Activity coefficient analytical expression

The situation for electrolyte solutions is more complex theory confimis the limiting expressions (originally from Debye-Htickel theory), but, because of the long-range interactions, the resulting equations are non-analytic rather than simple power series.) It is evident that electrolyte solutions are ideally dilute only at extremely low concentrations. Further details about these activity coefficients will be found in other articles. 

A graphical integration of the Gibbs-Duhem equation is not necessary if an analytical expression for the partial properties of mixing is known. Let us assume that we have a dilute solution that can be described using the activity coefficient at infinite dilution and the self-interaction coefficients introduced in eq. (3.64). 

The case of activity coefficients in solutions is easily but tediously implemented since well-constrained expressions exist, like those produced by the Debye-Hiickel theory for dilute solutions or the Pitzer expressions for concentrated solutions (brines). The interested reader may refer to Michard (1989) for a recent and still reasonably simple account. However simple to handle, activity coefficients introduce analytically cumbersome expressions incompatible with the size of a textbook. Real gas theory demands even more complicated developments. 

Equations 11 and 12 are not written for constant molality, and can not be easily used with the Gibbs-Duhem equation to obtain an analytical expression for the activity of water in the ternary solution. However, it is possible to propose a separate equation for the activity coefficient of water that is consistent with the proposed model of concentrated solutions. 

So far, we have expressed the number of ions per unit volume by concentration. Physical chemists prefer to use activity to characterize the behavior of solutes in solution. As a solution becomes more concentrated and as the ionic strength increases, ions behave as if there were fewer of them present than would be indicated by their analytical concentrations. The activity of a component is related to its concentration by a proportionality constant known as an activity coefficient . Considerations regarding ionic activity become particularly important, when fluids are quite concentrated. An extension of this treatment is the use of the activity product to... 

Determining solubility constants in aqueous solutions generally involves analytical work to determine concentrations [ ] or potentiometric measurements to obtain activities. The ratio of activity and concentration—i.e., the activity coefficient and its change with concentration— depends on the choice of the standard state. If pure water is chosen as a standard state, the activity coefficients approach unity only in dilute solutions. It is therefore necessary to express the so-called thermodynamic constants TK (48) in terms of activities. If, on the other hand, one chooses as reference an aqueous solution of comparatively high and constant ionic strength, the activity coefficients remain close to unity even at rather high concentrations of the reacting species. In this case, we may use stoichiometric constants K (48), expressed in molarities, M, and related to a particular ionic medium. 

One way of expressing the extent of the interaction between these analytes and the liquid phase is by activity coefficients. According to Raoult s law, the partial pressure of a solute like cyclohexane in a solvent like dinonylphthalate is given by... 

Analytical representation of the excess Gibbs energy of a system impll knowledge of the standard-state fugacities ft and of the frv. -xt relationshi Since an equation expressing /, as a function of x, cannot recognize a solubili limit, it implies an extrapolation of the /i-vs.-X[ curve from the solubility I to X) = 1, at which point /, = This provides a fictitious or hypothetical va for the fugadty of pure species 1 that serves to establish a Lewis/ Randall 1 for this species, as shown by Fig. 12.21. ft is also the basis for calculation of activity coefficient of species 1 ... 

WATEQ2 consists of a main program and 12 subroutines and is patterned similarly to WATEQF ( ). WATEQ2 (the main program) uses input data to set the bounds of all major arrays and calls most of the other procedures. INTABLE reads the thermodynamic data base and prints the thermodynamic data and other pertinent information, such as analytical expressions for effect of temperature on selected equilibrium constants. PREP reads the analytical data, converts concentrations to the required units, calculates temperature-dependent coefficients for the Debye-HKckel equation, and tests for charge balance of the input data. SET initializes values of individual species for the iterative mass action-mass balance calculations, and calculates the equilibrium constants as a function of the input temperature. MAJ EL calculates the activity coefficients and, on the first iteration only, does a partial speciation of the major anions, and performs mass action-mass balance calculations on Li, Cs, Rb, Ba, Sr and the major cations. TR EL performs these calculations on the minor cations, Mn, Cu, Zn, Cd, Pb, Ni, Ag, and As. SUMS performs the anion mass... 

Consolidation of (4-93) and (4-94) to solve for [SH2 ] yields a complex expression. Therefore the numerical values of analytical concentrations of salt Cbha initial base Cb along with the formation constant for the homoconjugate complex are inserted in (4-94), and [B] is obtained at a specified point in a titration. This value of [B] is substituted in (4-93) to obtain [SH2 ]. Fortunately, the activity coefficients drop out in this method oFcalculating hydrogen ion concentration. 

