Raoult’s law for fugacity

A form of Raoult s law with fugacities in place of partial pressures is often more useful /a = xa/a> where is the fugacity of A in the gas phase of system 2 at the same T and p as the solution. If this relation is found to hold over a given composition range, we will say the solvent in this range obeys Raoult s law for fugacity. 

These two forms of Raoult s law are equivalent when the gas phases are ideal gas mixtures. When it is necessary to make a distinction between the two forms, this book will refer specifically to Raoult s law for partial pressure or Raoult s law for fugacity. 

Raoult s law for fugacity can be recast in terms of chemical potential. Section 9.2.7 showed that if substance i has transfer equilibrium between a liquid and a gas phase, its chemical potential fit is the same in both equilibrated phases. The chemical potential in the gas phase is given by fXi — M°(g) +. KT ln fi/p° (Eq. 9.3.12). Replacing by xtf according to Raoult s law, and rearranging, we obtain... 

Equation 9.4.5 is an expression for the chemical potential in the liquid phase when Raoult s law for fugacity is obeyed. By setting Xi equal to 1, we see that ji represents the chemical potential of pure liquid i at the temperature and pressure of the mixture. Because Eq. 9.4.5 is valid for any constituent whose fugacity obeys Eq. 9.4.3, it is equivalent to Raoult s law for fugacity for that constituent. 

Depending on the temperature, pressure, and identity of the constituents of a liquid mixture, Raoult s law for fugacity may hold for constituent i at all liquid compositions, or over only a limited composition range when xt is close to unity. 

An ideal liquid mixture is defined as a liquid mixture in which, at a given temperature and pressure, each constituent obeys Raoult s law for fugacity (Eq. 9.4.3 or 9.4.5) over the entire range of composition. Equation 9.4.3 applies only to a volatile constituent, whereas Eq. 9.4.5 applies regardless of whether the constituent is volatile. 

If the liquid phase happens to be an ideal hquid mixture, then by definition constituent i obeys Raoult s law for fugacity at all values of x,. In that case, / h,/ is equal to fi, the fugacity when the gas phase is equilibrated with pure liquid i at the same temperature and pressure as the liquid mixture. 

Now let us assume the liquid mixture is an ideal Uquid mixture of nonelectrol5Tes in which /La obeys Raoult s law for fugacity, /jla = ii + RrinxA- The partial derivative (9/iA/9xA)r,p then equals RT/xa, and Eq. 12.5.19 becomes... 

This a Raoult s law-type fugacity statement with the Poynt-ing factor included.) We write this equation twice, once for pure water and once for the water in the ocean water, and equate the fugacities, finding... 

The activity coefficients and the fugacity coefficients consequently account for the deviation from Raoult s law for ammonia, and for the deviations from Henry s law for the dissolved gases. The validity of a published gas solubility for a binary system in a multicomponent system requires therefore that Henry s law... 

In vapor-liquid equilibria, it is relatively easy to start the iteration because assumption of ideal behavior (Raoult s law) provides a reasonable zeroth approximation. By contrast, there is no obvious corresponding method to start the iteration calculation for liquid-liquid equilibria. Further, when two liquid phases are present, we must calculate for each component activity coefficients in two phases since these are often strongly nonlinear functions of compositions, liquid-liquid equilibrium calculations are highly sensitive to small changes in composition. In vapor-liquid equilibria at modest pressures, this sensitivity is lower because vapor-phase fugacity coefficients are usually close to unity and only weak functions of composition. For liquid-liquid equilibria, it is therefore more difficult to construct a numerical iteration procedure that converges both rapidly and consistently. 

At equilibrium, a component of a gas in contact with a liquid has identical fugacities in both the gas and liquid phase. For ideal solutions Raoult s law applies ... 

IUPAC suggests fi for the activity coefficient with a Raoult s law standard state. We will use instead so as not to confuse the activity coefficient with the fugacity. which is also represented by the symbol f-... 

Figure 6.14 Graph of vapor fugacity /against. v, for. Vjl-TO +. y2HC1. The various curves are as follows . vapor fugacity of H Ot , vapor fugacity of HC1 . total vapor fugacity (H2O + HC1). The dashed line gives the Raoult s law limiting values for the vapor fugacity of H20.

