Equilibrium: chemical secular

The partition function is the key quantity in the calculation of the Boltzmann equilibrium distribution of all molecular energies, and paves the way to a calculation of the total internal energy of any molecular system at equilibrium. Consider a system made of N molecules, each of which has a set of quantized energy levels Sj, known by the solution of some quantum chemical secular equation, as shown in Chapter 3. Energies are distributed over the accessible energy levels. With a modicum of mathematical derivation the following basic equations can be obtained ... 

Tcan be seen as the global weathering rate, and (t- P) as the chemical weathering rate. The use of two nuclides, i and j, leads to simple relationships between the parameters. Assuming secular equilibrium for the unweathered material, the fraction of nuclide j in each phase of the system is inferred from the NijNj activity ratios in water and weathering product. 

Although there are three Rji isotopes in the U- and Th-decay series, only is sufficiently long lived tm= 3.8 days) to be a useful estuarine tracer. Radioactive decay of Ra continuously produces Rn, which because of its short half-life is generally in secular equilibrium in seawater. Being chemically non-reactive except for very weak Van der Waals bonding makes this isotope a unique marine tracer in that it is not directly involved in biogeochemical cycles. 

In the environment, thorium and its compounds do not degrade or mineralize like many organic compounds, but instead speciate into different chemical compounds and form radioactive decay products. Analytical methods for the quantification of radioactive decay products, such as radium, radon, polonium and lead are available. However, the decay products of thorium are rarely analyzed in environmental samples. Since radon-220 (thoron, a decay product of thorium-232) is a gas, determination of thoron decay products in some environmental samples may be simpler, and their concentrations may be used as an indirect measure of the parent compound in the environment if a secular equilibrium is reached between thorium-232 and all its decay products. There are few analytical methods that will allow quantification of the speciation products formed as a result of environmental interactions of thorium (e.g., formation of complex). A knowledge of the environmental transformation processes of thorium and the compounds formed as a result is important in the understanding of their transport in environmental media. For example, in aquatic media, formation of soluble complexes will increase thorium mobility, whereas formation of insoluble species will enhance its incorporation into the sediment and limit its mobility. 

During mantle partial melting, the partition coefficients of Th, Pa, and Ra are different from that of U. Assuming the melt and the mantle residue as a whole maintains secular equilibrium, if the melting process is slow, there is chemical equilibrium between the phases, which means each phase (such as the melt phase) is out of secular equilibrium because of different partition coefficients (McKenzie, 1985). 

The ratio of to would be expected to be unity as long as the uranium stays locked inside undisturbed crustal rock in secular equilibrium with its progeny, but measurements show that the ratio is typically different than unity (EPA 1994). This disequilibrium occurs when the rock is disturbed by chemical or physical changes involving water. In the environment, a portion of the separates from the by what is theorized to be a physical process (alpha recoil ejection of the Th decay product from surfaces of soil particles) or a combination of physical and chemical processes (a transformation at the soil particle surface fractures the surface allowing access for water to dissolve the more soluble Th product) (NCRP 1984a). These processes can change the uranium isotope ratios in air, soil, and water. 

This bulk state of secular equilibrium applies to the total amount of the U-series nuclides, but does not necessarily say where the different elements reside within the system. If the bulk system has a single phase (such as a melt or a monomineralic rock) then that phase will be in secular equilibrium. If the material has multiple phases with different partitioning properties, however, the individual phases can maintain radioactive dis-equilibria even when the total system is in secular equilibrium. There are two basic sets of models that exploit this fact, the first assumes complete chemical equilibrium between all phases and the second assumes transient diffusion controlled sohd exchange. 

Figure 2 Variations of counting statistical error (2cr or 95% confidence) for nuclides of different half-lives using different measurement techniques. Four different scenarios are shown with details listed in legend, aimed to cover a typical range of conditions. The values of ionization efficiencies span a full range of values appropriate for elements difficult (Th) and easy (Ra) to ionize thermally, see text. All scenarios assume a sample with MORB-hke U concentration (50 ngg ), but with all daughter nuclides in secular equilibrium (for illustrative simplicity). Calculations assume (unrealistic) 100% yields for chemical purification of the nuclide of interest and 40% counting...

It takes into account the counting efficiency of Y in W2 (7jgQ 2) and the chemical yield of analysis When secular equilibrium is reached, the activity per unit of volume V (or a mass, if necessary) will be calculated as follows ... 

Secular equilibrium will be obtained only if the system in question remains physically closed, such that radioactive decay is the only process by which a daughter is effectively removed from its parent. Any additional non-radiochemical process that physically removes the daughter will destroy secular equilibrium. We shall see that chemical separation mechanisms prevent a closed system, and this is necessary for useful rate information to be obtained from the radioactive decay series. 

Discuss factors governing the attainment of radioactive and chemical equilibrium in a water-rock system. What chemical and physical processes aid or hinder the attainment of radioactive (secular) equilibrium in a groundwater ... 

Trace analysis of uranium can be carried out using a variety of radiometric techniques. The least sensitive method is gamma analysis of short-lived daughter products, for example, the " Th daughter t /2 = 24.1 days) of U. Short-lived daughter isotopes are in secular equilibrium with their parent isotope, vm-less they have been chemically separated, and the activity of " Th is therefore exactly the same as that of the long-lived parent. 

Chemical yields are determined from tracer activities added to the analytical aliquots however, the yields can also be determined from short-lived activities that are present in the sample due to secular radioactive equilibrium. For example, the chemical yield of uranium can be determined from the recovery of an added tracer activity, but in plutonium samples it can also be determined from There is a 0.00245% branch for the decay of Pu by Qt-particle emission. If it is known that the plutonium sample was last purified more than a month ago, one can assume that 6.75-day in the analytical samples is in equilibrium with ""Pu, with a concentration defined by the branching ratio of activityf U)/activity( Pu) = 2.45 X 10 Similarly, radium is traced with 3.66-day " Ra, in secular equilibrium with the decay of 1.9-year Th. Use of intrinsic short-lived radionuclides as tracers requires that the time at which they were isolated from their parent activities is known precisely. 

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