Sample Equilibrium Calculations

Bubble point temperatures and vapor compositions on a stage are calculated iteratively by assuming a temperature and computing the bubble point pressure. A solution is reached when the calculated pressure matches the specified value (110 kPa). The calculations below demonstrate the final iteration for the solvent feed stage Ng. 

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EXAMPLE 9.7 Sample exercise Calculating the equilibrium composition by approximation... 

EXAMPLE 12.8 Sample exercise Calculating the equilibrium constant for a... 

With this relationship for all samples was calculated from ninh This M is used for evaluating the reaction data. The ultracen rifuge (u.c measurements were carried out in a Spinco model E analytical ultracentrifuge, with 0.4% solutions in 90% formic acid containing 2.3 M KCl. By means of the sedimenta- ion diffusion equilibrium method of Scholte (13) we determine M, M and M. The buoyancy factor (1- vd = -0.086) necessary for tSe calculation of these molecular weights from ultracentrifugation data was measured by means of a PEER DMA/50 digital density meter. 

While these calculations provide information about the ultimate equilibrium conditions, redox reactions are often slow on human time scales, and sometimes even on geological time scales. Furthermore, the reactions in natural systems are complex and may be catalyzed or inhibited by the solids or trace constituents present. There is a dearth of information on the kinetics of redox reactions in such systems, but it is clear that many chemical species commonly found in environmental samples would not be present if equilibrium were attained. Furthermore, the conditions at equilibrium depend on the concentration of other species in the system, many of which are difficult or impossible to determine analytically. Morgan and Stone (1985) reviewed the kinetics of many environmentally important reactions and pointed out that determination of whether an equilibrium model is appropriate in a given situation depends on the relative time constants of the chemical reactions of interest and the physical processes governing the movement of material through the system. This point is discussed in some detail in Section 15.3.8. In the absence of detailed information with which to evaluate these time constants, chemical analysis for metals in each of their oxidation states, rather than equilibrium calculations, must be conducted to evaluate the current state of a system and the biological or geochemical importance of the metals it contains. 

Equilibrium calculations suggested that Hg complexation varies greatly among redox and pH levels typical of the regions of lakes sampled during this study. In an oxic lake, pore water, and groundwater, Hg complexation with organic matter most likely dominates. Under anoxic conditions in the hypolimnion and pore waters, Hg most likely forms soluble bisulfide and polysulfide complexes. 

Fig. 14. Temperature dependence of the perturbation function 8Q(P)/K(P) of the flow-equilibrium calculated from PDC-measurements for four typical weight average degrees of polymerization Pw of the injected polystyrene sample 3), as indicated...

Dynamic surface tension is the time trajectory of surface tension before equilibrium is reached. Dynamic surface tension tracks the changes during surface formation when surfactants are added. The bubble pressure method is the one most commonly used for the determination of dynamic surface tension. The details of this method are described in ASTM D3825-90 (2000) [ 19]. In this method a capillary tube is immersed in a sample liquid and a constant flow of gas is maintained through the tube forming bubbles in the sample liquids. The surface tension of the sample is calculated from the pressure difference inside and outside the bubble and the radius of the bubble. 

Several chemical reactions, including calcium carbonate and hydroxyapatite precipitation, have been studied to determine their relationship to observed water column and sediment phosphorus contents in hard water regions of New York State. Three separate techniques have been used to Identify reactions important in the distribution of phosphorus between the water column and sediments 1) sediment sample analysis employing a variety of selective extraction procedures 2) chemical equilibrium calculations to determine ion activity products for mineral phases involved in phosphorus transport and 3) seeded calcium carbonate crystallization measurements in the presence and absence of phosphate ion. 

