Half-life equation for

The half-life equation for the general nth-order reaetion is... 

Aii radioactive decay processes foiiow first-order kinetics. What does this mean What happens to the rate of radioactive decay as the number of nuciides is haived Write the first-order rate law and the integrated first-order rate law. Define the terms in each equation. What is the half-life equation for radioactive decay processes How does the half-life depend on how many nuclides are present Are the half-life and rate constant k directly related or inversely related ... 

Compare the half-life equations for a first-order and second-order reaction. For which reaction order is the value of the half-life independent of the reactant concentration ... 

Table C2.6.5 Rapid coagulation half-life time for particles in water at 7 =300 K (equation (C2.6.16)).

Recall also from Chapter 15 that for first-order reactions, the time required for exactly half of the substance to react is independent of how much material is present. This constant time interval is the half-life, Equation... 

There is no single equation containing these four variables. For this reason, we need to use the two equations In (Nt/N0) = -kt and t1/2 = (In 2)/k. We will begin with the half-life equation t1/2 = (In 2)/k. We need to rearrange this equation and enter the half-life to determine the decay constant ... 

The half-life period for a first-order reaction may be obtained from equation (b) by substituting t = tm when x = all, i.e. 

Nitrosamine yields %) from reactions according to equations (4), (7), and (8) are at specific analysis times. The reported yields are not the maximum yields. The estimated half-life is for the parent nitrosamine (left column) and assumes conversion to VII only. Reactions were conducted in tetrahydrofuran at 70°C. Substrate, 0.42 M potassium t-butoxide, 0.56 M t -butyl alcohol,... 

If you recall, back in Chapter 5 we discussed half-life in the context of the decay of radioactive nuclei. In that chapter, we defined the half-life as the amount of time it took for one half of the original sample of radioactive nuclei to decay. Because the rate of decay only depends on the amount of the radioactive sample, it is considered a first-order process. Using the same logic, we can apply the concept of half-life to first-order chemical reactions as well. In this new context, the half-life is the amount of time required for the concentration of a reactant to decrease by one-half. The half-life equation from Chapter 5 can be used to determine the half-life of a reactant ... 

Equation 17.25 describes the half-life relationship for a drug that appears to follow one-compartment body kinetics that is, when the body is considered to be a single homogeneous pool of body fluids. However, many drugs appear to exhibit multiple distribution pools and therefore may have multiple half-lives (as was depicted in Fig. 17.3 for IFN-a). Drugs with multiple half-lives are usually reported in the literature as having "distribution" and "terminal elimination" half-lives. Defining the "relevant" half-life in such situations has been addressed by Benet and coworkers (2,14,15). 

You eould also solve for k using the half-life and eoneentration (2 ppm). Then substitute k and the new concentration (10 ppm) into the half-life equation to solve for the new half-life. Try it. 

The experimental rate law can be determined by monitoring the concentration of one of the reactants or products as a function of time using spectroscopic means. For instance, the Beer-Lambert law states that the absorbance of a colored compound is directly proportional to its concentration (for optically dilute solutions anyway), so that the absorbance can be measured as the course of the reaction proceeds. The data are then fit to a model, such as the function that results when integrating one of the differential rate law equations. The integrated rate laws for some commonly occurring kinetics are listed in Table 17.1. Half-life equations are also included for some of the reactions in this table, where the half-life ftyi) is defined as the length of time that it takes for half of the initial reactant concentration to disappear. 

The kinetic constant, k, for the model above, can be obtained from the half life where the half life equation is... 

STRATEGIZE Use the expression for half-life (Equation 19.1) to find the rate constant (k) from the half-life for C-14, which is 5730 yr (Table 19.3). 

AU radioactive elements decay according to first-order kinetics (Chapter 13) the half-life equation and the integrated rate law for radioactive decay are derived fiom the first-order rate laws. 

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. 

Equation (20.14) is valid only for first-order reactions. We will derive half-life expressions for other types of reactions as we encounter them. 

Equations 13.31 and 13.32 are only valid if the radioactive element in the tracer has a half-life that is considerably longer than the time needed to conduct the analysis. If this is not the case, then the decrease in activity is due both to the effect of dilution and the natural decrease in the isotope s activity. Some common radioactive isotopes for use in isotope dilution are listed in Table 13.1. 

