Thermodynamically Nonideal Conditions

It was mentioned in Sec. 1.3 that the rate constant defined there is actually only constant for thermodynamically ideal systems, and that in general it may vary with composition. Also, the classical form of the mass action law gives for the reaction 

Now we know from thermodynamics that the concentration equilibrium constant is not the proper one in the sense that it can be a function of concentrations in addition to temperature, especially for liquids and for gases at high pressure. Thus, in thermodynamics, the proper variable of activity is introduced  

This leads to an equilibrium constant that is a function only of temperature 

How can this be extended into the kinetic equation so that it has a proper driving force  

A useful way to do this is to use the transition state theory of chemical reaction rates (e.g., see Glasstone, Laidler, and Eyring [55] also, for a current review, sec Laidler [56]). This is based on the hypothesis that all elementary reactions proceed through an activated complex  

---

With thermodynamically nonideal conditions (e.g., high pressures) partial pressures may have to be replaced by fugacities. When use is made of mole fractions, the corresponding rate coefficient has dimensions hr" kmol m". According to the ideal gas law ... 

Physical Equilibria and Solvent Selection. In order for two separate Hquid phases to exist in equiHbrium, there must be a considerable degree of thermodynamically nonideal behavior. If the Gibbs free energy, G, of a mixture of two solutions exceeds the energies of the initial solutions, mixing does not occur and the system remains in two phases. Eor the binary system containing only components A and B, the condition (22) for the formation of two phases is... 

Given the limited data base from which solubility correlations can be drawn, it is essential to measure the solubility directly for the system of interest during process development. Since process conditions often favor operation with high concentrations of solute, such systems are often thermodynamically nonideal. It is necessary to measure the solubility in the solvent system(s) of interest in order to optimize the yield and the purity. To accomplish the latter relies upon the ability to measure the solubility of the key impurities as well as the product of interest. This requires the availability of both the key impurities and product however, the impurities often are not available as isolated solids. In that case, the solubility of impurities must be deduced from the purity profile of mother liquors taken from crystallizations of the actual process stream. It is often simplest and always fastest to measure the solubility and carry out crystallizations in a single-solvent system. However, working in multiple-solvent systems increases the likelihood of improving the yield, the separation factor, and the prospects of observing more of the possible crystal forms that may exist. 

Apart from a few exceptions, polyelectrolytes such as proteins and nucleic acids need a definite pH range and the presence of counterions for stability in aqueous solution. Usually this condition is reaUzed by use of dilute salt or buffer solutions. In this case only weak interactions between macromolecules and low-molecular solutes occur, thus reflecting only small contributions from thermodynamic nonideality. 

The method proposed in this monograph has a firm thermodynamic basis. For vapo/-liquid equilibria, the method may be used at low or moderate pressures commonly encountered in separation operations since vapor-phase nonidealities are taken into account. For liquid-liquid equilibria the effect of pressure is usually not important unless the pressure is very large or unless conditions are near the vapor-liquid critical region. 

With respect to an enzyme, the rate of substrate-to-product conversion catalyzed by an enzyme under a given set of conditions, either measured by the amount of substance (e.g., micromoles) converted per unit time or by concentration change (e.g., millimolarity) per unit time. See Specific Activity Turnover Number. 2. Referring to the measure of a property of a biomolecule, pharmaceutical, procedure, eta, with respect to the response that substance or procedure produces. 3. See Optical Activity. 4. The amount of radioactive substance (or number of atoms) that disintegrates per unit time. See Specific Activity. 5. A unitless thermodynamic parameter which is used in place of concentration to correct for nonideality of gases or of solutions. The absolute activity of a substance B, symbolized by Ab, is related to the chemical potential of B (symbolized by /jlb) by the relationship yu,B = RTln Ab where R is the universal gas constant and Tis the absolute temperature. The ratio of the absolute activity of some substance B to some absolute activity for some reference state, A , is referred to as the relative activity (usually simply called activity ). The relative activity is symbolized by a and is defined by the relationship b = Ab/A = If... 

This paper shows that the conditions of thermodynamic equilibrium in a mix-tine of chemically reacting ideal gases always have a solution for the concentrations of the mixture components and that this solution is unique. The paper has acquired special significance in the last few years in connection with the intensive study of systems in which this uniqueness does not occur. Such anomalies may be related either to nonideal components, or to treatment of stationary states, rather than truly equilibrium ones, in which the system exchanges matter or energy with the surrounding medium. 

