Standard state partial molar volume

Since Eqs. (5) and (6) are not restricted to the vapor phase, they can, in principle, be used to calculate fugacities of components in the liquid phase as well. Such calculations can be performed provided we assume the validity of an equation of state for a density range starting at zero density and terminating at the liquid density of interest. That is, if we have a pressure-explicit equation of state which holds for mixtures in both vapor and liquid phases, then we can use Eq. (6) to solve completely the equations of equilibrium without explicitly resorting to the auxiliary-functions activity, standard-state fugacity, and partial molar volume. Such a procedure was discussed many years ago by van der Waals and, more recently, it has been reduced to practice by Benedict and co-workers (B4). 

As seen from Eq. (130) an activity coefficient may deviate significantly from unity at higher salt concentrations. The activity coefficient can therefore also be used as a measure of the deviation of the salt solution from a thermodynamically ideal solution. If the chemical potential of a solute in a (pressure-dependent) standard state of infinite dilution is /x°, we find the standard partial molar volume from... 

Because for a pure phase partial molar volume is equivalent to molar volume Vg, and since we have defined as standard state the condition of the pure component at P = 1 bar and T of interest ... 

The term /x,(0, T) = /x,(0, T,Nk)) plays the role of a standard potential with respect to the stressed state. For practical applications of Eqn. (14.12) one assumes that Vh the partial molar volume of component 1, does not noticeably depend on concentration nor on stress. In the case of hydrostatic pressure, cr Ff is then jcr-dlf = 3 PdV,= -3.Plf. In equilibrium, it follows from Eqn. (14.13) that... 

In equations (18.91) and (18.92), C° 2 and V are the partial molar heat capacity and partial molar volume of the surfactant in the infinitely dilute solution (standard state values). 

Partial molar volumes and the isothermal compressibility can be calculated from an equation of state. Unfortunately, these equations require properties of the components, such as critical temperature, critical pressure and the acentric factor. These properties are not known for the benzophenone triplet and the transition state. However, they can be estimated very roughly using standard techniques such as Joback s modification of Lyderson s method for Tc and Pc and the standard method for the acentric factor (Reid et al., 1987). We calculated the values for the benzophenone triplet assuming a structure similar to ground state benzophenone. The transition state was considered to be a benzophenone/isopropanol complex. The values used are shown in Table 1. 

In Equations (10.43) and (10.44) Vf represents the partial molar volume of the component in the infinitely dilute solution, which is also the partial molar volume of the component in the standard state. The right-hand side of Equation (10.44) contains only quantities that can be determined experimentally, and thus A/j. [T, P, x] can be determined. However, just as in the previous case, the pressure is a function of the mole fraction. Therefore, if we require values of A/tf at some arbitrary constant pressure, the correction expressed in Equation (10.34) must be made with the substitution of Vf for... 

In order to evaluate each of the derivatives, such quantities as (V" — V-), (S l — Sj), and (dfi t/x t)T P need to be evaluated. The difference in the partial molar volumes of a component between the two phases presents no problem the dependence of the molar volume of a phase on the mole fraction must be known from experiment or from an equation of state for a gas phase. In order to determine the difference in the partial molar entropies, not only must the dependence of the molar entropy of a phase on the mole fraction be known, but also the difference in the molar entropy of the component in the two standard states must be known or calculable. If the two standard states are the same, there is no problem. If the two standard states are the pure component in the two phases at the temperature and pressure at which the derivative is to be evaluated, the difference can be calculated by methods similar to that discussed in Sections 10.10 and 10.12. In the case of vapor-liquid equilibria in which the reference state of a solute is taken as the infinitely dilute solution, the difference between the molar entropy of the solute in its two standard states may be determined from the temperature dependence of the Henry s law constant. Finally, the expression used for fii in evaluating (dx Jdx l)TtP must be appropriate for the particular phase of interest. This phase is dictated by the particular choice of the mole fraction variables. 

By applying equation 49 to standard-state partial molar volumes of species in a reaction, we obtain the basic expressions for describing the effect of change in pressure on the equilibrium constant of the reaction. 

Both these considerations would be taken into accoimt if the activation process were assumed to occur at a constant pressure, p, such that the partial molar volume of the solvent is independent of the temperature, though this possibility does not appear to have been considered. A full discussion is beyond the scope of this chapter, but the resulting heat capacities of activation are unlikely to differ greatly from those determined at a constant pressme of, say, 1 atm. (see p. 137). Unfortunately, this approach requires the definition of rather clumsy standard states for solutes, e.g., hypothetically ideal, 1 molal, under a pressure such that a given mass of the pure solvent occupies a particular volume. 

