Activity coefficients Raoultian

It can be immediately seen that for components exhibiting Raoultian behaviour, the activity coefficient is equal to unity. The Henry s law constant k is nothing but the activity coefficient Yj. Noting that Henrian behaviour is exhibited when the component i is present in very low concentrations, the constant is also expressed in this case as y and is known as the activity coefficient at infinite dilution. Henry s law may now be stated as... 

The activity of silicon in a binary Fe-Si liquid alloy containing Nsl = 0.02 is 0.000022 at 1,600°C relative to the Raoultian standard state. The Henrian activity coefficient y% is experimentally determined to be 0.0011. 

An activity coefficient of a constituent in a system is a number (always dimensionless) which when multiplied by the ideal activity (the concentration) gives the real activity. To illustrate this let s examine a system of the type in Figure 11.9 in more detail (Figure 11.10). In this binary system, B shows positive deviation from Raoultian behavior, so the Raoultian activity coefficient will be greater than one. Solutes that obey Raoult s Law have a, = X, so the Raoultian activity coefficient 7h is defined as... 

This leads to the easiest approach to understanding activities. The activity of a constituent is the ratio of the fugacity of that constituent to its fugacity in some other state, which we called a reference state. We then showed through consideration of the Lewis Fugacity Rule, which is an extension of Dalton s Law, that for ideal solutions of condensed phases, the activity of a constituent equals its mole fraction, if the reference state is the pure constituent at the same P and T. Deviations from ideal behaviour are then conveniently handled by introducing Henryan and Raoultian activity coefficients. 

Equation (12.10) can also be used for non-ideal gases as well as for liquid and solid solutions by introducing the Raoultian activity coefficient, thus... 

For solutions covering a wide range of compositions, such as many solid and liquid solutions, this equation can be used by introducing another correction factor, the Raoultian activity coefficient, Thus... 

But we do have a choice as to what kind of activity coefficient we want to use, Raoultian or Henryan. You might think that if the standard state consists of pure i, normally a pure solid or liquid, it might be difficult to use a dilute solution (Henryan) standard state. However, using hypothetical states makes it quite simple, and quite instmctive. But first we consider the Raoultian standard state. 

To illustrate the Raoultian activity-activity coefficient relationship we use activity coefficients defined by Equations (8.31), shown in Figure 8.2. In real systems these are measured quantities with associated uncertainties, and the shape of the activity curve may not fit any simple function. Figure 8.2 shows... 

In Figure 8.4, where the activities and activity coefficients of B are the previously defined Raoultian values, all divided by 2.2407. The data are shown in Table 8.2. 

Table 8.2 Raoultian and Henry an activities and activity coefficients for a regular solution having Wq = 2000Jmol. Raoultian data are from Table 10.2.

An interesting application of regular solution theory is presented by Nesbitt (1984). He shows that activity coefficients for COj in aqueous NaCl solutions to quite high temperatures ( 500°C) and NaCl concentrations ( 6 m) can be fit very well by a slight modification of (10.98). As written, the activity coefficients in (10.98) are based on Raoultian activities that is, 7b 1 as Xg 1. Solubility studies on the other hand normally use Henryan coefficients, where 7b 1 as Wb 0, where is the molality of the solute. 

The rest of this chapter is in two parts. First, we consider how to calculate mole fractions in solid solutions assumed to have only long range order, and how to combine these mole fractions into Raoultian activities. Then we consider the determination of activity coefficients in (binary) solid solutions, and how regular solution and Margules equations are used to systematize these. 

A non-ideal (or actual) solution is one for which Eq.(6.22) does not hold good for at least one component. A correction factor > known as the Raoultian activity coefficient of component i, is introduced to Eq.(6.22) so that it may also be applied to non-ideal solutions. Thus... 

Vanadium melts at 1720°C (1993 K). The Raoultian activity coefficient of vanadium at infinite dilution in liquid iron at 1620°C (1893 K) is 0.068. Calculate the free energy change accompanying the transfer of the standard state from pure solid vanadium to the infinitely dilute, weight percent solution of vanadium in pure iron at 1620°C. 

The excess term in the chemical potential = /xa — Ma hence, RT In fA is obtained from AmG and is ° W(1 —xa). This term goes to zero when xa 1 (cf. Raoult s law). The activity coefficient just considered therefore, is referred to as the Raoultian activity coefficient. When xa 0 it is constant (cf. Hemy s law). The displacement of the constant W from the term to the /x° term makes possible another normalization method, which is advantageous for describing dilute states. The first normalization chosen above and characterized by 1a —1 for xa 1 is known as the Raoultian normalization, and the second possibility with fA 1 for xa 0 as the Henryan normalization. There are then two equivalent representations of /xA) namely as + RTlnxA -I- RTln fA or as -h RTlnxA + RTln fA- In the regular model /Xa — W and, hence, In fA = W(xa — 2xa)/RT while... 

In both solid and gaseous solutions, virial equation-based Raoultian coefficients have often been proposed. For example, the Margules equations, often used in binary and sometimes in ternary solid solutions and which have a virial equation basis, were proposed originally for gaseous solutions. However, there is no satisfactory general model for Raoultian coefficients in multi-component solid solutions, and the tendency in modeling has been to treat these solutions as ideal (i.e., to use the mole fraction of a solid solution component as its activity see Equation (3.13)). 

Figure 8.7 Henryan activity of sucrose, Raoultian activity of water, and osmotic coefficients in sucrose solutions at 25°C. The inset is an enlargement of the region up to 1 molal, and the circle shows the ideal one molal standard state.

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