Activity pure condensed phases

With this definition, T is the numerical value of the activity for the substance under some pressure p. It is also the ratio of the fugacity of the pure condensed phase under pressure p to that of the phase under 1 bar pressure. 

If the gas phase activity of the host is controlled by the presence of a pure condensed phase, solid or liquid, the equilibrium between host and guest in a stoichiometric clathrate can be described in terms of the gas phase pressure of the guest. This is, in effect, a vapor pressure for the guest. At higher pressures the guest will condense to form clathrate, and at lower pressures the clathrate will decompose. Temperature variation of this pressure will follow the Clapeyron equation which, with the usual assumptions (ideal gas behavior of the vapor and negligible volume of the condensed phase), reduces to the Clausius-Clapeyron equation ... 

Because, at constant temperature, dGm = Vm dP and the molar volumes of condensed phases are very small, it is usually sufficiently accurate to take their molar free energy as pressure independent and the same as that at the 1.0-bar standard state. This is equivalent to setting the activity of pure, condensed phases equal to unity. (See Problem 9.) The activity of a condensed phase is also independent of just how much of the phase is present. As a result of these considerations, no variable describing the condensed phase appears in the equilibrium constant and the equilibrium is independent of just how much condensed phase is present. 

One should note that X(T) here involves the relative activities. When mole fractions are adopted as composition variables Eq. (3.7.6b) exhibits the special feature that whenever q indexes a pure condensed phase (denoted by s) the corresponding factor in the product reduces to the form [7s(T Piqs)a (T,P) ] - [rs(T,p,qs) ]", which will later be shown not to differ significantly from unity. Thus, the only terms that contribute substantially to (3.7.6b) are factors relating to species actually dissolved in solutions. The above scheme has the further advantage that only a single K(T) is invoked, regardless of what composition variable is selected, and that according to Eq. (3.7.6a) this quantity varies only with temperature. 

We shall now show that the products in (i) generally differ only slightly from unity, so that for all but the most accurate calculations, the contributions 7sa,q are customarily omitted from the right-hand side of Eqs. (3.7.6b) and (3.7.8b). This step is summarized by the generally cited statement that "the activity of all pure condensed phases is unity." The procedure discussed here also offers a mechanism to determine the activity of condensed phases, which quantity must be experimentally determined in accurate analyses of experimental data. 

The mathematical treatment of Sections 3.4—3.10 has been largely formal so far because only the activity or activity coefficients of pure condensed phases has been discussed up to this point In Eqs. (3.7.16) through (3.7.18) a procedure was set up for their determination by experiment. No method has yet been provided by which the activity coefficients or activities of species in solution can be determined experimentally. We now turn our attention to these matters. 

Activities and Equilibrium Constants for Pure Condensed Phases... 

By way of review we see from Eq. (3.7.8b) that the equilibrium constant involves products of factors of the following forms (i) the quantities rs(T,P,q ) = Ys T, P, q )a (T, P), which relate to pure condensed phases, (ii) terms involving the activity coefficients yj(T, P,qj), which correct for gross deviations from ideal properties of species making up homogeneous solutions, (iii) terms involving a (7, P), which, by (3.5.17) or (3.5.20), relate to the activities and activity coefficients of jiure j at pressures other than one atmosphere, and (iv) the usual products [j leq that involve concentration units and which constitute the equi-... 

As discussed earlier, the activity of a pure condensed phase is ordinarily close to unity, and the ratio of fugacities may usually be replaced by the ratio of vapor pressures. In these circumstances one finds the approximate relations... 

Phase equilibria between condensed phases, like melting and crystal polymorphic transitions, have no mass-dependent terms (no equilibrium constants) since the activity of pure condensed phases is unity, and hence the equilibrium thermodynamics is represented by the simple relationships ... 

Because both metal and metal oxide are pure condensed phases, and their activities are constant and defined to be unity, the oxygen activity at the phase boundary is fixed by the temperature according to the phase law or, expressed in a different way, its temperature dependence is given by that of the equilibrium constant. 

When the standard states for the solid and liquid species correspond to the pure species at 1 atm pressure 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... 

As we have already seen, the standard potentials are relative to standard reference conditions—i.e., one-molal solutions at 2) = 25 °C and /) = 1 bar, in equilibrium with pure metals or pure gases. Applying the Nernst relation to a redox equilibrium such as reaction 8.163 and assuming unitary activity for the condensed phases (i.e., pure metals), we have... 

As we did for equilibria between solute species, we can also define the boundaries between solute species and condensed phases. Assuming the condensed forms to be pure phases (i.e., assuming unitary activity), in the presence of metallic cerium we have... 

The vaporization of a pure liquid or the reverse process, the condensation of the liquid, provides an interesting test of this delayed equilibration hypothesis. Thus as a hydrogen-bonded molecule, which vibrates in the liquid, separates from the surface it frees itself from the potential energy restrictions which prevented rotation. However, in so far as the evaporating molecule has insufficient collisions with neighbors to equilibrate to the free rotational partition function, fgy of the gas, it will retain substantially the partition function, fb of the condensed phase even in the activated complex. Consequently for the condensation process the usual... 

For a pure substance in condensed phase (solid or liquid) their vapour pressure is taken to be related to their activities. When we measure equilibrium vapour pressure, the vapour and the solid (or liquid) are at equilibrium and hence the activities of the constituent in both phases are the same. Hence, the activity of the vapour phase would represent the activity of the solid or liquid. 

The activity of a condensed phase is defined as its fugacity relative to the fugacity at its standard state, i.e., pure constituent at 25 °C (298 °K) and 1 atmosphere pressure. 

Equation 1 relates the force fields describing the motions of the molecule in the condensed and in the gaseous phase with the activity ratio. These fields are different owing to the effect of the intermolecular forces which are operative in the condensed phase. The intermolecular forces are exclusively solute-solute forces in the pure state (where the ratio P /P reduces to the vapor pressure isotope effect, VPIE),... 

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