Activity coefficients definitions

It is strictly for convenience that certain conventions have been adopted in the choice of a standard-state fugacity. These conventions, in turn, result from two important considerations (a) the necessity for an unambiguous thermodynamic treatment of noncondensable components in liquid solutions, and (b) the relation between activity coefficients given by the Gibbs-Duhem equation. The first of these considerations leads to a normalization for activity coefficients for nonoondensable components which is different from that used for condensable components, and the second leads to the definition and use of adjusted or pressure-independent activity coefficients. These considerations and their consequences are discussed in the following paragraphs. 

The difference on the left is the partial excess Gibbs energy G y the dimensionless mXio J on the right is called the activity coefficient of species i in solution, y. Thus, by definition. 

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,... 

For a substance in a given system the chemical potential gi has a definite value however, the standard potentials and activity coefficients have different values in these three equations. Therefore, the selection of a concentration scale in effect determines the standard state. 

The standard state given by the unsymmetric convention for normalization has one very important advantage it avoids all arbitrariness about/2°, which is an experimentally accessible quantity the definition off2° given by Eq. (37) assures that the activity coefficient of component 2 is unambiguously defined as well as unambiguously normalized. There is no fundamental arbitrariness about f2° because Hl2p(M) can be determined from experimental measurements. 

At constant temperature, the activity coefficient depends on both pressure and composition. One of the important goals of thermodynamic analysis is to consider separately the effect of each independent variable on the liquid-phase fugacity it is therefore desirable to define and use constant-pressure activity coefficients which at constant temperature are independent of pressure and depend only on composition. The definition of such activity coefficients follows directly from either of the exact thermodynamic relations... 

In the reference state the activity coefficients are, by definition, unity. The reference state may be that in the limit of infinite dilution, but the more conventional reference state is C° = 1 M. With the -y s = 1,... 

This is an equation for calculating the activity coefficient of an individual ion m (i.e., a parameter inaccessible to experimental determination). Let us, therefore, change to the values of mean ionic activity. By definition [see Eq. (3.27)],... 

Thermodynamics describes the behaviour of systems in terms of quantities and functions of state, but cannot express these quantities in terms of model concepts and assumptions on the structure of the system, inter-molecular forces, etc. This is also true of the activity coefficients thermodynamics defines these quantities and gives their dependence on the temperature, pressure and composition, but cannot interpret them from the point of view of intermolecular interactions. Every theoretical expression of the activity coefficients as a function of the composition of the solution is necessarily based on extrathermodynamic, mainly statistical concepts. This approach makes it possible to elaborate quantitatively the theory of individual activity coefficients. Their values are of paramount importance, for example, for operational definition of the pH and its potentiometric determination (Section 3.3.2), for potentiometric measurement with ion-selective electrodes (Section 6.3), in general for all the systems where liquid junctions appear (Section 2.5.3), etc. 

In view of the definition of the mean activity coefficient and of the electroneutrality condition, v+z+ = -v z, the limiting law also has the form... 

This notional definition of the pH scale can, however, not be used for practical measurements, as it contains the activity coefficients of the individual ions, y(H30+). 

In view of the term containing activity coefficients, the acidity function depends on the ionic type of the indicator. The definition of H0 is combined with the assumption that the ratio Yb/Ybh+ is constant for all indicators of the same charge type (in the present case the base is electroneutral hence the index 0 in //0). Thus, the acidity function does not depend on each individual indicator but on the series of indicators. 

It is very often necessary to characterize the redox properties of a given system with unknown activity coefficients in a state far from standard conditions. For this purpose, formal (solution with unit concentrations of all the species appearing in the Nernst equation its value depends on the overall composition of the solution. If the solution also contains additional species that do not appear in the Nernst equation (indifferent electrolyte, buffer components, etc.), their concentrations must be precisely specified in the formal potential data. The formal potential, denoted as E0, is best characterized by an expression in parentheses, giving both the half-cell reaction and the composition of the medium, for example E0,(Zn2+ + 2e = Zn, 10-3M H2S04). 

