Reaction order half-life method

Good introduction to order of reaction, including half-life method. Most of the topic is covered. www.chemistryrules.me.uk/hfhf/hfhf3.htm title... 

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 ... 

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. 

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... 

We see that the half-life is always inversely proportional to k and that its dependence on [A]o depends on the reaction order. Thereby the method can be used to determine both the rate constant and the reaction order, even for reactions with noninteger reaction order. Similar to the integral method, the half-life method can be used if concentration data for the reactant are available as a function of time, preferably over several half-lives. Alternatively the half-life can be determined for different initial concentrations in several subsequent experiments. 

The half-life method requires data from several experiments, each at different initial concentration. The method shows that the fractional conversion in a given time rises with increased concentration for orders greater than one, drops with increased concentration for orders less than one, and is independent of the initial concentration for reactions of first order. This also applies to the reaction A + B — products when Cao = Cbo-... 

If the reaction rate depends on more than one species, use the method of excess coupled either with the half-life method or the differential method. If the method of excess is not suitable, an initial rate plot may be constructed by varying the concentration of one reactant while the concentrations of the others are held constant. This process is repeated until the orders of reaction of each species and the specific reaction rate are evaluated. At level 5, the least-squares analysis can be employed. 

The half-life method (in general, the fractional-life method) is very useful in preliminary estimates of the order of the reaction. A series of experimental runs is carried out with different initial concentrations. If an irreversible reaction is considered ... 

Determine the order of the degradation reaction using the half-life method. 

The Half Life Method We have shown above that, provided all reactants are present in the same molar concentrations, the half-life, t1/2, of nth-order reaction is given by Eq. 2.38. 

The Half-Life Method The half-life is the batch time required to get 50 percent conversion. For an nth-order reaction,... 

Many authors propose alternative mathematical treatments for kinetics equations. Some examples are a general approach based on a matrix formulation of the differential kinetic equations (Berberan-Santos Martinho, 1990) spreadsheets in which rate equations are integrated using the simple Euler approximation (Blickensderfer, 1990) a method for the accurate determination of the first-order rate constant (Borderie, Lavabre, Levy Micheau, 1990) a simplification of half-life methods that provides a fast way of determining reaction orders and rate constants (Eberhart Levin, 1991) a general approach to reversible processes, the special cases of which are shown to be equivalent to basic kinetic equations (Simonyi Mayer, 1985) an equation from which zero-, first- and higher order equations can be derived (Tan, Lindenbaum Meltzer, 1994). 

The data for this experiment do not extend much beyond one half-life Therefore the half-life method of predicting the order of the reaction as described in the solutions to Problems 22.1 and 22.2 cannot be used here. However, a similar method based on three-quarters lives will work. For a first-order reaction, we may write (analogous to the derivation of eqn 22.13)... 

IRREVERSIBLE REACTIONS OF ORDER n-HALF-LIFE METHOD... 

Wilkinson s method allows the evaluation of the reaction order from data taken during the first half-life. This, as we saw, was not possible from treatment by the integrated rate law. Note, however, that relatively small errors in [A] can lead to a larger error in E at small conversions.17... 

A particularly useful application of MW-assisted synthesis at elevated pressure has been in the preparation of radiopharmaceuticals containing isotopes with short half-lives, such as C-ll (half-life 20 min) and F-18 (half-life 110 min) [25-27]. Clearly, these compounds have to be synthesized very rapidly in order to give products with high radiochemical yield. For example, [1-11C] tyrosine 12 was synthesized using the two step Bucher-Strecker method by the reaction of p-hydroxyphenylacetaldehyde bisulfite adduct 11 with K11CN and (NH4)2C03 followed by hydrolysis with aqueous NaOH (Scheme 4.7)... 

Very rarely are measurements themselves of much use or of great interest. The statement "the absorption of the solution increased from 0.6 to 0.9 in ten minutes", is of much less use than the statement, "the reaction has a half-life of 900 sec". The goal of model-based analysis methods presented in this chapter is to facilitate the above translation from original data to useful chemical information. The result of a model-based analysis is a set of values for the parameters that quantitatively describe the measurement, ideally within the limits of experimental noise. The most important prerequisite is the model, the physical-chemical, or other, description of the process under investigation. An example helps clarify the statement. The measurement is a series of absorption spectra of a reaction solution the spectra are recorded as a function of time. The model is a second order reaction A+B->C. The parameter of interest is the rate constant of the reaction. 

Fractional Change Method From the equations of half life tor reactions of various orders except first order reaction, time required to complete a definite fraction of the reaction is inversely proportional to af- where n is the order of reaction and a is initial concentration. 

For n t- 1 it is important to keep in mind that the half-life is a function of the initial concentration c(A)a. Knowing the order of reaction and the rate constant allows the half-life to be calculated. Or it can be determined experimentally and used to calculate the other parameters, e. g. by the trial and error method. 

Sol. 1st Method Since the half life periods are the same, irrespective of the initial concentrations or pressures, the reaction is of the first order. 2nd Method We know that ... 

A fifth method is the least accurate of all but it is extremely simple and it is adequate for many purposes. It depends on the fact that if the time taken for three-fourths of the material to decompose is twice as long as the time taken for half of it to decompose the reaction is first order. Under these conditions the velocity constant k can be evaluated by simply dividing the period of half-life into 0.69 as previously explained (page 22). 

Oxidation rate constant k, for gas-phase second order rate constants, kon for reaction with OH radical, k os with NOj radical and koj with O3 or as indicated, data at other temperatures see reference photooxidation half-life of 0.24-2.4 h in air for the gas-phase reaction with OH radical, based on the rate of disappearance of hydrocarbon due to reaction with OH radical (Darnall et al. 1976) k < 0.8 M s for the reaction with ozone in water at pH 2 and 20-23°C (Hoignd Bader 1983) koH = (13.6 1.3) X cm molecule s at 298 K (Wallington et al. 1988a Atkinson 1989) kojj = (14.4 1.5) X 10 2 cm molecule s at 298 2 K (relative rate method, Nelson et al. 1990) koe(calc) = 11.67 X 10 cm molecule s (molecular orbital calculations, Klamt 1996)... 

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