The unit of the veloeity eonstant k is see Many reaetions follow first order kineties or pseudo-first order kineties over eertain ranges of experimental eonditions. Examples are the eraeking of butane, many pyrolysis reaetions, the deeomposition of nitrogen pentoxide (NjOj), and the radioaetive disintegration of unstable nuelei. Instead of the veloeity eonstant, a quantity referred to as the half-life iyj is often used. The half-life is the time required for the eoneentration of the reaetant to drop to one-half of its initial value. Substitution of the appropriate numerieal values into Equation 3-33 gives... 

In a two-compartment model, /3 is equivalent to k in the one-compartment model. Therefore, the terminal half-life for the elimination of a chemical compound following two-compartment model elimination can be calculated from the equation (i = 0.693/ti/i ... 

Do you think it would be possible to resolve DCBP into different enantiomers at room temperature Answer this question by calculating the effective energy barrier, AE, for internal rotation (choose the lowest possible barrier), and then calculating the half-life of the favored conformers at 298 K (use equation 1). 

For each molecule, calculate the overall energy barrier for ring inversion in each direction. Use this barrier to calculate the half-life (t./,) of an individual molecule at 298 K (use equation 2). Which molecule inverts most rapidly Most slowly Why (Hint What geometrical changes are required for inversion )... 

The experiments were carried out with two initiators. According to published data (B.), at the base temperature, Tb, the fast initiator, II, has a half-life of 3.5 minutes, and the slow initiator, 12, has a half-life of 95 minutes. A minor modification of the monomer mass balance (Equation 7) is required for the case of two initiators. 

Equation 6 would hold for a family of free radical initiators of similiar structure (for example, the frarw-symmetric bisalkyl diazenes) reacting at the same rate (at a half-life of one hour, for example) at different temperatures T. Slope M would measure the sensitivity for that particular family of reactants to changes in the pi-delocalization energies of the radicals being formed (transition state effect) at the particular constant rate of decomposition. Slope N would measure the sensitivity of that family to changes in the steric environment around the central carbon atom (reactant state effect) at the same constant rate of decomposition. 

The slopes, Y-intercepts and squares of correlation coefficients for the linear regression analyses of the T versus AE(ir) plots (equation 7) for reactions 1-4 for one-hour and ten-hour half life rates of decomposition to form free radical products are given in Table II. 

Another characteristic of first-order reactions is that the time it takes for half the reactant to disappear is the same, no matter what the concentration. This time is called the half-life ( 1/2). Applying Equation to a time interval equal to the half-life results in an equation for / i 2 When half the original concentration has been consumed,... 

As shown in Example, Equation is used to find a nuclear half-life from measurements of nuclear decays. Equation is used to find how much of a radioactive substance will remain after a certain time, or how long it will take for the amount of substance to fall by a given amount. Example provides an illustration of this t q)e of calculation. In Section 22-1. we show that Equation also provides a way to determine the age of a material that contains radioactive nuclides. 

The question asks for the time it takes for 99% of a sample of plutonium to decay. The half-life is known from the previous Example. Equation relates the ratio Nq / //to time and the half-life for decay. This equation can be solved for t, the time at which the ratio reaches the desired value ... 

As a noble gas, Rn in groundwater does not react with host aquifer surfaces and is present as uncharged single atoms. The radionuclide Rn typically has the highest activities in groundwater (Fig. 1). Krishnaswami et al. (1982) argued that Rn and all of the other isotopes produced by a decay are supplied at similar rates by recoil, so that the differences in concentrations are related to the more reactive nature of the other nuclides. Therefore, the concentration of Rn could be used to calculate the recoil rate for all U-series nuclides produced by a recoil. The only output of Rn is by decay, and with a 3.8 day half-life it is expected to readily reach steady state concentrations at each location. Each measured activity (i.e., the decay or removal rate) can therefore be equated with the input rate. In this case, the fraction released, or emanation efficiency, can be calculated from the bulk rock Ra activity per unit mass ... 

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