It has been emphasized repeatedly that the individual activity coefficients cannot be measured experimentally. However, these values are required for a number of purposes, e.g. for calibration of ion-selective electrodes. Thus, a conventional scale of ionic activities must be defined on the basis of suitably selected standards. In addition, this definition must be consistent with the definition of the conventional activity scale for the oxonium ion, i.e. the definition of the practical pH scale. Similarly, the individual scales for the various ions must be mutually consistent, i.e. they must satisfy the relationship between the experimentally measurable mean activity of the electrolyte and the defined activities of the cation and anion in view of Eq. (1.1.11). Thus, by using galvanic cells without transport, e.g. a sodium-ion-selective glass electrode and a Cl -selective electrode in a NaCl solution, a series of (NaCl) is obtained from which the individual ion activity aNa+ is determined on the basis of the Bates-Guggenheim convention for acr (page 37). Table 6.1 lists three such standard solutions, where pNa = -logflNa+, etc. 

The mathematical definition of bh+ is like that of Ka (now right-to left, see equation (5)) writing a for activities and / for molar activity coefficients, as is commonly done in strong acid work, equation (6) is obtained ... 

The second way, called the Bunnett-Olsen method,30 makes the less drastic assumption that log activity coefficient ratios such as those in equation (7) are linear functions of one another, rather than cancelling out. From the definition of H0 in equation (8) we can write equation (11), where Am refers to the primary aromatic amines used in the determination of Ho, and then any specific activity coefficient ratio, say for the weak base B, is assumed to be linear in this according to equation (12) ... 

Kinetic theory indicates that equation (32) should apply to this mechanism. Since the extent of protonation as well as the rate constant will vary with the acidity, the sum of protonated and unprotonated substrate concentrations, (Cs + Csh+), must be used. The observed reaction rate will be pseudo-first-order, rate constant k, since the acid medium is in vast excess compared to the substrate. The medium-independent rate constant is k(), and the activity coefficient of the transition state, /, has to be included to allow equation of concentrations and activities.145 We can use the antilogarithmic definition of h0 in equation (33) and the definition of Ksh+ in equation (34) ... 

The ratio of perceived to real concentrations is called the activity coefficient y (because, from Equation (7.25), y = a -=- c). Furthermore, from the definition of activity in Equation (7.20), y will have a value in the range zero to one. The diagram in Figure 7.9 shows the relationship between y and concentration c for a few ionic electrolytes. 

The problem at hand is the evaluation of the activity coefficient defined in Eq. (76). It will be assumed that only pairwise interactions between the defects need be considered at the low defect concentrations we have in mind. (The theory can be extended to include non-pairwise forces.23) Then the cluster function R(n) previously defined in Eq. (78) is the sum of all multiply connected diagrams, in which each bond represents an /-function, which can be drawn among the set of n vertices, the /-function being defined by Eqs. (66), (56), and (43). The Helmholtz free energy of interaction of two defects appearing in this definition can be written as... 

The previous derivation was made under the implicit assumption that the activity coefficients of A and B are both equal to unity. This assumption matches the definition of E° as a standard potential. There are two cases of practical interest, where these conditions are not fulfilled. One is when the activity coefficients differ from unity but do not depend on the relative amounts of A and B in the film. This type of situation may arise when the interactions between the reactants are weak but the presence of the supporting electrolyte decreases the activity coefficients of A and/or B, yA and yB, to below 1 while they remain constant over the entire voltammo-gram. The only change required is thus to replace the standard potential by the formal potential ... 

Which if we combine with equation (6) we can develop a definition of the activity coefficient of a single ion in a multi-ion solution as... 

For concentrated solutions, the activity coefficient of an electrolyte is conveniently defined as though it were a nonelectrolyte. This is a practical definition for the description of phase equilibria involving electrolytes. This new activity coefficient f. can be related to the mean ionic activity coefficient by equating expressions for the liquid-phase fugacity written in terms of each of the activity coefficients. For any 1-1 electrolyte, the relation is ... 

Although we cannot determine its absolute value, the chemical potential of acomponent of a solution has a value that is independent of the choice of concentration scale and standard state. The standard chemical potential, the activity, and the activity coefficient have values that do depend on the choice of concentration scale and standard state. To complete the definitions we have given, we must define the standard states we wish to use. 

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