It is found empirically that the vapor pressure of component 2 nonideal solution may be specified by P2 = 2x2 — x )P2- (a) Determine its activity coefficient relative to the standard state, (b) Find an analytic expression for P in terms of x. (c) Plot out the partial pressures as a function of composition and note their shape. 

In the above equations denotes the vapor mole fraction of component i, Pf is the vapor pressure of the pure component i, Bu is the second virial coefficient of component i, dn = 2Bn — Bn — B22 and B 2 is the crossed second virial coefficient of the binary mixture. The vapor pressures, the virial coefficients of the pure components and the crossed second virial coefficients of the binary mixtures were taken from [32], The Wilson [38], NRTL [39] and the Van Ness-Abbott [40] equations were used for the activity coefficients in Eq. (17). The expressions for the activity coefficients provided by these three methods were differentiated analytically and the obtained derivatives were used to calculate D = 1 -I- Xj(9 In Yil 2Ci)pj. There is good agreement between the values of D obtained with the three expressions for the systems V,V-dimethylformamide-methanol and methanol-water. For the system V,V-dimethylformamide-water, the D values calculated with the Van Ness-Abbott equation [40] were found in good agreement with those obtained with the NRTL equation, but the agreement with the Wilson expression was less satisfactory. 

The analytical expressions obtained using the two-suffix Margules equationsfor the activity coefficients and eq 20 for the molar volume are given in Appendix 2. 

In this paper, the Kirkwood—Buff formalism was used to relate the Henry s constant for a binary solvent mixture to the binary data and the composition of the solvent. A general equation describing the above dependence was obtained, which can be solved (analytically or numerically) if the composition dependence of the molar volume and the activity coefficients in the gas-free mixed solvent are known. A simple expression was obtained when the mixture of solvents was considered to be ideal. In this case, the Henr/s constant for a binary solvent mixture could be expressed in terms of the Henry s constants for the individual solvents and the molar volumes of the individual solvents. The agreement with experiment for aqueous solvents is better than that provided by any other expression available, including an empirical one involving three adjustable parameters. Even though the aqueous solvents considered are nonideal, their degrees of nonideality are much lower than those of the solute gas in each of the constituent solvents. For this reason, the assumption that the binary solvent behaves as an ideal mixture constitutes a reasonable approximation. 

The activity coefficients are affected from differences in size, the short-range lateral interactions among the adsorbed particles and in the case of adsorbed ions from the repulsive Coulombic interactions among these ions. Analytical expressions for the activity coefficients may be obtained either from statistical mechanical models or experiment. Thus, an approximate expression, arising from monolayer models under mean field approximation, is the following [12,13,21] ... 

If the temperature dependence of the activity coefficients can be expressed analytically. Equation 1.40 may be used directly to calculate the excess enthalpy. 

Once the analytical expression for AG ix is known, the calculation of chemical potentials and other thermodynamic functions (activities, activity coefficients, virial coefficients, etc.) is straightforward. For polymer solutions, we must apply Eq. (3.8) to Floiy-Huggins equation (3.50), keeping in mind that volume fractions i are functions of the number of moles, as given by... 

Ef is the Fermi energy, Ec and Ev are the energies of the conduction and valence band edges, respectively, and F1/2(rj) is the Fermi integral of order one-half for the argument 17. The activity coefficients approach values of unity at dilute carrier concentration because the value of Fi/2(t7) approaches exp(Tj) at dilute carrier concentrations. The concentration dependency of Eqs. (5a)-(5d) can be obtained explicitly through analytic expressions relating exp(r7) to F M -50... 

Best practice is to assess one single correlation, i.e., between the free excess enthalpy of the whole system g /(R-T) and the state variables, and then to derive the activity coefficients for each component consistently by means of (2.1-132). Thus the data drawn in Fig. 2.1-26 have to be approximated by an analytical function, which is differentiable with respect to the concentration and which complies with the boundary condition g / R T) = 0 for each of the pure substances. The literature provides a huge variety of such expressions, e.g., Margules (Margules 1895), van Laar (van Laar 1935), Wilson (Wilson 1964), NRTL (Renon and Praus-nitz 1968), or UNIQUAC (Abrams and Prausnitz 1975). The procedure is elucidated following the method after Margules for a binary mixture ... 

Equation 10.11 can be, however, integrated using any of the analytical expressions available for the activity coefficient In 73 , such as the Van Laar, Margules, Wilson, NRTL, and so forth. To take into account the nonideality of the molar volume, one can use the expression,... 

As discussed in Section 5.1, besides the vapor pressure of the pure compounds an activity coefficient model is required, which allows the calculation of the VLE behavior using only binary experimental data. Using Eq. (4.85) an analytical expression for the activity coefficients can be derived if an expression for the excess Gibbs energy is available. By definition, the expression for the excess Gibbs energy... 

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