Solids calculating concentrations

Knowing the experimental retention times, the previous equation allows the calculation of experimental concentration on the solid phase. Parameters of adsorption isotherms, can then be determined by fitting experimental and calculated concentrations. 

In Table 10.2, this correlation is shown, comparing solid phase concentration calculated from the retention times of the fronts, and using the adsorption isotherm equation. 

Table 10.2 Correlation experimental and calculated concentrations in solid phase...

Fig. 5. Concentration profiles of three species involved in a reaction of PADA (pyridine-2-azo-p-dimethylaniline) with nickel nitrite to form a complex within a micro-channel. Solid black lines, reactant Ni2+ concentration red points and solid lines, reactant PADA measured and calculated concentrations blue points and solid lines, product complex measured and calculated concentrations...

Fig. 7.5 Calculated concentration profiles for oxygen in Clark electrode consisting of (a) electrolyte (b) membrane and sample solution. The smooth solid curve is for bare Pt electrode of the same dimensions (adapted from Fatt, 1976)...

In addition to ice formation, salts also precipitate as these solutions are lofted to higher altitudes. A consequence of the formation of these solid phases (ice and salts) and the low-temperature eutectics of strong acids (Fig. 3.5) is that the atmospheric solutions become more and more acidic with altitude (Fig. 5.8). For example, the final elevation (temperature) examined is 11.54 km (—50 °C). At this point, the calculated concentrations of the Hubbard Brook solution are H+ = 7.55m with acid anions (Cl-, NO3, SO4-, HSOJ) = 7.91m. Similarly, for the Mt. Sonnblick solution, H+ = 6.50 m and acid anions = 6.90 m. These acidic trends are in line with stratospheric chemistries, which are predominantly sulfuric/nitric acid aerosols (Carslaw et al. 1997). For example, the total acid concentration at 20.7km in the stratosphere is 10.17m (calculated from fig. 7 in Carslaw et al. 1997), which is in line with our lower atmospheric concentrations. 

The model balance equation for each metal and ligand (e.g., Eqs. 2.49 and 2.52) is augmented to include formally the concentration of each possible solid phase. By choosing an appropriate linear combination of these equations, it is always possible to eliminate the concentrations of the solid phases from the set of equations to be solved numerically. Moreover, some of the free ionic concentrations of the metals and ligands also can be eliminated from the equations because of the constraints imposed by on their activities (combine Eqs. 3.2 and 3.3), which holds for each solid phase formed. The final set of nonlinear algebraic equations obtained from this elimination process will involve only independent free ionic concentrations, as well as conditional stability and solubility product constants. The numerical solution of these equations then proceeds much like the iteration scheme outlined in Section 2.4 for the case where only complexation reactions were considered, with the exception of an added requirement of self-consistency, that the calculated concentration of each solid formed be a positive number and that IAP not be greater than Kso (see Fig. 

For laminar flow, the characteristic time of the fluid phase Tf can be deflned as the ratio between a characteristic velocity Uf and a characteristic dimension L. For example, in the case of channel flows confined within two parallel plates, L can be taken equal to the distance between the plates, whereas Uf can be the friction velocity. Another common choice is to base this calculation on the viscous scale, by dividing the kinematic viscosity of the fluid phase by the friction velocity squared. For turbulent flow, Tf is usually assumed to be the Kolmogorov time scale in the fluid phase. The dusty-gas model can be applied only when the particle relaxation time tends to zero (i.e. Stp 1). Under these conditions, Eq. (5.105) yields fluid flow. This typically happens when particles are very small and/or the continuous phase is highly viscous and/or the disperse-to-primary-phase density ratio is very small. The dusty-gas model assumes that there is only one particle velocity field, which is identical to that of the fluid. With this approach, preferential accumulation and segregation effects are clearly not predicted since particles are transported as scalars in the continuous phase. If the system is very dilute (one-way coupling), the properties of the continuous phase (i.e. density and viscosity) are assumed to be equal to those of the fluid. If the solid-particle concentration starts to have an influence on the fluid phase (two-way coupling), a modified density and viscosity for the continuous phase are generally introduced in Eq. (4.92). 

