the rate of reaction to be analytically useful it is usually required that the reaction be rapid. An important classification of complexes is based upon the rate at which they undergo substitution reactions, and leads to the two groups of labile and inert complexes. The term labile complex is applied to those cases where nucleophilic substitution is complete within the time required for mixing the reagents. Thus, for example, when excess of aqueous ammonia is added to an aqueous solution of copper(II) sulphate, the change in colour from pale to deep blue is instantaneous the rapid replacement of water molecules by ammonia indicates that the Cu(II) ion forms kinetically labile complexes. The term inert is applied to those complexes which undergo slow substitution reactions, i.e. reactions with half-times of the order of hours or even days at room temperature. Thus the Cr(III) ion forms kinetically inert complexes, so that the replacement of water molecules coordinated to Cr(III) by other ligands is a very slow process at room temperature. 

The filtration conditions (i.e., the rate and time of filtration) should be established in preliminary experiments. The two major considerations are (1) filter washing must be sufficient to minimise the nonspecific retention of radiologand by the filter and (2) the time course of filtration should be sufficiently short that dissociation of the radioligand from its receptor is avoided. In these studies, it is important to have a preliminary estimate of the dissociation rate, i.e., k i in equation23. For a simple exponential reaction, the half-time of dissociation (t./,) is related to the dissociation rate constant by t./, = 0.693/k i... 

The fastest steps in an enzymatic process cannot be observed by conventional steady-state kinetic methods because the latter cannot be applied to reactions with half-times of less than about 10 s. Consequently, a variety of methods have been developed18 56-593 to measure rates in the range of 1 to 1013 s... 

Table 14 shows the calculation of the reaction rate, the time law, and the half-life depending on the reaction s order. The order results from the sum of the exponents of the concentrations. The number does not necessarily have to be an integer. The half-life states in which time half of the reactants is converted into the products. Reaction rate constants k are 1012 to 10"11 L/s for first order reactions and 1010 to 10"11 L/(mol s) for second order reactions. 

Pulsed continuous flow is a method in which continuous flow is established for a short time. This method can reduce reagent consumption to 5 ml, and fast jet mixers have lowered the accessible reaction half-time to the 10 ps range. Pulsed accelerated flow may be viewed as an adaptation of pulsed continuous flow in which the flow rate through the mixer and observation chamber is varied during the course of one kinetic run. This method can be used for reactions with half-times down to 10 ps. This method is limited to first-order reaction conditions. 

The following is a quote from an article in the August 18, 1998, issue of The New York Times about the breakdown of cellulose and starch A drop of 18 degrees Fahrenheit [from 77 °F to 59 °F] lowers the reaction rate six times a 36-degree drop [from 77 °F to 41 °F] produces a fortyfold decrease in the rate. (a) Calculate activation energies for the breakdown process based on the two estimates of the effect of temperature on rate. Are the values consistent (b) Assuming the value of calculated from the 36° drop and that the rate of breakdown is first order with a half-life at 25 °C of 2.7 yr, calculate the half-life for breakdown at a temperature of — 15 °C. 

Everyday laboratory experience suggests that, with very few exceptions, reactions between acids and bases are extremely fast, since no time lag is observable in the dissociation of acids or bases, buffer action, hydrolysis, etc. In fact, for many purposes proton-transfer reactions involving simple acids and bases are fast enough to be treated as equilibrium processes. However, there are two reasons why the rates of these processes are of interest. In the first place modem techniques have made it possible to measure the rates of extremely fast reactions, with half-times down to about 10" second, and hence to obtain information about the mechanism of such reactions. In the second place, when proton-transfer reactions are coupled with other chemical processes they may lead to slow observable changes, in particular to the catalysis of reactions by acids and bases. The latter type of approach is historically the older, but it is more logical to consider first the direct observation of reactions between simple acids and bases, as will be done in this chapter. Some general features of the experimental results will be described, but detailed consideration of the relations between rates, equilibria, and structures will be deferred until Chapter 10, so as to include the information obtained less directly from studies of acid-base catalysis, described in Chapters 8 and 9. ... 

