Constant fractional life

This method is based on the principle of constant fractional life (usually called half-life ), which applies to a species undergoing reaction in such a way that, after any time interval, a constant fraction of the amount left unreacted at the end of the previous interval has reacted (or a constant fraction remains unreacted), irrespective of the initial concentration. This property is associated with first-order or pseudo-first-order reactions—such as radioactive decay—for which half-lives are often quoted as a measure of reaction rate. This property of constant fractional life also applies to more complex reactions—such as successive and parallel reaction-sequences—involving first-order reactions. The initial concentration of a species reacting with constant fractional life is directly proportional to the amount of product formed at any given time. 

Experiments have been carried out on the mass transfer of acetone between air and a laminar water jet. Assuming that desorption produces random surface renewal with a constant fractional rate of surface renewal, v, but an upper limit on surface age equal to the life of the jet, r, show that the surface age frequency distribution function, 4>(t), for this case is given by ... 

The fractional life approach is most useful as a means of obtaining a preliminary estimate of the reaction order. It is not recommended for the accurate determination of rate constants. Moreover, it cannot be used for systems that do not obey nth order rate expressions. 

Fractional Life Relations for Constant Volume Systems... 

Fractional-life methods. If a reaction is known to be first order and at constant fluid density, its apparent rate coefficient can be found very quickly. For batch and differential recycle reactors, the relationship between the rate coefficient and the time ty required for all but a fraction y of the reactant to be consumed is... 

If quick inspection shows a reaction at constant volume to be first order, an approximate value of the rate coefficient can be found immediately by application of the fractional-life equation 3.24 or 3.25 to a data point in the mid-conversion range. 

As the steady state is approached, the curves in the first-order plots become parallel straight lines. Accordingly, the fractional-life equation 5.8 can be used to obtain an estimate of from the time required to approach this condition closely. The individual coefficients kn and can then be obtained from their sum and the equilibrium constant Kn = k12 /k21 ... 

The fractional life methods can also be used to determine a. As already indicated, the fractional life method of treating the data of a single experiment has few advantages over the superimposition method and consequently need not be considered further. On the other hand, the measurement of a particular fractional life of the reactant A in a series of experiments in which its initial concentration is varied provides a means by which a can be determined which sometimes offers experimental advantages. The only point to remember is that throughout the series of experiments [BJo must be kept constant at such a value that [BJo/[A]o,i Vb/Va. From the condition that... 

The time necessary for a given fraction of a limiting reagent to react will depend on the initial concentrations of the reactants in a manner that is determined by the rate expression for the reaction. This fact is the basis for the development of the fractional life method (in particular, the half-life method) for the analysis of kinetic data. The half-life, or half-period, of a reaction is the time necessary for one-half of the original reactant to disappear. In constant-volume systems it is also the time necessary for the concentration of the limiting reagent to decline to one-half of its original value. 

Thus the three-quarters life (or any given fractional life) is also independent of concentration for a first-order reaction. Examination of the data shows that the first three-quarters life (time to [A] = 0.237 mol dm -) is about 80 min and by inteipolation the second (lime to [A] = 0.178 mol dm - ) is also about 80 min. Therefore the reaction is first-order and the rate constant is approximately... 

This method attempts to relate the capital allowance to the total life of the assets (i.e. the field s economic lifetime) by linking the annual capital allowance to the fraction of the remaining reserves produced during the year. The capital allowance is calculated from the unrecovered assets at the end of the previous year times the ratio of the current year s production to the reserves at the beginning of the year. As long as the ultimate recovery of the field remains the same, the capital allowance per barrel of production is constant. However, this is rarely the case, making this method more complex in practice. 

The nondepreciable investments, ie, land and working capital, are often assumed to be constant preoperational costs that are fully recoverable at cost when the project terminates. Equipment salvage is another end-of-life item that can represent a significant fraction of the original fixed capital investment. However, salvage occurs at the end of life, can be difficult to forecast, and is partially offset by dismantling costs. Eor these reasons, a zero salvage assumption is a reasonable approximation ia preliminary analysis. 

In the total-life plan the amount of depreciation is obtained by multiplying the total amount that can be depreciated by a fraction. The numerator of this fraction is the number of years of useful life remaining. The denominator is the sum of the digits from one through the total estimated number of years of useful life. The denominator is a constant. 

The answer is b. (Katzang, p 40.) The fraction change in drug concentration per unit of time for any first-order process is expressed by fte. This constant is related to the half-life (ti,2) by the equation Ktul — 0 693. The units of are time-1, while the tm is expressed in units of time. By substi-... 

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. 

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