Decay constant from activity

Calculating the Decay Constant and Activity from the Half-Life... 

Calculating the decay constant from the activity Given the activity (disintegrations per second) of a radioactive isotope, obtain the decay constant. (EXAMPLE 21.7)... 

Various investigators have tried to obtain information concerning the reaction mechanism from kinetic studies. However, as is often the case in catalytic studies, the reproducibility of the kinetic measurements proved to be poor. A poor reproducibility can be caused by many factors, including sensitivity of the catalyst to traces of poisons in the reactants and dependence of the catalytic activity on storage conditions, activation procedures, and previous experimental use. Moreover, the activity of the catalyst may not be constant in time because of an induction period or of catalyst decay. Hence, it is often impossible to obtain a catalyst with a constant, reproducible activity and, therefore, kinetic data must be evaluated carefully. 

Let us consider an atmosphere where the (activity) concentrations of radon, 222Rn, and its three short-lived daughters, 218Po, 21ifPb and 21I+Bi, are CQ, Ci, C2 and C3, the decay constants and unattached fractions of the daughter products, X, X2, X3, fi, f2 and f3 respectively, and E and E" the energies of the alpha particles from 2 8Po and 21I+Po (the fourth short(est)-lived radon daughter). 

Figure 1-4 (a) Logarithm of the activity of Be (counts per minute) in Be(OH)2 versus time to demonstrate that the decay is a first-order reaction and that the decay constant is 0.012977 day (or half-life is 53.41 days). Error is about the size of the points, (b) ln(j4Mo)- -0.012977t versus t to compare the decay constant of Be in different compounds. The error bars are at the 2a level. The data indicate that the decay constant of Be depends on the chemical environment. Data from Huh (1999). 

Suppose a decay system is disturbed so that Th activity differs from activity. Based on Equation 2-100, the excess " Th activity, i.e., 2 " 238U would decay away with the decay constant of and after about 5 half-lives... 

Because of differences in chemical properties of U, Th, and Pa, the elements are fractionated in many geochemical processes, such as sedimentation, mantle partial melting, and coral precipitation from water. With fractionation, the nuclide activities of Th, and Pa do not equal one another. Define the time of disturbance to be time zero. Use hi, A2, and A3 to denote the decay activity of Th, and Pa, respectively, and Xy X2, and X3 to denote the decay constants of Th, and Pa. Start from the full evolution equation for Pa in Box 2-6,... 

For the next nuclide in the decay series, °Th, because Xs (decay constant of °Th) is not smaller but larger than X4, the activity evolution equation is more complicated. Starting from the fuU equation (Box 2-6), with X3 X2 Xs X4 X4, the following may be derived ... 

The decay constant of 220Rn is 6000 times greater than that of 222 Rn, so from (1.4) L is about 80 times less, that is only about 1 cm. The fraction of 220Rn atoms which escape from the rock crystals to the interstitial air is apparently about the same as for 222Rn, and since the specific activities of the thorium and radium chains are similar, equation (1.6) implies that the emanation of 222Rn should be 80 times greater... 

For example, suppose the eluant milked from the "Tcm generator is calibrated at 8 00 to contain 5 mCi/mL of eluant. If a physician orders a procedure to be performed at 12 00, what is the activity of the eluant Well, A0 = 5 mCi/mL, and the time is 4 hr. From our previous discussion, the decay constant is 0.115 hr-1. So we find the activity of the eluant has fallen by about 40%. 

Pfister (1977) measured hole mobilities of TPA doped PC. Figure 51 shows the temperature dependencies for different concentrations. The field was 7.0 x 105 v/cm. The concentration is expressed as the weight ratio X of TPA to PC. The mobilities were thermally activated with activation energies that increase with decreasing TPA concentration. The concentration dependence was described by the lattice gas model with a wavefunetion decay constant of 1.3 A. Figure 52 shows the field dependencies at different temperatures for X - 0.40. The solid lines were derived from the Scher-Montroll theoiy (1975) using the listed parameters. Pfister concluded that the theoiy provides a self-consistent interpretation of all experimental observations if field-induced barrier lowering and temperature-dependent dispersion are formally introduced into the expression for the transit time. 

Since dead sites have zero activity, the overall activity of the catalyst is a(n) = a (n), where a is the vector giving the activity levels of the active states. In order to see how overall activity a changes over time, first consider what happens if one starts with a "quasi-steady-state" distribution of active states, i.e., let v(0) = Cj, the eigenvector of Pn corresponding to the dominant eigenvalue Aj of Pn. In this case Piiei=AiCj, so s(n) = Pn"ei = Ai"ei = Aj"s(0). Thus the relative proportions of sites in active states remain unchanged over time there is simply an overall exponential decrease in the total population of active sites. Similarly, in this quasi-steady-state case we have a(n) = a s(n) = A "a ei = Ai"a(0) i.e., the overall activity decreases exponentially. The decay constant Aj is very close to 1 since the columns of Pu all have a sum very close to 1. In fact, if the columns of Pu all have identical sums P, then Aj = P this corresponds to the situation where the probability of sudden death is the same from each active state, namely b = 1-P. 

The half-lives (decay constants) that appear in many of the equations used to calculate CRE ages undergo revision from time to time. For example, at this writing (April, 2003), the half-lives of Be and of Mn are under active scrutiny. In all cases, the absolute production rates depend inversely on the half-life of the relevant radionuclide of interest. Thus, an increase of 10% in the half-life of Kr would decrease by 10% any value of based on Kr measurements. It follows that all CRE ages would then shift upward by 10%. While such changes may be important for understanding the dynamics of meteoroid delivery to Earth, they do not affect the relative values of the CRE ages, and hence the characteristic shapes of CRE age distributions. 

Figure 11 Conceptual models of thorium scavenging (Coale and Bmland, 1985 Bruland and Coale, 1986 Clegg et al., 1991). (a) The surface water Th net scavenging model. This model incorporates two different size classes of particles, small suspended particles and large sinking particles with the various sources and sinks for the activity (A) of Th depicted. Axh is the decay constant of Th, k is the net rate transfer of Th from dissolved to suspended particles and is the net rate of transfer of Th from small suspended particles to large sinking particles, (b) A reversible scavenging model including desorption, particle disaggregation and remineralization for the deep sea.

A concentration in activity units (decays/ substance from the water column assuming that a single constant applies to all... 

---