Rates of chemical reactions vary within a wide region of magnitude. Due to diffusion limitations, the rate constant of bimolecular reaction cannot exceed 10 ° mol dm s [28] therefore, this value can be used as an upper limit of Reactions whose half-times are less than 10 s are considered fast [29] then, the rate constant for monomolecular reaction should exceed 0.1 s L In accordance with the reaction rate, metal complexes are classified as labile and inert, but it is not possible to draw a sharp boundary line between the two groups. 

Other blend time correlations were presented by Penney (1971), Khang and Levenspiel (1976), and Fasano and Penney (1991). Use of these correlation equations allows the estimation of blending times, which can be compared to molecular reaction times for all the reactions in the reactor. Even though local mixing time is the critical time for determining apparent reaction rate, blend time can be used in an approximate manner. If the characteristic molecular reaction time (e.g., the half-life) is much greater than the characteristic blend time... 

It was pointed out that a bimolecular reaction can be accelerated by a catalyst just from a concentration effect. As an illustrative calculation, assume that A and B react in the gas phase with 1 1 stoichiometry and according to a bimolecular rate law, with the second-order rate constant k equal to 10 1 mol" see" at 0°C. Now, assuming that an equimolar mixture of the gases is condensed to a liquid film on a catalyst surface and the rate constant in the condensed liquid solution is taken to be the same as for the gas phase reaction, calculate the ratio of half times for reaction in the gas phase and on the catalyst surface at 0°C. Assume further that the density of the liquid phase is 1000 times that of the gas phase. 

The earliest examples of analytical methods based on chemical kinetics, which date from the late nineteenth century, took advantage of the catalytic activity of enzymes. Typically, the enzyme was added to a solution containing a suitable substrate, and the reaction between the two was monitored for a fixed time. The enzyme s activity was determined by measuring the amount of substrate that had reacted. Enzymes also were used in procedures for the quantitative analysis of hydrogen peroxide and carbohydrates. The application of catalytic reactions continued in the first half of the twentieth century, and developments included the use of nonenzymatic catalysts, noncatalytic reactions, and differences in reaction rates when analyzing samples with several analytes. 

For this to be constant, if L/dp doubles then must become one half, or u to be (1/2)° = 0.71 over the smaller pellets. Since the reaction rate is the same and the catalyst volumes are the same, the number of moles converted are the same. This constant number of moles will result in the 71% flow over the smaller pellets in 1/0.71 = 1.41 times larger reactant decrease than over the larger pills. The flow volumes are relative, not absolute values. 

Several products resulted when 4-HMP was allowed to condense under the same conditions. These are provided in Scheme 6. These products indicate that a p-hydroxymethyl group can react with either an unsubstituted ortho position or through an ipso-attack on a hydroxymethyl-substituted para position. The two condensation products were formed at approximately equal rates. Since there were two unoccupied orthos, it seems that the specific reaction rate for the ortho site is about half that of the occupied para position in this situation. The condensation rate for 4-HMP was about 5 times the rate for 2-HMP overall. 

The only kinetic data reported are in a Ph.D. thesis (41). Integral order kinetics were usually not obtained for the reaction of a number of ketones with piperidine and a number of secondary amines with cyclohexanone. A few of the combinations studied (cyclopentanone plus piperidine, pyrrolidine, and 4-methylpiperidine, and N-methylpiperazine plus cyclohexanone) gave reactions which were close to first-order in each reactant. Relative rates were based on the time at which a 50% yield of water was evolved. For the cyclohexanone-piperidine system the half-time (txn) for the 3 1 ratio was 124 min and for the 1 3 ratio 121 min. It appears that an... 

Another way to describe reaction rates is by half-life, t/, the amount of time it takes for the reactant concentration to drop to one half of its original value. When the reaction follows a first-order rate law, rate = -krxn[reactant], ti is given by ... 

