Activity coefficient pure-component standard state

In Equation (16), it has been assumed that the pure component standard state is used for all components and that the activity coefficients of the solutes are normalized according to Equation (4). The right-hand side of Equation (13) becomes... 

In a generic pure-component standard state, the activity coefficient is expressed as... 

Since the value of an activity coefficient depends on the standard state, an activity coefficient based on (10.2.21) will differ numerically from one that is based on a pure-component standard state. To emphasize that difference, we make a notational distinction between the two we use y for an activity coefficient in a pure-component standard state and use y for an activity coefficient in the solute-free infinite-dilution standard state. Then for y, the generic definition of the activity coefficient (5.4.5) gives... 

Using the gamma-phi form (12.1.7) with the activity coefficient based on a pure-component standard state, we can determine the limiting behavior of K,. In the pure limit. 

Limiting behaviors. The gamma-phi form (12.1.15) is convenient for determining limiting behaviors of relative volatilities. In the following we use activity coefficients in a pure-component standard state. First consider the pure-1 limit of ai2 in a multi-component mixture, taking the limit with T fixed, VLB maintained (so P ), and... 

When a pure-component standard state is used for both activity coefficients, then the pure-component and infinite-dilution limits are straightforward. Taking the pure-component limit, with T held fixed, (12.1.24) becomes... 

Fluctuation Solution Theory (FST) At infinite dilution the solubility expression contains no hypothetical chemical potential of the solute [4, 45], For dilute solutions, the Henry s law standard state can be more reliable than the pure component standard state since the unsymmetric convention activity coefficients, designated by y., are often very close to unity, y is related to y. by... 

Here /g,hq and y ,ss are the activity coefficients of component B in the liquid and solid solutions at infinite dilution with pure solid and liquid taken as reference states. A fus A" is the standard molar entropy of fusion of component A at its fusion temperature Tfus A and AfusGg is the standard molar Gibbs energy of fusion of component B with the same crystal structure as component A at the melting temperature of component A. 

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. ... 

In a binary liquid solution containing one noncondensable and one condensable component, it is customary to refer to the first as the solute and to the second as the solvent. Equation (13) is used for the normalization of the solvent s activity coefficient but Equation (14) is used for the solute. Since the normalizations for the two components are not the same, they are said to follow the unsymmetric convention. The standard-state fugacity of the solvent is the fugacity of the pure liquid. The standard-state fugacity of the solute is Henry s constant. 

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... 

The most frequently used standard state is the pure liquid (x = 1) at the system temperature and pressure. When this standard state is used for all components in the mixture, the activity coefficients are said to be symmetrically normalized, because in this case, for every component /,... 

Jaques and Furter (37,38,39,40) devised a technique for treating systems consisting of two volatile components and a salt as special binaries rather than as ternary systems. In this pseudo binary technique the presence of the salt is recognized in adjustments made to the pure-component vapor pressures from which the liquid-phase activity coefficients of the two volatile components are calculated, rather than by inclusion of the salt presence in liquid composition data. In other words, alteration is made in the standard states on which the activity coefficients are based. In the special binary approach as applied to salt-saturated systems, for instance, each of the two components of the binary is considered to be one of the volatile components individually saturated with the... 

The thermodynamic reaction equilibrium constant K, is only a function of temperature. In Equation 4.18, m, the activity of the guest in the vapor phase, is equal to the fugacity of the pure component divided by that at the standard state, normally 1 atm. The fugacity of the pure vapor is a function of temperature and pressure, and may be determined through the use of a fugacity coefficient. The method also assumes that an, the activity of the hydrate, is essentially constant at a given temperature regardless of the other phases present. 

Christian et al base their measurements on the pure component as the standard state. They define the activity coefficient of the organic solute as... 

Tki = activity coefficient of group k in the standard state of pure component i... 

Here the standard state for the ionic species is a 1-molal ideal solution the enthalpies and Gibbs energies of formation for some ions in this standard state at 25 C are given in Table 13.1-4. In Eq. 13.1-27 the standard state for the undissodated molecule has also been chosen to be the ideal 1-molal solution (see Eq. 9.7-20), although the pure component state could have been used as well (with appropriate changes in AfG, b aA B ). Finally, we have used the mean molal activity coefficient, y , of Eq. 9.10-11. Also remember that for the 1-molal standard state, y ° 1 as the solution becomes veiy dilute in the component. 

