and dX, decay dt
where udcc is a decay rate constant and Yx is a stoichiometric factor. During growth (vm > 0), both organic substrate and electron acceptors are consumed at rates that are proportional to vm.
Thus, for a known reaction stoichiometry, the actual rates can be easily determined. For example, in the previously discussed case of toluene degradation under sulfate-reducing conditions (equation 4), the complete mineralization of 1 mol of toluene consumes 4.15 mol of sulfate (during growth) and yields 0.14 mol of sulfate-reducing bacteria (SRB), thus:
and dc sulf dt
where Csuif is the concentration of sulfate. As shown in Fig. 4.1, for given initial concentrations (at time t = 0 [see Table 4.2]), we can now compute the temporal development of toluene and sulfate and of the microbial mass in a closed batch-type system (case la). This can be done, for example, with the geochemical model PHREEQC-2 (45) by using its capability to compute arbitrary user-defined kinetic reactions. Notable in the upper plot is the lag period of several days before the degradation affects the aqueous concentrations of toluene and sulfate. Its length depends largely on the initial bacterial concentration and on vniax, the maximum uptake rate (values given in Tables 4.2 and 4.3). The removal of the initial toluene mass (0.1 mmol) is reached after 14 days, at a time when approximately 0.415 mmol of sulfate is depleted. The microbial (net) growth then stops immediately (vm = 0) and the microbial mass is subsequently changing at the rate given by equation 18, thereby consuming the remaining 0.035 mmol of sulfate, as discussed previously (Fig. 4.1, case la). By comparison, Fig. 4.1 shows also the temporal development of the sulfate concentration if the effects of the different valence states of bacteria (compared to the end product C02) and the related geochemical changes are not considered (Fig. 4.1, case lb). In the latter case, all sulfate is consumed during bacterial growth and none
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