The main reason that the only really useful choice is to have 7 equal to 1.0 in the standard state is that values of 7// in real solutions approach 1.0 in very dilute solutions, so that many properties of ideal solutions can be estimated by measuring them as a function of concentration, and then extrapolating to zero concentration to get their value in an ideal solution, i.e. where yn = 10. These are then called standard state values of partial molar volume, or partial molar enthalpy, etc. There is no way of deriving properties of solutions with other values of 7f/. The choice of m as 1.0 does not have the same degree of necessity. Any value could be used, adding a constant factor to all attendant values of solute activities and (/u /i°) values. The very dilute or infinitely dilute solutions themselves (m very small or zero) would make rather poor standard states, because although values would be 1.0, consideration of... 

Since the pressure dependence of this unusual standard-state chemical potential of species i at constant temperature and composition yields the partial molar volume of species i, Wi, one obtains the final result for the pressure dependence of kinetic rate constants ... 

One advantage offered by (5.5.11) is that it collects all the pressure effects into a single term. As we shall see in 5.6, many models for contain no pressure dependence hence, those models provide no pressure dependence for the activity coefficient, and such models are strictly valid only at the standard-state pressure P°. To include pressure in those models, we could use (5.5.11), if we have a reliable estimate for the partial molar volume—say, from a PvTx equation of state. 

Note that in FFF 5 neither the activity coefficient nor the standard-state fugacity depends on the mixture pressure. The Po)mting factor in FFF 5 can be computed, provided we can evaluate the partial molar volume for the real substance along the isotherm T from Pj to P. In contrast, the Po)mtmg factor appearing in FFF 3 applies to component i in its standard state and involves an integral over the partial molar volume of that standard-state substance. 

The advantage to this choice is that the activity coefficient is now to be evaluated at the standard-state pressure (P ), but the disadvantage is that the Poynting factor requires an integration over the partial molar volume for i in the reaction mixture. 

Extfapolating values of back to m = 0 will therefore provide a value of V, the partial molar volume at infinite dilution, which is the standard state value. 

Because partial molar volume, enthalpy, and heat capacity are the same anywhere on the Henry s law tangent, including both the state of infinite dilution and the ideal one molal solution, either of these states can serve as the standard state for these properties. We have chosen to say that the infinitely dilute solution is the standard state, but many treatments prefer to say that the standard state for these properties, as well as for the Gibbs energy and entropy, is the ideal one molal solution. For some reason, these treatments (e.g., Klotz, 1964, p. 361) then define the reference state for enthalpy, volume and heat capacity... 

In solutions, particularly electrolyte solutions, the standard state for the solvent is always the pure phase (pure water), so that, for example, refers to the molar volume of pure component 1, that is, pure water. For the solute, the standard state for most properties is, as just mentioned, the state of infinite dilution, so we could use for tho partial molar volume of the solute in the standard state. However, this proves a bit confusing, so for clarity we introduce superscript °° to indicate the infinite dilution state (1 ), and we understand that this is also the standard state for most properties. This raises the question of what symbol to use for the solute in its pure state. The lUPAC recommends the use of for pure substances, but our examples involve only minerals so we will just use the mineral name. Thus we use for the molar volume of pure NaCl. 

Because it applies mostly to electrolytes, it is discussed in Chapter 15. Briefly, Helgeson models the behavior of solutes by developing equations for the standard state partial molar volume (Helgeson and Kirkham 1976) and standard state partial molar heat capacity (Helgeson et al. 1981) as a function of P and T, with adjustable constants such that they can be applied to a wide variety of solutes. If you know these quantities (V°, C°p), you can calculate the variation of the standard state Gibbs energy, and that leads through fundamental relationships to equilibrium constants, enthalpies, and entropies. 

In the unsymmetric standard state convention, where the solvent is referenced to its pure state (Raoult s law reference state) and the solutes to the infinite dilution state (Henry s law reference state), the standard partial molar volume is equal to the pressure derivative of the chemical potential at... 

The standard state partial molar volume is the value of V2 at infinite dilution, 1 2°, and it is obtained by extrapolation of (j)y at infinite dilution. 2° represents the solute-solvent interaction and is related to the salt-solvent direct correlation function, Ci2(r), through the relationship (Brelvi and O Connell, 1971) ... 

Sedlbauer and Wood (2004) used the MSA model to describe the tiiermodynamics properties of NaCl near the critical point. In this case the crystallographic diameters of the ions were used along with a model (Sedlbauer et al, 2000) for the standard state term. The MSA model without adjustable parameters provides a better fit of the partial molar volume than the Pitzer model. 

The standard state partial molar volume or partial molar at infinite dilution of electrolytes, 3°, reflects the solute-solvent interaction and it is additive that is, it is the sum of the anion and cation contributions ... 

Table 2.3 Test of the different equations for the standard partial molar volume of aqueous ions (Reproduced from Chemical Geology, A new equation of state for correlation and prediction of standard molal thermodynamic properties of aqueous species at high temperatures and pressures with permission from Elsevier)...

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