Half-life equations

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

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 problem specifies that this is a second-order process. Thus, you must simply enter the appropriate values into the second-order half-life equation ... 

X10 g. Table 20-1 tells you that thorium-234 has a half-life of 24.1 days, so that s T. The time elapsed (t) is 365 days, and the original sample was 1 g (Aq). Plugging these values into the half-life equation gives you... 

Volume of distribution and clearance are both properties of a drug. These two properties determine a drug s elimination rate constant and half-life (Equation 7.12). 

Absolute bioavailability (Equation (8.28)) and half-life (Equation (8.5)) as a measure of the rate of elimination. These parameters would give an indication of the relative effectiveness of each of the compounds. Absolute bioavailability would indicate the compound with the best absorption characteristics, whilst half-life would show which compound was the most stable in situ and so would have the best chance of being therapeutically effective. 

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 ... 

Order Integrated Rate Equation Half-Life Equation... 

The reason that Pd(ss) is smaller than either of these two estimates is that neither the half-life equation used to calculate Vd(area) I ior the single-compartment model implied in calculating naakes any provision... 

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 ... 

Am. We can calculate the number of half-lives in 2.5 days and then use our half-life equation to calculate the remaining dose ... 

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. 

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 ... 

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. 

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