Standard state pure-component

Choosing the pure component standard state, we obtain the following expression for the interfacial tension by combination of Equations 1-4. 

The only information needed to predict the mixture surfactant concentration to attain a specified adsorption level is the pure component adsorption isotherms measured at the same experimental conditions as the mixture isotherms. These isotherms are needed to obtain the pure component standard states. 

The method of predicting the mixture adsorption isotherms is to first select the feed mole fractions of interest and to pick an adsorption level within Region II. The pure component standard states are determined from the total equilibrium concentration that occurs at that set level of adsorption for the pure surfactant component adsorption isotherms. The total equilibrium mixture concentration corresponding to the selected adsorption level is then calculated from Equation 8. This procedure is repeated at different levels of adsorption until enough points are collected to completely descibe the mixture adsorption isotherm curve. 

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... 

Develop an expression for the Henry s law constant as a function of the A parameter in the Marguies expression, the vapor pressure, and composition. Compare the hypothetical pure component fugacity based on the Henry s law standard state with that for the usual pure component standard state. 

The equilibrium constants (based on the pure component standard states at P = the reaction temperature) for the set of independent reactions... 

The equilibrium constants above can be calculated from heat/free energy of formation data based on the pure component standard states at A = 1 atm and the temperature of interest. These results are shown in Figure 1.13. As in Illustration 1.6, we make a table for the extent of reaction based on one mol of steam for the gaseous species. This gives the following set of coupled nonlinear... 

For the reference state, lets us choose a pure-component standard state the real (or hypothetical) pure substance at the temperature of the mixture and at some convenient... 

In the land of pure-component standard states, the Lewis-Randall rule (5.1.5) is but a district. The two differ in their standard-state pressures and phases. The Lewis-Randall standard-state pressure and phase are always those of the mixture, but in a generic pure-component standard state, the standard-state pressure and phase need not be the same as those of the mixture. In general, the choice for standard-state is dictated by the availability of a value for the pure-component fugacity either from a reduction of experimental data, or from a correlation, or from an estimate. We caution that other authors may make other distinctions, and some may make no distinction between the Lewis-Randall rule and the pure-component standard state. 

In 10.1 we present the basic thermodynamic relations that are used to start phase-equilibrium calculations we discuss vapor-liquid, liquid-liquid, and liquid-solid calculations. We have seen that the most interesting phase behavior occurs in nonideal solutions, but when we describe nonidealities using an ideal solution as a basis, we must select an appropriate standard state. Common options for standard states are discussed in 10.2 they include pure-component standard states and dilute-solution standard states. 

Fugacities Based on Pure-Component Standard States... 

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

For pure condensed phases, the Poynting factor in (10.2.8) is straightforward to compute, but unless P P, it is usually small enough to neglect. For example, for liquid water at 25°C and 10 bar, the error introduced by neglecting the Poynting factor is less than 1%. Therefore, for condensed phases at low pressures, we usually approximate the pure-component standard-state fugadty (10.2.7) by... 

Although pure-component standard states are the ones most commonly used, situations arise in which a pure-liquid fugacity is unknown or difficult to determine. These situations occur, for example, when the mixture temperature T is above the critical temperature of the pure component (the gas solubility problem) and when T is below the pure-component melting temperature (the solid solubility problem). In such cases, we seek alternatives to the pure-component standard state. One way is to exploit any data available for mixtures that contain only small amounts of the component however, we emphasize that this approach does not require the real mixture to be dilute in that component. We are merely seeking an alternative to pure-component data to use as a basis for defining an ideal solution. 

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... 

When using FFF 5, we choose a pure-component standard-state for all condensed components i (solid and liquid) whose pure critical temperatures are not much beyond the system temperature T say, T < 1.2T. We choose one of the Henry s law standard states for any component i that has %, < 0.01 and T > l.bT, . Exceptions to this might include liquid-solid equilibria or situations in which a PvTx equation of state has been directly fit near the conditions of interest. 

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... 

Gamma-phi. In the gamma-phi approach to gas solubilities, FFF 1 is always used for the vapor, so the issues center on appropriate expressions for the fugacities of components in the liquid phase. Let component 1 be the supercritical solute and 2 be the subcritical solvent. For liquid fugacities we often use FFF 5 and for the solvent we would likely choose a pure-component standard state with the standard-state pressure equal to the vapor pressure of the pure liquid (P2 = P2). Then FFF 5 becomes... 

Pure-component standard states. When we take the standard state to be based on a pure component, then / = /p j-e i the derivatives in (12.5.1) and (12.5.2) become... 

Since v/RT > 0, f° must always increase with an isothermal increase in pressure, no matter which pure-component standard state we use. 

Figure 12.20 Effect of temperafure on Henry s constants for several gases(l) in liquids(2). These are the same data as plotted in Figure 12.10, but here we plot 1 /T on the abscissa to emphasize that the temperature dependence of a dilute-solution standard-state fugacity differs from that for a pure-component standard state cf. this with Figure 12.19.

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... 

Note the difference between = (set operation) and = (list operation) as used in the last two comparisons. The first test ensures that the union of the component sets of Cj=i,n equals the component set of B, while the second test ensures that each component is repeated k e. 1,2, n) times in the ciunulated list. For pure component standard states k = 1, for geometric activity models like e.g. the Kohler expression (Bertrand, Acree Burchfield 1983), k = 2, and finally for a full mixture contribution k = n. 

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