Composition and Temperature Dependence

Cullinan presented an extension of Cussler s cluster diffusion the-oiy. His method accurately accounts for composition and temperature dependence of diffusivity. It is novel in that it contains no adjustable constants, and it relates transport properties and solution thermodynamics. This equation has been tested for six very different mixtures by Rollins and Knaebel, and it was found to agree remarkably well with data for most conditions, considering the absence of adjustable parameters. In the dilute region (of either A or B), there are systematic errors probably caused by the breakdown of certain implicit assumptions (that nevertheless appear to be generally vahd at higher concentrations). 

In chemical laboratories, small flasks and beakers are used for liquid phase reactions. Here, a charge of reactants is added and brought to reaction temperature. The reaction may be held at this condition for a predetermined time before the product is discharged. This batch reactor is characterized by the varying extent of reaction and properties of the reaction mixture with time. In contrast to the flasks are large cylindrical tubes used in the petrochemical industry for the cracking of hydrocarbons. This process is continuous with reactants in the tubes and the products obtained from the exit. The extent of reaction and properties, such as composition and temperature, depend on the position along the tube and does not depend on the time. 

Section 5.1 shows how nonlinear regression analysis is used to model the temperature dependence of reaction rate constants. The functional form of the reaction rate was assumed e.g., St = kab for an irreversible, second-order reaction. The rate constant k was measured at several temperatures and was fit to an Arrhenius form, k = ko exp —Tact/T). This section expands the use of nonlinear regression to fit the compositional and temperature dependence of reaction rates. The general reaction is... 

S02 vapor pressure (pgn2) was measured by dynamic saturation and by a gas-sensing S02 electrode over solutions containing 0.5 to 2.0 M sodium citrate at pH 3.5 to 5 with up to 1 M NaHSOj, Na2S04, and NaCl. Pgo2 was measured at 25° to 168°C pH at 25° to 95°C. Both pH and the vapor pressure ratio Ps02/pH20 were independent of temperature. The composition and temperature dependence of the data are correlated by the semiempirical expressions ... 

Tang, I. N., and H. R. Munkelwitz, Composition and Temperature Dependence of the Deliquescence Properties of Hygroscopic Aerosols, Atmos. Environ., 27A, 467-473 (1993). 

Resistivities and conductivities are the numerical inverses of one another, unless both are listed, indicating two separate sources for the information. Both are highly temperature sensitive. Values listed here are nominally at room temperature. Consult Section 6.1 for further information on compositional and temperature dependences. 

A general formulation of the problem of solid-liquid phase equilibrium in quaternary systems was presented and required the evaluation of two thermodynamic quantities, By and Ty. Four methods for calculating Gy from experimental data were suggested. With these methods, reliable values of Gy for most compound semiconductors could be determined. The term Ty involves the deviation of the liquid solution from ideal behavior relative to that in the solid. This term is less important than the individual activity coefficients because of a partial cancellation of the composition and temperature dependence of the individual activity coefficients. The thermodynamic data base available for liquid mixtures is far more extensive than that for solid solutions. Future work aimed at measurement of solid-mixture properties would be helpful in identifying miscibility limits and their relation to LPE as a problem of constrained equilibrium. 

Simultaneous measurements of the rate of change, temperature and composition of the reacting fluid can be reliably carried out only in a reactor where gradients of temperature and/or composition of the fluid phase are absent or vanish in the limit of suitable operating conditions. The determination of specific quantities such as catalytic activity from observations on a reactor system where composition and temperature depend on position in the reactor requires that the distribution of reaction rate, temperature and compositions in the reactor are measured or obtained from a mathematical model, representing the interaction of chemical reaction, mass-transfer and heat-transfer in the reactor. The model and its underlying assumptions should be specified when specific rate parameters are obtained in this way. 

A further issue for review is the treatment of attractive interactions. The treatment here was limited to consideration of the second virial coefficient as in Eq. (4.46), and this implies the composition and temperature dependences exhibited in Eq. (4.49). Those composition and temperature dependences are certainly the leading factors, but a more general evaluahon of first-order perturbahon theory could result in subtle corrections to those dependences. Additionally, some implicit temperature and pressure dependence is implied by the variations of the pure liquid properties in those factors. Finally, the limitation of the hrst-order perturbation theory must also be borne in mind there are experimental cases where hrst-order perturbation theory appears to be unsatisfactory (Lefebvre et al, 1999). 

