Ideal solution equations

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 is the basis for development of expressions for all other thermodynamic properties of an ideal solution. Equations (4-60) and (4-61), apphed to an ideal solution with replaced by Gj, can be written... 

Unfortunately, phases of geochemical interest are not ideal. As well, aqueous species do not occur in a pure form, since their solubilities in water are limited, so a new choice for the standard state is required. For this reason, the chemical potentials of species in solution are expressed less directly (Stumm and Morgan, 1996, and Nordstrom and Munoz, 1994, e.g., give complete discussions), although the form of the ideal solution equation (Eqn. 3.4) is retained. 

We have derived an expression for the free energy of mixing two pure substances to form one mole of an ideal solution [Equation (14.35)],... 

Thermodynamic Properties of Ideal Solutions Equation (11.79) is the starting point for deriving equations for AmjxZ, the change in Zm for forming an ideal mixture.hh For the ideal solution, 7ri(- = 1 and equation (11.79) becomes... 

Here R(T )2/A/fv ] is a property of the pure solvent. Equation (10.92) is the basic equation for the simpler expressions involving the mole fraction or molality of the solute of the equation for the boiling point elevation, but it must be emphasized that it is only an approximate equation, valid in the limit as xx approaches unity. Even for the approximation of ideal solutions, Equation (10.90) should be used for the calculation of the boiling point when Xj is removed from unity. 

Thus, the canonical form (2.5.1) does indeed meet the requirements (a)-(c) for ideal solutions. Equation (2.5.1) represents one of the most basic relationships in chemical thermodynamics. 

These equations are alternative forms of Eqs. (11.76) tlirough (11.79). As written here tliey apply to ideal-gas mixtures as well as to ideal solutions. Equation (12.29) may be writtenfor an ideal solution ... 

Subtracting off the Gibbs energy of mixing for the ideal solution (equation (1.6.11)), one obtains for Aj ixC ... 

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. 

A solution is ideal if it satisfies the ideal-solution law. No real solution is rigorously ideal, but solutions of similar substances approach ideal-solution behavior as the similarity increases. Solutions of xylene isomers, for example, deviate from ideal-solution law by about 1% at the maximiun. Close members of the same homologous series are often assumed to be ideal. It is not unusual to calculate mixtures of paraffin hydrocarbons with the ideal-solution equation. Ideal-solution law is the basis for ideal K values often used in industry. However, ideal-solution law is of great value in another way, and that is to provide a basis for introducing a correction factor, known as the activity coefficient. 

These are usually heat effects during dissolution of solids (and liquids) in solvents, and the ideal solution relation is modified by an energy term that takes into account heats of mixing. The term modifies the ideal solution equation to give the regular solution theory prediction of solubility... 

At this point we can write out the defining equations of a truly non-ideal solution (equations (15.3)). This would be one for which all properties in addition to the enthalpy differ from the ideal values. Here again, the subscript non-ideal dissol n refers to the difference between a non-ideal solution and a mechanical mixture of its pure components ... 

For processes that involve phase transformation, the general approach is to use the ideal-solution equation for the enthalpy of the liquid and treat the vapor phase as an ideal-gas mixture. This reduces the problem to a calculation of the enthalpies of pure liquid and pure vapor components. If the calculation involves states near the phase boundary, hypothetical states maybe involved, whose properties must be calculated by extrapolation from known real states. As an example, consider the constant-pressure heating of a solution that contains 30% acetonitrile in nitromethane, at 1 bar. This is shown by the line LVon the Txy graph in Figure 11-1. The enthalpy change for this process is... 

An excess property is the difference between the property of solution and the same property calculated by the ideal-solution equations ... 

By definition, the excess properties of ideal solution are zero. We may view the excess properties as corrections that must be added to the ideal-solution equations in order to obtain the correct property of solution. 

Calculation of the freezing-point depression of the solvent and hence the molecular weight of the solute by this method proceeds exactly the same way as for the boiling-point elevation. For cryoscopy of ideal solutions, equations corresponding to those for Ar and ke are AT/ = -kftnz and = (RT Mi)l(l000Lf), where ATf = T - Tf Hhe freezing-point depression. Tf is the... 

The parent supersaturated phase is supposed to behave like an ideal solution (Equation 13.15) ... 

We assume that the solution is sufficiently dilute that the solvent obeys the ideal solution equation ... 

For non-ideal solutions Equation (2.23) must be modified in terms of the activity of i rather than its mole fraction, thus... 

In order to deal with the nonideality of real solutions, the mole fraction variable, Xy in Eq. (12), was replaced by Lewis [1—4] with a fictitious mole fraction-like variable, ay- The variable ay was defined by Lewis as the activity of solute, Y ", and can be regarded as the variable for real solutions which replaces Xy fo foe ideal solution equation (12). Thus, for real solutions, py can be written in analogy to Eq. (12) as... 

If both solutions behave ideally, so that Raoult s law is applicable, the vapour pressure is proportional to the mole fraction of the solvent in the particular solution (Equation 1.45) hence, for ideal solutions. Equation 1.47 becomes... 

This equation becomes equivalent to that for an ideal solution (Equation 3.24) when jc = 1 and x 0. 

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