Half-lives, method

The half-life method requires data from several experiments, eaeh at different initial eoneentration. The method shows that the fraetional eonversion in a given time rises with inereased eoneentration for orders greater than one, drops with inereased eoneentration for orders less than one, and is independent of the initial eoneentration for reaetions of first order. This also applies to the reaetion A -i- B —> produets when... 

If the reaetion rate depends on more than one speeies, use the method of exeess eoupled either with the half-life method or the differential method. If the method of exeess is not suitable, an initial rate plot may be eonstrueted by varying the eoneentration of one reaetant while the eoneentrations of the others are held eonstant. This proeess is repeated until the orders of reaetion of eaeh speeies and the speeifie reaetion rate are evaluated. At level 5, the least-squares analysis ean be employed. 

Experimental results O 107ppm A 50ppm 35ppm Calculation results Half-life method... 

And here is another variation of the half-life method. 

Fractional Life Method The half-life method can be extended to any fractional life method in which the concentration of reactant drops to any fractional value F = C /Cao in time The derivation is a direct extension of the half-life method giving... 

Example 3.1c showed how to find a rate equation by using the fractional life method where F = 80%. Take the data from that example and find the rate equation by using the half-life method. As a suggestion, why not take Cao = 10, 6, and 2 ... 

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. 

Half-life method Two different initial concentrations of A are used. The decrease in A over time is measured. The half-life x = tl/2 is defined as the time where c(A) = c(A)0/2. The order can be determined with the following equation (see Figure 4-la, b)... 

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 ... 

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,... 

To determine the Cl-nt of compound X, we are able to use the in vitro half-life method, which is simpler than finding all the component Cl nt values. When the substrate concentration is much smaller than Km, the Michaelis-Menten equation simplifies from velocity (V) = Vmax([S])/(Km + [S]), because [S] (substrate concentration) becomes negligible. Furthermore, under these conditions, the in vitro half-life (7) 2 = 0.693/Xel) can be measured, and this, in turn, is related to the Michaelis-Menten equation through the relationship velocity (V) = volume x Kel (where volume is standardized for the volume containing 1 mg of microsomal protein). When both V and Vmax are known, then Km is also found. Although simpler than finding a complicated Cint, one caveat of the in vitro half-life method is that one assumes that the substrate concentration is much smaller than Km. It may be necessary to repeat the half-life determinations at several substrate concentrations, and even model the asymptote of this relationship, because very low substrate concentrations that are beneath biochemical detection may be needed to fulfill the assumptions needed to simplify the Michaelis-Menten equation. 

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). 

To determine the Cl[ of compound X, we are able to use the in vitro half-life method, which is simpler than finding all the component Cl[ values. When the substrate concentration is much smaller... 

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. 

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... 

The radionuclidic purity can be determined by gamma spectrometry, or by the determination of the half-life. The half-life method is only a qualitative assessment but is useful for very short lived isotopes for use in PET, such as oxygen ( O) (half-life 2 min). 

Equations (1.2.18) and (1.2.19) give identical numerical values. The former is obtained from separation of variables, while the latter from Laplace transforms. From Eq. (1.2.16), D can be obtained through the half-life method at a relative uptake (right-hand side value) of 0.5,... 

Are these values consistent with first-order kinetics If so, determine the rate constant by plotting the data as shown in Figure 14.7(b). (c) Determine the rate constant by the half-life method. 

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