Temperature activity coefficient

The solubilities of the scale-forming salts barium and strontium sulphates in aqueous solutions of sodium chloride have been reviewed by Raju and Atkinson (1988, 1989). Equations were proposed for the prediction of specific heat capacity, enthalpy and entropy of dissolution, etc., for all the species in the solubility equilibrium, and the major thermodynamic quantities and equilibrium constraints expressed as a function of temperature. Activity coefficients were calculated for given temperatures and NaCl concentrations and a computer program was used to predict the solubility of BaS04 up to 300 °C and SrS04 up to 125 °C. 

As indicated in the previous section, the ion interaction parameters i(K, OH") and 2(K" > OH") have been used, at 25 C, as the starting point for the calculation of other ion interaction parameters. The same strategy will also be used for other temperatures. Activity coefficient data for other temperatures have been provided by Li and Pitzer (1996) and are reproduced here in Table 2.5 but in the moles per kilogram scale rather than mole fraction. 

For such components, as the composition of the solution approaches that of the pure liquid, the fugacity becomes equal to the mole fraction multiplied by the standard-state fugacity. In this case,the standard-state fugacity for component i is the fugacity of pure liquid i at system temperature T. In many cases all the components in a liquid mixture are condensable and Equation (13) is therefore used for all components in this case, since all components are treated alike, the normalization of activity coefficients is said to follow the symmetric convention. ... 

The standard-state fugacity of any component must be evaluated at the same temperature as that of the solution, regardless of whether the symmetric or unsymmetric convention is used for activity-coefficient normalization. But what about the pressure At low pressures, the effect of pressure on the thermodynamic properties of condensed phases is negligible and under such con-... 

If we vary the composition of a liquid mixture over all possible composition values at constant temperature, the equilibrium pressure does not remain constant. Therefore, if integrated forms of the Gibbs-Duhem equation [Equation (16)] are used to correlate isothermal activity coefficient data, it is necessary that all activity coefficients be evaluated at the same pressure. Unfortunately, however, experimentally obtained isothermal activity coefficients are not all at the same pressure and therefore they must be corrected from the experimental total pressure P to the same (arbitrary) reference pressure designated P. This may be done by the rigorous thermodynamic relation at constant temperature and composition ... 

Equations (2) and (3) are physically meaningful only in the temperature range bounded by the triple-point temperature and the critical temperature. Nevertheless, it is often useful to extrapolate these equations either to lower or, more often, to higher temperatures. In this monograph we have extrapolated the function F [Equation (3)] to a reduced temperature of nearly 2. We do not recommend further extrapolation. For highly supercritical components it is better to use the unsymmetric normalization for activity coefficients as indicated in Chapter 2 and as discussed further in a later section of this chapter. 

In some cases, the temperature of the system may be larger than the critical temperature of one (or more) of the components, i.e., system temperature T may exceed T. . In that event, component i is a supercritical component, one that cannot exist as a pure liquid at temperature T. For this component, it is still possible to use symmetric normalization of the activity coefficient (y - 1 as x - 1) provided that some method of extrapolation is used to evaluate the standard-state fugacity which, in this case, is the fugacity of pure liquid i at system temperature T. For highly supercritical components (T Tj,.), such extrapolation is extremely arbitrary as a result, we have no assurance that when experimental data are reduced, the activity coefficient tends to obey the necessary boundary condition 1... 

Since we make the simplifying assumption that the partial molar volumes are functions only of temperature, we assume that, for our purposes, pressure has no effect on liquid-liquid equilibria. Therefore, in Equation (23), pressure is not a variable. The activity coefficients depend only on temperature and composition. As for vapor-liquid equilibria, the activity coefficients used here are given by the UNIQUAC equation. Equation (15). ... 

Evaluation of the activity coefficients, (or y for noncondensable components),is implemented by the FORTRAN subroutine GAMMA, which finds simultaneously the coefficients for all components. This subroutine references subroutine TAUS to obtain the binary parameters, at system temperature. 

The molar excess enthalpy h is related to the derivatives of the activity coefficients with respect to temperature according to... 

In typical situations, we do not have the necessary experimental data to find constants b... To obtain these constants, we need experimental vapor-liquid equilibria (i.e. activity coefficients) as a function of temperature. 

GIVEN TEMPERATURE T K) AND ESTIMATES OF PHASE COMPOSITIONS XR AND XE (USED WITHOUT CORRECTION TO EVALUATE ACTIVITY COEFFICIENTS GAR AND GAE), LILIK NORMALLY RETURNS ERR=0, BUT IF COMPONENT COMBINATIONS LACKING DATA ARE INVOLVED IT RETURNS ERR=l, AND IF A K IS OUT OF RANGE THEN ERR=2 key SHOULD BE 1 ON INITIAL CALL FOR A SYSTEM, 2 (OR 6)... 

GAMMA calculates activity coefficients for N components (N 20) at system temperature. For noncondensable components effective infinite-dilution activity coefficients are calculated. 