A relevant characteristic of the technology should be the ability to remove the water selectively and continuously in order to shift the chemical equilibrium to full conversion. Because the presence of a liquid water phase will lead to rapid deactivation of the solid catalyst, operating conditions for water-free organic liquid should be found. In addition, the thermodynamic behavior of the reaction mixture is nonideal, particularly with respect to the couple alcohol-water. 

Third, a serious need exists for a data base containing transport properties of complex fluids, analogous to thermodynamic data for nonideal molecular systems. Most measurements of viscosities, pressure drops, etc. have little value beyond the specific conditions of the experiment because of inadequate characterization at the microscopic level. In fact, for many polydisperse or multicomponent systems sufficient characterization is not presently possible. Hence, the effort probably should begin with model materials, akin to the measurement of viscometric functions [27] and diffusion coefficients [28] for polymers of precisely tailored molecular structure. Then correlations between the transport and thermodynamic properties and key microstructural parameters, e.g., size, shape, concentration, and characteristics of interactions, could be developed through enlightened dimensional analysis or asymptotic solutions. These data would facilitate systematic... 

Aqueous electrolytes and the equilibrium constants that define various reactions in low-temperature geochemistry are inexorably linked with the problem of activity coefficients, or the problem of nonideality for aqueous electrolyte solutions. Thermodynamic equilibrium constants, defined by an extrapolation to infinite dilution for the standard state condition (not the only standard state), require the use of activity coefficients. Unfortunately, there is neither a simple nor universal nonideality method that works for all electrolytes under all conditions. This section provides a brief overview of a major subject still undergoing research and development but for... 

Also, it is customary to refer all thermodynamic properties to chemical potentials of all species, whether in the pure state or in solution, to their values under standard conditions. In that case the equilibrium constant will be designated, as before, by fCx and the pressure in the above equations is set at P = bar. Finally, it is possible to specify compositions in terms of molarity c, or molality m, leading to the specification of Kc and Km or Kc and Km - The resulting analysis becomes somewhat involved and will not be taken up here interested readers should read Section 3.7 for a full scale analysis of the treatment of nonideal solutions. 

Conversely, the correct approach to formulate the diffusion of a single component in a zeolite membrane is to use the MaxweU-Stefan (M-S) framework for diffusion in a nonideal binary fluid mixture made up of species 1 and 2 where 1 and 2 stands for the gas and the zeohtic material, respectively. In the M-S theory it is recognized that to effect relative motions between the species 1 and 2 in a fluid mixture, a force must be exerted on each species. This driving force is the chemical potential gradient, determined at constant temperature and pressure conditions [68]. The M-S diffiisivity depends on coverage and fugacity, and, therefore, is referred to as the corrected diffiisivity because the coefficient is corrected by a thermodynamic correction factor, which can be determined from the sorption isotherm. 

The activity coefficients, 7 , depend on the concentration. For nonideal systems, it is necessary to develop a model and derive an equation for the activity coefficients in the adsorbed phase. The definition of these coefficients must satisfy three thermodynamic conditions ... 

Although the procedure described above is effective for controlling for most reaction materials, there are some exceptions. First, water-soluble solvents and substrates, particularly saccharides and glycols, continuously adsorb water without reaching an equilibrium condition. ° ° In addition, salts may react with substrates. Furthermore, certain salts exhibit either nonideal thermodynamic behavior or too slow a rate of equilibration. ... 

The enhancement factors here are very large, in fact, among the larger nonideal corrections encountered in chemical engineering thermodynamics. (Enhancement factors for other mixtures at cryogenic conditions of 10 and larger have been reported, i Note that at T = 35°C and P = 255.3 bar, the solubility of naphthalene is enhanced by a factor of more than 17 700 above its ideal value however, its total solubility is still small at less than 2 mol %. 

Comparison of the differences between pure and mixed compaient cases for the permealnlities of each component calculated in the standard fashion and in the thermodynamically normalized Esshkm (P") is revealing. The difference between the pure and mixed gas cases for the P columns of each con xment corresponds to the apparent total depression in flux resulting from both true dual-mode competition effects and nonideal gas-phase effects. The differences between the pure and mixed gas cases for the P cohmms are free of complications arising from nonideal gas-phase effects. Therefore, the differences b ween these columns are manifestations of the rather small competition effect due to dual-mode sorption under these conditions. 

The application of thermodynamics to electrochemical systems also helps us understand potentials at nonstandard conditions and gives us a relationship with the equilibrium constant and reaction quotient. However, we understand now that concentration is not necessarily the best unit to relate to the properties of a solution. Rather, activity of ions is a better unit to use. Using Debye-Hiickel theory, we have ways of calculating the activities of ions, so we can more precisely model the behavior of nonideal solutions. 

---