Ideal solution fugacities

When intermolecular forces are independent of composition, each fugacity deviates from its ideal-gas value by an amount that is also independent of composition. This means each ideal-solution fugacity coefficient does not depend on composition. 

Since the ideal-gas fugacity is linear in the mole fraction x,- while q) is independent of mole fraction, the ideal-solution fugacity must also be linear in x,-. We write that linearity in this form ... 

This shows that, although the ideal-solution fugacity coefficient is independent of composition, it does depend on the choice made for the standard state. Consequently, the ideal-solution fugacity coefficient is not the same as the standard-state fugacity coefficient unless we choose = P. That is, in general... 

To obtain expressions for the partial molar properties of ideal solutions, we first determine the chemical potential. Using the ideal-solution fugacity (5.1.6) in the integrated definition of fugacity (4.3.12) we find... 

In 5.1 we observed that every ideal-solution fugacity (5.1.2) is linear in its mole fradion. We now write (5.1.2) in a more explicit form. 

Then, substituting this into (5.4.7), we obtain the ideal-solution fugacity, which is that of a Lewis-Randall ideal solution. 

But while we can pick any straight line and use it to represent an ideal-solution fugacity, in practice we always choose a line that intersects or lies tangent to the curve for the real fugacity at the standard-state pressure. This means that we choose fi%T, P°, x° ) = fi(T, P°, x° ) at some composition x then, at that composition, the activity coefficient must be unity. At other mole fractions, the fugacity of the ideal solution is given by the equation for the straight line. 

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 ... 

A simple equation for the fugacity of a species in an ideal solution follows from equation 190. Written for the special case of species / in an ideal solution, equation 160 becomes equation 195 ... 

This equation, known as the Lewis-RandaH rule, appHes to each species in an ideal solution at all conditions of temperature, pressure, and composition. It shows that the fugacity of each species in an ideal solution is proportional to its mole fraction the proportionaUty constant is the fugacity of pure species i in the same physical state as the solution and at the same T and P. Ideal solution behavior is often approximated by solutions comprised of molecules similar in size and of the same chemical nature. 

Thus, the fugacity 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. 

The residual Gibbs energy and the fugacity coefficient are useful where experimental PVT data can be adequately correlated by equations of state. Indeed, if convenient treatment or all fluids by means of equations of state were possible, the thermodynamic-property relations already presented would suffice. However, liquid solutions are often more easily dealt with through properties that measure their deviations from ideal solution behavior, not from ideal gas behavior. Thus, the mathematical formahsm of excess properties is analogous to that of the residual properties. 

When Eq. (4-282) is applied to XT E for which the vapor phase is an ideal gas and the liquid phase is an ideal solution, it reduces to a veiy simple expression. For ideal gases, fugacity coefficients and are unity, and the right-hand side of Eq. (4-283) reduces to the Poynting factor. For the systems of interest here this factor is always veiy close to unity, and for practical purposes <1 = 1. For ideal solutions, the activity coefficients are also unity. Equation (4-282) therefore reduces to... 

Activity coefficients are equal to 1.0 for an ideal solution when the mole fraction is equal to the activity. The activity (a) of a component, i, at a specific temperature, pressure and composition is defined as the ratio of the fugacity of i at these conditions to the fugacity of i at the standard state [54]. 

In Chapter 6, we showed that the total vapor fugacity/above an ideal solution was linearly related to composition by... 

Activity ax is termed the rational activity and coefficient yx is the rational activity coefficient This activity is not directly given by the ratio of the fugacities, as it is for gases, but appears nonetheless to be the best means from a thermodynamic point of view for description of the behaviour of real solutions. The rational activity corresponds to the mole fraction for ideal solutions (hence the subscript x). Both ax and yx are dimensionless numbers. 

The fugacity of species B in an ideal solution of gases is given by the Lewis and Randall rule... 

Figure 14.1. Dependence of fugacities in the gas phase on the composition of the condensed phase for an ideal solution.

Fugacity may be defined as a substitute for pressure to explain the behaviour of real gas and activity may be defined as the substitute for the concentration to explain the behaviour of a non-ideal solution. 

Let us now consider two special cases. In the first case, we assume that the compound of interest forms an ideal solution or mixture with the solvent or the liquid mixture, respectively. In assuming this, we are asserting that the chemical enjoys the same set of intermolecular interactions and freedoms that it has when it was dissolved in a liquid of itself (reference state). This means that Yu is equal to 1, and, therefore, for any solution or mixture composition, the fugacity (or the partial pressure of the compound i above the liquid) is simply given by ... 

Equation (12.89) allows one to calculate the surface concentration of solute from the vapor fugacity/ It can be simplified more for the ideal solution where... 

Amagat s law of additive volumes holds for all pressures, which means that the ideal-solution law holds for the gaseous mixture, but not necessarily for the pure gases per se, and therefore the fugacity (/) is given by... 

The fugacity in Equation 2-39 is that of the component in the equilibrium mixture. However, fugacity of only the pure component is usually known. It is also necessary to know something about how the fugacity depends on the composition in order to relate the two, therefore, assumptions about the behavior of the reaction mixture must be made. The most common assumption is that the mixture behaves as an ideal solution. In this case, it is possible to relate the fugacity, f, at equilibrium to the fugacity of the pure component, f, at the same pressure and temperature by... 

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 ... 

The nature of imaginary (or ficticious or hypothetical) standard states is most easily explained by reference to Fig. 12.18. The two dashed lines shown both conform to ideal-solution behavior as prescribed by Eq. (12.45). The points labeled fl(LR) and f°(HL) are both fugacities of pure i, but only Jf(LR) is the fugacity of pure i as it actually exists at the given T and P. The other point P(HL) represents an imaginary state of pure i in which its imaginary properties are fixed at values other than those of the real fluid. Either choice of value for f° fixes the entire line which represents ff = xj°. 

The ideal solution is introduced to provide a model of solution behavi which we may compare actual solution behavior. Such a model is arbitrary, as an idealization it should be simple, and at the same time it should confi to actual solution behavior over some limited range of conditions. The defini of Eq. (12.45) ensures that the ideal solution exhibits simple behavior. Moreo the two standard-state fugacities chosen, f°(LR) and f° HL), ensure that models represent real-solution behavior at a limiting condition. 

Figure 12.18 is drawn for a species that shows positive deviations from ideality in the sense of the Lewis/ Randall rule. Negative deviations from ideality are also common, and in this case the /j-vs.-x, curve lies below the Lewis/Randall line. In Fig. 12.19 we show the composition dependence of the fugacity of acetone in two different binary solutions at 50°C. When the second component is methanol, acetone shows positive deviations from ideality. On the other hand, when the second component is chloroform, acetone shows negative deviations from ideality. The fugacity of pure acetone f—<— is of course the same regardless of the second component. However, Henry s constants, represented by the slopes of the two dotted lines, are very different for the two cases. 

Because of the complex functionality of the K-values, these calculations in general require iterative procedures suited only to computer solution. However, in the case of mixtures of light hydrocarbons, in which the molecular force fields are relatively weak and uncomplicated, we may assume as a reasonable approximation that both the liquid and the vapor phases are ideal solutions. By definition of the fugacity coefficient of a species in solution, =fffxtP. But by Eq. (11.61), f f = xj,. Therefore... 

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