Assume that an aqueous solute adsorbs at the mercury-water interface according to the Langmuir equation x/xm = bc/( + be), where Xm is the maximum possible amount and x/x = 0.5 at C = 0.3Af. Neglecting activity coefficient effects, estimate the value of the mercury-solution interfacial tension when C is Q.IM. The limiting molecular area of the solute is 20 A per molecule. The temperature is 25°C. 

It is not necessary to limit the model to idealized sites Everett [5] has extended the treatment by incorporating surface activity coefficients as corrections to N and N2. The adsorption enthalpy can be calculated from the temperature dependence of the adsorption isotherm [6]. If the solution is taken to be ideal, then... 

The Debye-Htickel limiting law predicts a square-root dependence on the ionic strength/= MTLcz of the logarithm of the mean activity coefficient (log y ), tire heat of dilution (E /VI) and the excess volume it is considered to be an exact expression for the behaviour of an electrolyte at infinite dilution. Some experimental results for the activity coefficients and heats of dilution are shown in figure A2.3.11 for aqueous solutions of NaCl and ZnSO at 25°C the results are typical of the observations for 1-1 (e.g.NaCl) and 2-2 (e.g. ZnSO ) aqueous electrolyte solutions at this temperature. 

Concentration and purity can both be traced to the minimum work of separation, W, where Tq is the sink temperature Ai is the number of moles of a species present in the feed x = and y is an activity coefficient. The value of Wprovides a target that is easily calculated and approachable in... 

Journals for the pubHcation of VLE data are available as is a comprehensive tabulation of a2eotropic data (28) if the composition and temperature of the a2eotrope are known (at a given pressure), then such information may be used to calculate activity coefficients. At the a2eotropic point, by definition, y. = xc, from equation 6,... 

A.ctivity Coefficients. Activity coefficients in Hquid mixtures are directiy related to the molar excess Gibbs energy of mixing, AG, which is defined as the difference in the molar Gibbs energy of mixing between the real and ideal mixtures. It is typically an assumed function. Various functional forms of AG give rise to many of the different activity coefficient models found in the Hterature (1—3,18). Typically, the Hquid-phase activity coefficient is a function of temperature and composition expHcit pressure dependence is rarely included. 

The solubihty parameter, 5, is a function of temperature, but the difference 6 — 6) is only weaMy dependent on temperature. By convention, both 5 and IV are evaluated at 25°C and are treated as constants independent of both T and P. The activity coefficients given by equation 30 are therefore functions of Hquid composition and temperature, but not of pressure. 

Fiend s Constant. Henry s law for dilute concentrations of contaminants ia water is often appropriate for modeling vapor—Hquid equiHbrium (VLE) behavior (47). At very low concentrations, a chemical s Henry s constant is equal to the product of its activity coefficient and vapor pressure (3,10,48). Activity coefficient models can provide estimated values of infinite dilution activity coefficients for calculating Henry s constants as a function of temperature (35—39,49). 

Outlined below are the steps required for of a X T.E calciilation of vapor-phase composition and pressure, given the liquid-phase composition and temperature. A choice must be made of an equation of state. Only the Soave/Redlich/Kwong and Peng/Robinson equations, as represented by Eqs. (4-230) and (4-231), are considered here. These two equations usually give comparable results. A choice must also be made of a two-parameter correlating expression to represent the liquid-phase composition dependence of for each pq binaiy. The Wilson, NRTL (with a fixed), and UNIQUAC equations are of general applicabihty for binary systems, the Margules and van Laar equations may also be used. The equation selected depends on evidence of its suitability to the particular system treated. Reasonable estimates of the parameters in the equation must also be known at the temperature of interest. These parameters are directly related to infinite-dilution values of the activity coefficients for each pq binaiy. 

In systems that exhibit ideal liquid-phase behavior, the activity coefficients, Yi, are equal to unity and Eq. (13-124) simplifies to Raoult s law. For nonideal hquid-phase behavior, a system is said to show negative deviations from Raoult s law if Y < 1, and conversely, positive deviations from Raoult s law if Y > 1- In sufficiently nonide systems, the deviations may be so large the temperature-composition phase diagrams exhibit extrema, as own in each of the three parts of Fig. 13-57. At such maxima or minima, the equihbrium vapor and liqmd compositions are identical. Thus,... 

It follows that die separation of cadmium must be carried out in a distillation column, where zinc can be condensed at the lower temperamre of each stage, and cadmium is preferentially evaporated. Because of the fact that cadmium-zinc alloys show a positive departure from Raoult s law, the activity coefficient of cadmium increases in dilute solution as the temperature decreases in the upper levels of the still. The separation is thus more complete as the temperature decreases. 

The last term in Eq. (6-32) describes the temperature dependence of the molar concentration in water, this contributes only about —45 cal mol to E at room temperature. In a strong mineral acid solution, the temperature dependence of the activity coefficient term contributes about —90 cal mol . These are small quantities relative to the uncertainty in E s-... 

---