Figure 8.17 Vapor fugacity for component 2 in a liquid mixture. At temperature T, large positive deviations from Raoult s law occur. At a lower temperature, the vapor fugacity curve goes through a point of inflection (point c), which becomes a critical point known as the upper critical end point (UCEP). The temperature Tc at which this happens is known as the upper critical solution temperature (UCST). At temperatures less than Tc, the mixture separates into two phases with compositions given by points a and b. Component 1 would show similar behavior, with a point of inflection in the f against X2 curve at Tc, and a discontinuity at 7V...

The difference between Raoult s law [Equation (14.2)] and Henry s law lies in the proportionality constant relating the fugacity to the mole fraction. For Raoult s law, this constant is /, g, the fugacity of the vapor in equilibrium with the pure solute. Generally, however, for Henry s law. 

For the predominant component of a solution, i. e. the solvent, the 3tate of the pure liquid at the temperature of the systom and the pressure of 1 atm. is chosen as the standard state.. In so far as sufficiently diluted solutions are concerned (i. e. such solutions the composition of which differs but slightly from the pure solvent) the solvent can be considered to follow approximately Raoult s law valid for ideal solutions, according to which the fugacity / of the solvent in a solution can be expressed as the product of its molar fraction N-, ) and of the fugacity of the pure liquid substance f at the same temperature, thus ... 

Thus tlie fngacity coefficient of species i in an ideal solution is equal to the fugacity coefficient of pure species i in the same physical state as the solution and at the same T and P. Since Raoult s law is based on tlie assninptionof ideal-solution behavior for the hquid phase, the same systems tliat obey Raoult s law form ideal solutions. 

Attention may be < lled here to the restriction in connection with Raoult s law that the total (external) pressure, under which the system is in equilibrium, should remain constant as the composition is changed, at a given temperature. This has been implied in the foregoing treatment. At moderate pressures, however, the fugacity (or vapor pressure) is virtually independent of the external pressure ( 27m) hence, in many of the practical applications of Raoult s law given below, the restriction of constant total pressure is not emphasized, although for strict accuracy it should be understood. ... 

The foregoing conclusions are depicted graphically in Fig. 24, which shows the partial vapor pressure (or fugacity) curves of three types. Curve I is for an ideal system obeying Raoult s law over the whole concentration range for such solutions k in equation (36.3) is equal to / , and Henry s law and Raoult s law are identical. For a system exhibiting positive deviations, curve II may be taken as typical in the dilute range,... 

If the solution were ideal over the whole range of composition, k in equation (37.2) would, of course, be equal to the fugacity of the pure liquid solute at 1 atm. pressure, and Henry s law and Raoult s law would be identical ( 36a). However, although the behavior of a soluie in solution may deviate considerably from Raoult s law, it almost invariably satisfies Henry s law at high dilutions. Consequently, for the study of not too concentrated solutions, the standard state under consideration has some advantages over that in III. A. 

Raoult s law is the simplest quantitative expression for vapor-liquid equilibrium. This law is based on the vapor phase being an ideal gas and the liquid phase being an ideal solution. Therefore, the vapor-phase fugacity is given by Eq. (2). The effect of pressure on the liquid phase is very small, and since the vapor is ideal, the liquid fugacity may be written as in Eq. (4). Here, is the vapor pressure of component i at the temperature of the solution. 

For liquids that cannot be represented by equations of state, the liquid fugacities are expressed in terms of activity coefficients, discussed in Section 1.3.3. For ideal solutions. Equation 1.24 reduces to Raoult s law, presented in Section 1.3.2. 

The goal of a subctitical VLE calculation is to quantitatively predict or correlnte the various kinds of behavior illustrated by Fig. 1.5-1 or by its iso baric or multicomponent coimlerparta. The basis for the calculation is phase-equilibrium formulation of Section 1.2-5, whare liquid-phase fogaciries are elimienled in favor of liquid-phase activity coefficients, aed vapor-pbase fugaciries in favor of vapor-phase fugacity coefficients. Raoult s Law standard states are chosen (tor all components in the liquid phase hence, (fTf- = f > and Eq. (1.2-63) becomes... 

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