First-order reactions mean life

Notice that the first order reaction half-life is unique in having no dependence on the initial concentration, [A]q. These half-life equations highlight a means for determining reaction order. If t is determined for several different initial concentrations, [A]q, then the data can be used to determine if x varies linearly with [A]q as in Equation 6.25 or inversely as in Equation 6.27, or if it is independent as in Equation 6.26. Once the order of the reaction is determined, the measurement of t and [A]q means that the value of the rate constant, k, is known via one of these three equations or by a corresponding equation for a higher order reaction. 

Explain what the term half-life means. Calculate the half-life of a first-order reaction. 

Important quantities characteristic of a first-order reaction are the half-life of the reaction, which is the value of t when [A], = [A](,/2, and t, the relaxation time, or mean lifetime, defined as k. ... 

The concentration evolution curves of Figures 2-la and 2-lb may be used to estimate the half-life or mean reaction time. When Figures 2-la and 2-lb are compared, the mean reaction time is found to differ by four orders of magnitude Hence, for second-order reactions, the timescale to reach equilibrium in general depends on the initial conditions. This is in contrast to the case of first-order reactions, in which the timescale to reach equilibrium is independent of the initial conditions. 

In Chapter 12, the concept of half-life was used in connection with the time it took for reactants to change into products during a chemical reaction. Radioactive decay follows first order kinetics (Chapter 12). First order kinetics means that the decay rate... 

The characteristic life time of a reaction is a measure of the time required after initiation for it to reach completion. This period is frequently related to the rate constant for the reaction in a veiy clear and specific way. Solutions to some of the common zero-, first- and second-order rate equations are presented in Table 9.5. Examples of zero- and first-order reactions are discussed in this section application of the second-order equations to general catalytic processes will be presented in the section on catalysis. The last column of Table 9.5 lists the relations between r, the characteristic life time of the reactant with respect to the chemical reaction, and the rate constant for the reaction. The meaning of the characteristic life time depends upon the order and reversibility of the reaction. 

At fixed conditions, the half-life of a first-order reaction is a constant, independent of reactant concentration. For example, the half-life for the first-order decomposition of N2O5 at 45 C is 24.0 min. The meaning of this value is that if we start with, say, 0.0600 mol/L of N2O5 at 45 C, after 24 min (one half-life). 

This relationship does not depend on another variable than these three parameters and will be used as a tool to detect conditions in which self-organisation based on proto-metabolic fluxes of energy can take place. Selecting a value of 1 for the transmission coefficient k (meaning that there is no possibility of reverting to the reactants after the system has crossed the transition state), the value of the free energy of activation can then be deduced as a function of the half-life of a first-order (or pseudo-first-order) reaction at different values of temperature (Fig. 8.6). 

Note 2 Lifetime is sometimes used for the processes that are not of first order. However, in such cases, the lifetime depends on the initial concentration of the entity or of a quencher, and therefore, only an initial or a mean lifetime can be defined. In this case it should be called decay time. Note 3 Occasionally, the term half-life (q/2) is used, representing the time needed for the concentration (c) of an entity to decrease to one half of its original value that is, c(l= i/2) =c(f=0)/2 (i.e., concentration c at ri/2 = half of the original concentration - at t=0). For the first-order reactions, Ti/2 = ln2r. 

You can see that the successive half-lives have values that are fairly close to each other (17.0,17.3,16.7). The mean half-life is 17.0 minutes for this reaction. We can tell that this reaction is first order because the successive half-lives are more or less constant. In a first-order reaction like this the half-life is independent of the original concentration of reactant. This means that whatever the starting concentration of cyclopropane, the half-life will always be 17 minutes. 

If an intermediate such as a radical or an excited molecule is produced by absorption of radiation and disappears at a rate proportional to the first power of the intensity, it is possible to interrupt the light beam and the rate of disappearance integrated over time will then vary merely as the fraction of the time the light reaches the reaction vessel. If this is 0.25, the net rate with interruption will be 0.25 times the rate with full illumination. If, on the other hand, the intermediates disappear by a second order reaction (i.e. the rate is proportional to the square of their concentration) the situation is different. At very high rates of interruption the light will appear to be on all of the time but with one quarter of the intensity. At very slow rates of interruption the light will appear to be on one quarter of the time with full intensity. In a plot of rate vs. duration of flash the asymptotes for very slow and for very rapid interruption will differ by a factor of four. The inflection point will come at about the mean life of the intermediate being studied. 

The thermal scission of a compound to yield two radicals was used for illustration in reaction (6-6), because this is the most common means of generating radicals. Thermal decomposition is ideally a unimolecular reaction with a first-order constant, kd, which is related to the half-life of the initiator, / /2, by Eq. (6-33). 

The length of time for which a substance persists in the environment is often stated in the form of a half-life of degradation. This is the time required for an initial concentration to be reduced by half. Strictly speaking, it is only possible to define of a half-life if degradation follows first order kinetics. This is the case if the speed of degradation is proportional to the current concentration of the substance. Since at least one reaction partner is necessary for the (bio)chemical conversion of a substance, this condition can only be fulfilled if an excess of such a reaction partner is present. This is usually the case with water and oxygen in suitable environmental media, which means that the reaction does in fact follow first order kinetics. 

The decay of elements by radioactivity, although not a chemical reaction, follows first-order kinetics. For example, the half-life of is 4.5 X 10 years. This means... 

What does the term half-life mean Write the expressions for the half-lives of zero-order, first-order, and second-order reactions. 

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