Figure 6. Calculated concentration profile for the decay using the A mechanism. [A] is shown in solid circles, [B] is shown in open circles and [C] is shown in plus sign. Note that the decrease in concentration of species A, while concentrations of species B and C increase, albeit differently. In this experiment the reaction was followed for 120 min.

Fig. 52 Surface permeabilities determined from the calculated concentration profiles. The solid line (uy) and the dotted line (Uz) represent the analytical dependence of the surface permeabilities which yields excellent agreement with the measmed concentration integrals (standard deviation a = 0.006)...

FIGURE 2.3 Comparison of observed and calculated concentration-time relationships for caprolactam (x), polymer chains (i ), and amino caproic acid (5i) fVo = 0.82 mol/kg, T = 259°C. Solid lines represent experimental data broken lines calculated using set-II parameters dotted lines calcnilated using set I parameters. (From Reimschuessel, H.K. and Nagasubramanian, K., Chem. Eng. ScL, 1972, 27, 1119. With permission.)... 

The use of soluble and polymeric Mn02 allows for direct analysis of Mn02 as a reactant. In previous studies (21, 26-28), the solution had to be filtered to remove solid Mn02 in order to determine soluble Mn. Any solid Mn02 remaining in the reaction vessel was calculated by the difference between the initial solid phase concentration and the Mn. ... 

Fig. 5 Theoretical and experimental descriptions of the impact of uptake inhibition (a) and enhancement of release (b) of the responses recorded at microdisk electrodes. Theoretical curves Numerical solutions of Eq. (3) were used to generate predicted concentration profiles at various times during a simulated period of stimulation. The calculated concentration profiles shown in the main panel of the top and bottom portion of this figure were obtained at the end of the simulated stimulus. The top panel shows how an increase in the Michaelis constant (/fm) changes the concentration profile, while the bottom panel shows the effect of an increase in the magnitude the simulated stimulus (further details can be found in Ref [25]). Stimulation responses The inset panels show experimental stimulus responses recorded in the rat brain with microdisk electrodes. Open circles denote the beginning and end of the electrical stimulation. Predrug responses (solid lines) were recorded prior to systemic administration of either 20 mg kg nomifensine (a) or 250 mg kg L-DOPA (b). Postdrug responses (dotted lines) were recorded 25 min after nomifensine administration or 55 min after L-DOPA administration. Note that the trends in the amplitude of the experimental signals correspond very well to those apparent in the theoretical concentration profiles.

A sintered solid of silica 2.0 mm thick is porous with a void fraction e of 0.30 and a tortuosity t of 4.0. The pores are filled with water at 298 K. At one face the concentration of KCl is held at 0.10 g mol/liter, and fresh water flows rapidly by the other face. Neglecting any other resistances but that in the porous solid, calculate the diffusion of KCl at steady state. 

Another frequent mistake among students is to try to apply the ideal gas law to calculate the concentrations of species in condensed-matter phases (e.g., liquid or solid phases). Do not make this mistake] The ideal gas law only applies to gases. To calculate concentrations for liquid or solid species, information about the density (pj) of the liquid or solid phase is required. Both mass densities and molar densities (concentrations) as well as molar and atomic volumes may be of interest. The complexity of calculating these quantities tends to increase with the complexity of the material under consideration. In this section, we will consider three levels of increasing complexity pure materials, simple compounds or dilute solutions, and more complex materials involving mixtures of multiple phases/compounds. 

Calculating concentration of a species that is present at low levels (say, <1%) as a solute in a solid solution is also fairly straightforward. The typical approach is to assume that the density of the host material (p i) is not changed by the presence of the solute species. In this case, the mass density (/J ,m) molar concentration (C m) of the solute species i in the host material, M, can be calculated as... 

When calculating concentrations for liquid or solid species (which will be frequently encountered in materials kinetics problems), the ideal gas law DOES NOT APPLY Instead, information about the density (or atomic structure and packing) is needed. These calculations can become increasingly complicated depending on the number of phases/components involved. The last section of the chapter provides detailed examples of such calculations. Mastering these concepts will be extremely useful as we move forward in our exploration of materials kinetics, as most kinetic equations involve species concentration. 

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