RELATIVE RATES OF REACTION OF Ac20 WITH R CjH4FeCjHs IN CH2C12 CATALYSED BY BF3Et20 AT 0 AND 25 °C (HALF TIME AT EACH TEMPERATURE)431... 

Let us now assume that these matters have been attended to properly. At this stage we can but assume that the reaction orders were correctly identified and correct mathematical procedures followed. During the course of the work, the investigator should make the occasional quick calculation to show the values are roughly correct. (Does the rate constant yield the correct half-time ) Also, one should examine the experimental data fits to see that the data really do conform to the selected rate equation. Deviations signal an incorrect rate law or complications, such as secondary reactions. 

The half-time (or half-life) of the reaction is independent of [A]o. The reciprocal of the rate constant, t = l/k, is referred to as the lifetime or the mean reaction time. In that time [A] falls to l/e of its initial value. The pharmaceutical industry refers to the shelf life or t90, the time at which [A]/[A]o reaches 0.90. Both t and t90 are also independent of [A]o. 

The time required to convert a given fraction of the limiting reagent is a characteristic of the rate equation. A comparison of successive half-times, or any other convenient fractional time, reveals whether a reaction follows any simple-order rate law. Thus, the ratio of the time to reach 75 percent completion to that for 50 percent is characteristic of the reaction order. Values of this ratio for different orders are as follows ... 

Second-order kinetics, (a) Derive expressions for the half-time and lifetime of A if the rate law for its disappearance is v = fc[A]2 (b) calculate t]/i and t for the data presented in Section 2.2 (c) calculate the second half-life, t /i(2), i.e., the time elapsed between 50 percent and 75 percent completion, for the same reaction (d) compare fj/2(l) and fi/>(2), and contrast this result with that from first-order kinetics. 

The first of these reactions was carried out in 1,4-cyclohexadiene over a temperature range of 39 to 100 °C. It is fairly slow the half-times were 20 h and 3.4 min at the extremes. Reaction (7-11) is quite fast the second-order rate constant, kn, was evaluated over the range 6.4 to 47.5 °C. Values of feio and fen are presented in Table 7-1. The temperature profiles are depicted in Fig. 7-1 from their intercepts and slopes the activation parameters can be obtained. A nonlinear least-squares fit to Eq. (7-1) or... 

Under conditions where the dismutation reaction is slow the exchange between Au(III) and Au(I) has been shown to proceed at a measurable rate at 0 °C in 0.09 M HCl, an exchange half-time of about 2 min was observed. The isotopic method ( Au) and a separation method based on the precipitation of dipyridine -chloroaurate(III) was used to obtain data. An acceleration in the exchange rate was observed as the HCl concentration was increased. ... 

Deamination, the hydrolytic loss of exocyclic amino groups on the DNA bases, is typically a very slow reaction. For example, deamination of cytosine residues in dnplex DNA occnrs with a half-life of about 30,000 years under physiological conditions, and the deamination of adenine residues is still more sluggish. " Alkylation at the N3-position of cytosine (Scheme 8.5) greatly increases the rate of deamination (ty2 = 406 h). Deamination of 3-methyl-2 -deoxycytidine proceeds 4000 times faster than the same reaction in the unalkylated nucleoside. Alkylation of the N3-position in cytosine residues also facilitates deglycosylation (Jy2 = 7700 h, lower pathway in Scheme 8.5), but the deamination reaction is 20 times faster and, therefore, predominates. ... 

Addition of a strong acid snch as methanesnlfonic acid (MSA) to the reaction mixture has a positive impact on the reactivity, as shown in Figure 3.8. The induction time is shortened by 10 minutes and the reaction rate almost doubled. Due to the reaction rate increase from the acid addition, the catalyst loading could be lowered. In addition, the hydrogen pressnre conld be donbled to rednce the reaction time by half. However, improvements from addition of acid and pressure increase are not sufficient to make this process commercially viable because the catalyst loading and the TOF are significantly lower than the criteria listed in Table 3.n. Therefore, we initiated a search for catalysts more active than Et-DnPhos-Rh catalyst. 

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 ... 

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