Polythermal path

Reaction paths can be traced at steady or varying temperature the latter case is known as a polythermal path. Strictly speaking, heat transfer occurs even at constant temperature, albeit commonly in small amounts, to offset reaction enthalpies. For convenience, modelers generally define polythermal paths in terms of changes in temperature rather than heat fluxes. 

Fig. 2.2. Example of a polythermal path. Fluid from a hydrothermal experiment is sampled at 300 °C and analyzed at room temperature. To reconstruct the fluid s pH at high temperature, the calculation equilibrates the fluid at 25 °C and then carries it as a closed system to the temperature of the experiment.

Polythermal reaction models (Section 14.1), however, are commonly applied to closed systems, as in studies of groundwater geothermometry (Chapter 23), and interpretations of laboratory experiments. In hydrothermal experiments, for example, researchers sample and analyze fluids from runs conducted at high temperature, but can determine pH only at room temperature (Fig. 2.2). To reconstruct the original pH (e.g., Reed and Spycher, 1984), assuming that gas did not escape from the fluid before it was analyzed, an experimentalist can calculate the equilibrium state at room temperature and follow a polythermal path to estimate the fluid chemistry at high temperature. 

A second type of polythermal path traces temperature as reactants mix into the... 

The time-stepping proceeds as previously described (Chapter 13), with the slight complication that the surface areas As of the kinetic minerals must be evaluated after each iteration to account for changing mineral masses. For polythermal paths, each rate constant k+ must be set before beginning a time step according to the Arrhenius equation (Eqn. 16.3) to a value corresponding to the temperature at the new time level. 

Finally, we define a polythermal path by equilibrating the hot hydrothermal fluid (Table 22.3)... 

Since we have provided initial and final temperatures but have not specified any reactants, the program traces a polythermal path for a closed system (see Chapter 14). The fluid s pH (Fig. 23.1) changes with temperature from its initial value of 5 at 250 °C to less than 4 at 25 °C. The change is entirely due to variation in the stabilities of the aqueous species in solution. As shown in Figure 23.2, the H+ concentration increases in response to the dissociation of the HC1 ion pair,... 

To invoke our geothermometer, we need to recombine the vapor and fluid phases and then heat the mixture to determine saturation indices as functions of temperature. We could do this in two steps, first titrating the vapor phase into the liquid and then picking up the results as the starting point for a polythermal path. We will employ a small trick, however, to accomplish these steps in a single reaction path. The trick is to add the vapor phase quickly during the first part of the reaction path but use the cutoff option to prevent mass transfer over the remainder of the path. The commands to set the mass transfer are... 

A second type of polythermal path traces temperature as reactants mix into the equilibrium system. This case differs from a sliding temperature path only in the manner in which temperature is determined. The modeler assigns a temperature To to the initial system, as before, and a distinct temperature Tr to the reactants. By assuming that the heat capacities CP, CPk, and CPr of the fluid, minerals, and reactants are constant over the temperature range of interest, we can calculate temperature (< ) from energy balance and the temperature T(c ) at the onset of the step according to... 

To invoke our geothermometer, we need to recombine the vapor and fluid phases and then heat the mixture to determine saturation indices as functions of temperature. We could do this in two steps, first titrating the vapor phase into the liquid and then picking up the results as the starting point for a polythermal path. 

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