Schott B., Yuen D. A., and Braun A. (2002) The influences of composition and temperature dependent rheology in thermal-chemical convection on entrainment of the D"-layer. Phys. Earth Planet. Inter. 129, 43 - 65. 

The excess volume for liquid mixtures is usually small, and in accord with Eq. (4-249) the pressure dependence of is usually ignored. Thus, engineering efforts to model G center on representing its composition and temperature dependence. Eor binary systems at constant T, C becomes a function of just Xi, and the quantity most conveniently represented by an equation is G /xpc RT. The simplest procedure is to express this quantity as a power series in Xi ... 

S. Paul, J. B. Roy and P. K. Basu, Empirical expressions for the alloy composition and temperature dependence of the band gap and intrinsic carrier density in Ga Ini-xAs, J.Appl.Phys.69,827-829 (1991). 

Thus the results exhibit strong composition and temperature dependence. For an ideal solution... 

Higashi S (1982) Tobelite, a new ammonium dioctahedral mica. Mineral J 11 138-146 Hirschmaim M, Evans BV, Yang H (1994) Composition and temperature dependence of Fe-Mg ordering in cummingtonite-granerite as determined by X-ray diffraction. Am Mineral 79 862-877 Hoppe R (1979) Effective coordination numbers (ECON) and mean fictive ionic radii (MEFIR). Z Kristallogr 150 23-52... 

FIGURE 10.6 Deliquescence RH as a function of temperature for (NH SO (Reprinted from Atmos. Environ. 27A, Tang, I. N., and Munkelwitz, H. R., Composition and temperature dependence of the deliquescence properties of hygroscopic aerosols, 467-473. Copyright 1993, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington OXS 1GB, UK.)... 

HEN/RED] Henderson, C. M. B., Redfem, S. A. T., Smith, R. 1., Knight, K. S., Chamock, J. M., Composition and temperature dependence of cation ordering in Ni-Mg olivine solid solutions a time-of-flight neutron powder diffraction and EXAFS study. Am. Mineral, 86, (2001), 1170-1187. Cited on pages 239,438. 

Converter models require reliable kinetic expressions that account for the composition and temperature dependence of the reaction rates. The temperature dependence is essential for predicting light-off performance, whereas the consequences of the concentration dependence are more difficult to assess. 

It Is Interesting to note that. In some of the examples above, a kinetic approach Is appropriate which differs from the traditional strategy of defining composition and temperature dependence of reaction rates functions. Certain assemblies of complicated cellular reaction and regulation processes may be represented reasonably accurately under a variety of growth conditions In terms of timers, the Initiation points and durations of which may be dependent upon growth conditions. However, the above examples show that certain parameters associated with starting... 

Haley, J. C., T. P. Lodge, Y. Y. He, M. D. Ediger, E. D. von Meerwall, and J. Mijovic. 2003. Composition and temperature dependence of terminal and segmental dynamics in polyisoprene/poly(vinylethylene) blends. Macromolecules 36 6142-6151. 

In the case of complete data, this means VLE data, where P, T, x, y,- is given, also the deviation between the experimental and predicted activity coefficients or excess Gibbs energies can be used to fit the required binary parameters. Furthermore the parameters can be determined by a simultaneous fit to different properties to cover properly the composition and temperature dependence of the activity coefficients. For example, the deviation of the derived activity coefficients can be minimized together with the deviations of the activity coefficients at infinite dilution, excess enthalpies, and so on. Accurate activity coefficients at infinite dilution measured with sophisticated experimental techniques are of special importance, since they deliver the only reliable information about the real behavior in the dilute range [23], for example, at the top or the bottom of a distillation column. Excess enthalpies measured using flow calorimetry are important too, since they provide the most reliable information about the temperature dependence of the activity... 

The temperature dependence of the separation factor (see Eq. (5.18)) and of the azeotropic composition of binary systems depends on the type of azeotrope (pressure maximum, pressure minimum), the temperature dependence of the vapor pressures, and the composition and temperature dependence of the activity coefficients. These dependencies can be described with the help of the heats of vaporization and partial molar excess enthalpies following the Clausius-Clapeyron respectively the Gibbs-Helmholtz equation [38] (derivation see Appendix C, B9) ... 

---