Solubility of a Pure Component Strong Electrolyte. The calculation of the solubility of a pure component solid in solution requires that the mean ionic activity coefficient be known along with a thermodynamic solubility product (a solubility product based on activity). Thermodynamic solubility products can be calculated from standard state Gibbs free energy of formation data. If, for example, we wished to calculate the solubility of KCI in water at 25 °C,... 

For the i surface active components, infinite dilution (x 0) is experimentally better accessible than the pure state. It should be mentioned that setting the activity coefficients to 1 at infinite dilution is not necessarily consistent with setting the activity coefficient for pure components to unity. Therefore, for the case of infinite dilution of a multicomponent system, an additional normalisation of the potentials of the components should be performed [49]. This yields unity for the activity coefficient of pure components, while the activity coefficients at infinite dilution, in general should not be equal to 1. Indicating parameters at infinite dilution by the subscript (0), and those in the pure state by the superscript 0, the two standard potentials are interrelated by... 

It is seen that the additional (nonnalised) activity coefficients introduced in Eq. (2.10) to establish the consistency between the standard potentials of the pure components and those at infinite dilution, can be incorporated into the constant Kj in Eq. (2.15). Therefore, if a diluted solution with activity coefficients of unity is taken as the standard state, the form of Eqs. (2.13) and (2.14) remains unchanged. The equations (2.14) and (2.15) are the most general relationships from which meiny well-known isotherms for non-ionic surfactants can be obtained. For further derivation it is necessary to express the surface molar fractions, x-, in terms of their Gibbs adsorption values Tj. For this we introduce the degree of surface coverage, i.e. 9j = TjCOj or 0j = TjCO. Here to is the partial molar area averaged over all components or all... 

The first addend characterizes the electric potential of oxidizing-reducing reaction under standard conditions when all its components have activities equal to 1, and the second addend is equal 0. This means that the first addend determines voltage between atoms of redox-couples j represented under standard conditions by pure substances. For ions such pure substances are their water solutions with concentration 1 mole/kg (molarity), and for gas components, their gaseous state with partial pressure of 1 bar (100 kPa). At that, activities coefficients in them are considered equal to 1. The value of this first addend... 

When the standard states for the solid and liquid species correspond to the pure species at a pressure of 1 bar or at a low equilibrium vapor pressure of the condensed phase, the activities of the pure species at equilibrium are taken as unity at all moderate pressures. Consequently, the gas-phase composition at equilibrium will not be affected by the amount of solid or liquid present. At very high pressures, equation (2.8.1) must be used to calculate these activities. When solid or liquid solutions are present, the activities of the components of these solutions are no longer unity even at moderate pressures. In this case, to determine the equilibrium composition of the system, one needs data on the activity coefficients of the various species and the solution composition. 

Using the Lewis-Randall rule (5.4.11) for the standard state fugacity in (5.4.5), the resulting expression for the activity coefficient jj approaches unity as the mixture is made more nearly pure in component i ... 

Activity coefficients can display wide variations in response to changes in composition. For example, consider the three binaries that can be extracted from a ternary mixture of acetone, chloroform, and methanol. Figure 5.7 shows the composition dependence of activity coefficients in those three binary mixtures. Since all these Yi are in the Lewis-RandaU standard state, each Y satisfies the pure-component limit given in (5.4.12). But, depending on the kinds of molecules present, Yi may be greater than unity or less than unity for example, the acetone-chloroform mixtures have Y/ < L but the mixtures containing the alcohol have Yi > 1- Furthermore, the values of the Yi in... 

These activity coefficients are relative to the Lewis-RandaU standard state (5.1.5) hence, they must satisfy the pure-component limit given in (5.4.12). That is, Yi —> 1 as... 

The Margules expressions for activity coefficients are based on the Lewis-Randall standard state (5.1.5), and therefore they must obey the pure-component limit (5.4.12). In addition, as with Porter s equations, the parameters A and A2 are simply related to the activity coefficients at infinite dilution. In particular, when we apply the dilute-solution limit (5.4.13) to (5.6.12) and (5.6.13), we obtain... 

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