Polysulfides can be generated via two major pathways. First, polysulfides can be formed by the oxidation of dissolved sulfide and sulfide minerals(l, 2). Second, they can be formed by the reaction of elemental sulfur with bisulfide ion(35). Polysulfide levels can be predicted for the second process as described in previous studies(, 36-38). Equilibrium calculations as described in a previous study(22) were performed for the polysulfide levels in these samples. The ratio of S(0) experimental to S(0) calculated for all samples from Great Sippewissett were 0.145 (4-8 cm), 0.137 (8-13 cm) and 0.128 (23-28 cm). Because these ratios are less than 1.0, these results indicate that polysulfides should form primarily from the reaction of bisulfide ion with elemental sulfur(5) rather than sulfide oxidation. This data set is... 

The downside to the power of Car-Parrinello calculations is that they are computationally costly. Typically, the largest systems that can be treated are of the order of 100 atoms, and the time scale of the simulations is of the order of picoseconds. The purposes of Car-Parrinello simulations generally fall under three main categories (i) simulations as a means of optimization to determine structural properties, (ii) direct simulations of processes occurring over short time scales, and (iii) simulations to sample equilibrium properties. 

In summary, equilibrium calculations based on pore-water composition indicate that Fe carbonate or phosphate are unlikely to form below the top few centimeters of sediment and that siderite formation is unlikely generally. Undersaturations by a factor of A, p/IAP —100 are found. The presence of solid-phase sulfide is evidence for the formation of Fe sulfides. However, pore waters are not always saturated with respect to the common Fe-sulfide minerals and under-or supersaturations by a factor of —10 are calculated. These deviations may be due to such problems as organic complexing or cumulative analytical and sampling errors, but the possibility that other phases are influencing Fe concentrations cannot be excluded on the basis of these data. 

Mobility spectra will exhibit protonated monomers for most polar or strongly polarizable molecules when the reactant ions are hydrated protons and vapor levels of analyte are more than 10 to 100 ppb, the detection limits for most such compounds. Mobility spectra may contain a proton-bound dimer when vapor levels are increased to 0.5 to 1 ppm, yet a proton-bound trimer or tetramer is never observed, even if vapor concentrations exceed those needed to form these higher cluster ions according to equilibrium calculations. In mobility spectrometers today, ions are formed and then drawn into purified air or gases excluding neutrals of sample. Thus, equilibrium does not exist in analytical mobility spectrometers, and ion passage through purified gas should be seen as a kinetic experiment. 

Proceeding in the same way, the values of for the other samples are calculated. Note from Table 15.1 that the value for is constant (within the limits of experimental error) even though the initial concentrations vary. Furthermore, Experiment 4 shows that equilibrium can be achieved beginning with N2O4 rather than with NO2. That is, equilibrium can be approached from either direction. FIGURE 15.5 shows how Experiments 3 and 4 result in the same equilibrium mixture even though the two experiments start with very different NO2 concentrations. 

Equation 17.9 is known as the Henderson-Hasselbalch equation. Biologists, biochemists, and others who work frequently with buffers often use this equation to calculate the pH of buffers. In doing equilibrium calculations, we have seen that we can normally neglect the amounts of the acid and base of the buffer that ionize. Therefore, we can usually use the initial concentrations of the acid and base components of the buffer directly in Equation 17.9, as seen in Sample Exercise 17.3. However, the assumption that the initial concentrations of the acid and base components in the buffer are equal to the equilibrium concentrations is just that an assumption. There maybe times when you will need to be more careful, as seen in Sample Exercise 17.4. 

The Sj 2 reaction, X + RY XR + Y", has been simulated with MC equilibrium calculations by Jorgensen and coworkers [81, 82]. The procedure used by these authors involves three steps i) the lowest energy reaction path is determined for the in vacuo system by using ab initio molecular orbital calculations ii) inter-molecular potential functions are obtained to describe the interactions between the substrate and a solvent molecule these potentials depend on the internal structure of the substrate iii) MC simulations are carried out to determine the free energy profile for the reaction in solution. This is a difficult computational task since importance sampling methods are required to explore all the values of the reaction coordinate. A similar technique was used by Madura and Jorgensen [83] in simulating the nucleophilic addition of hydroxide ion to formaldehyde in the gas phase and in aqueous solution. 

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