Ideal Solutions Defined

According to the scenario, the calculation of the order of precedence always teikes place in the same manner. First plans are evaluated in view of first-order criteria (FOQ. If plans still show the same ranking, second-order criteria (SOQ are hiken to refine the order. If a fined order of precedence is still not possible, the ideal solution, defined by the best characteristic numbers of edl plems, determines the order. The plan with the least difference from the optimal plan will be preferred. 

Motivatedby (4.432) we use in classical thermodynamics (e.g. [129, 138, 152, 155]) the ideal mixture or the ideal solution defined by the following expression for molar chemical potential of gas or liquid... 

That is, the excess properties are the differences between the real and ideal-solution changes of properties on mixing. The result (5.2.2) can be used for any ideal solution defined relative to any standard state for example, when excess properties are relative to the Lewis-Randall ideal solution, we substitute the ideal-solution expressions (5.1.22)-(5.1.27) into (5.2.2) to obtain the following relations between/ and/ . First-law excess properties are identical to the changes on mixing. 

A problem of obvious importance is the determination of the chemical potential of constituents that form a liquid or solid solution. We proceed by analogy to Eq. (2.4.15) for the ideal gas mixture. This objective is sensible, at least for ideal solutions defined below, because the different constituents in an ideal condensed phase do not interact, so they form an analog to the ideal gas mixture for which the partial pressure F,- constimtes the independent variable. The corresponding composition variable is the mole fraction x,. The solutions must be sufficiently dilute for the ideal Uquid model, discussed below, to apply. 

We regularly use y,-, for mol fractions in the gas phase and x,-for mol fractions in the liquid phase, P for the total pressure of the gas and pi for the pure-species vapor pressure of species i in the liquid.) For a pure liquid the partial vapor pressure is equal to the pure liquid s vapor pressure at that temperature (because x, = 1.0 for a pure liquid). For one kind of ideal solution (defined in Chapter 7) the sum of the partial vapor pressures equals the total vapor pressure of the liquid. The partial vapor pressure is defined only for liquids it is occasionally used for solids and never for gases. 

We define Fj to be the mole fraction of component 1 in the vapor phase and fi to be its mole fraction in the liquid solution. Here pj and p2 are the vapor pressures of components 1 and 2 in equihbrium with an ideal solution and Pi° and p2° are the vapor pressures of the two pure liquids. By Dalton s law, Plot Pi P2 Pi/Ptot these are ideal gases and p is propor-... 

If M represents the molar value of any extensive thermodynamic property, an excess property is defined as the difference between the actual property value of a solution and the value it would have as an ideal solution at the same temperature, pressure, and composition. Thus,... 

Foi an ideal solution, G, = 0, and tlieiefoie 7 = 1- Compatison shows that equation 203 relates to exactiy as equation 163 relates ( ) to GG Moreover, just as ( ) is a partial property with respect to G /E.T, so In y is a partial property with respect to G /RT. Equation 116, the defining equation for a partial molar property, in this case becomes equation 204 ... 

All three quantities are for the same T, P, and physical state. Eq. (4-126) defines a partial molar property change of mixing, and Eq. (4-125) is the summability relation for these properties. Each of Eqs. (4-93) through (4-96) is an expression for an ideal solution property, and each may be combined with the defining equation for an excess property (Eq. [4-99]), yielding ... 

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]. 

Units It should be noted that in the S.I. the activity of a solute is defined with reference to a standard state, i.e. an ideal solution of molality 1 mol kg". Thus the relative activity of a metal ion in solution is given by... 

There are several different scales 011 which the activity of a solute may be defined.1 In thermodynamic expressions for a solute in a non-ideal solution the activity on the molality scale plays the same part that is played by the molality of a solute in an ideal solution. Since the activity is expressed in the same units as the molality, the ratio of the activity to the molality—the activity coefficient—is a pure number whose value is independent of these units it is also indopendont of the particular b.q.s. that has been adopted. Thus the numerical values of all activities and molalities would change in the same ratio, if at any time a new choice were made for the b.q.s. 

The Disparity of a Solution. We may begin to use the word disparity in a technical sense, for the quantity defined above, and to speak of d as the disparity of the solution when the mole fraction of the solute is x. In dilute ionic solutions the sign of d is always negative. The effect of the interionic forces is that ions added to a dilute solution always lose more free energy than they would when added to the corresponding ideal solution hence the total communal term is less than the cratic term. 

Equations (6.65) and (6.66) are statements of Raoult s law. They form the basis for defining an ideal solution. 

The excess molar thermodynamic function Z is defined as the difference in the property Zm for a real mixture and that for an ideal solution. That is,... 

For a solution of a non-volatile substance (e.g. a solid) in a liquid the vapour pressure of the solute can be neglected. The reference state for such a substance is usually its very dilute solution—in the limiting case an infinitely dilute solution—which has identical properties with an ideal solution and is thus useful, especially for introducing activity coefficients (see Sections 1.1.4 and 1.3). The standard chemical potential of such a solute is defined as... 

An ideal solution is defined as one for which the chemical potential of every component (/ , ) is related to its mole fraction by... 

Most real solutions cannot be described in the ideal solution approximation and it is convenient to describe the behaviour of real systems in terms of deviations from the ideal behaviour. Molar excess functions are defined as... 

The activity coefficient of component i, y(-, is now defined as a measure of the deviation from the ideal solution behaviour as the ratio between the chemical activity and the mole fraction of i in a solution. 

In this particular case of ideal solutions the phase diagram is defined solely by the temperature and enthalpy of fusion of the two components. 

If an ideal solution is formed in the octanol phase, and the solute in the aqueous phase is not affected by the dissolved octanol, then the last two factors in Equation 10 equal zero. Under these assumptions, an ideal octanol/water partition coefficient (K w) can be defined by (19) ... 

The previous summary of activities and their relation to equilibrium constants is not intended to replace lengthier discussions [1,18,25,51], Yet it is important to emphasize some points that unfortunately are often forgotten in the chemical literature. One is that the equilibrium constants, defined by equation 2.63, are dimensionless quantities. The second is that most of the reported equilibrium constants are only approximations of the true quantities because they are calculated by assuming the ideal solution model and are defined in terms of concentrations instead of molalities or mole fractions. Consider, for example, the reaction in solution ... 

C-t, which means, of course, that the ideal solution model is adopted, no matter the nature or the concentrations of the solutes and the nature of the solvent. There is no way of assessing the validity of this assumption besides chemical intuition. Even if the activity coefficients could be determined for the reactants, we would still have to estimate the activity coefficient for the activated complex, which is impossible at present. Another, less serious problem is that the appropriate quantity to be related with the activation parameters should be the equilibrium constant defined in terms of the molalities of A, B, and C. As discussed after equation 2.67, A will be affected by this correction more than A f//" (see also the following discussion). 

Results. Our experimental observations appear in Table I and Figure 6. We have defined as K JP/Z2PivaP. The slopes from Figure 6 inserted into Equation 16 provide values for B -Unfortunately, we were not able to find the fiduciary data necessary to evaluate (V ),. anc must leave the calculation unfinished. The ideal solution assumption is totally inadequate in this particular experiment. 

Historically, an ideal solution was defined in terms of a liquid-vapor or solid-vapor equilibrium in which each component in the vapor phase obeys Raoult s law. 

If Equation (14.2), Equation (14.6), or Equation (14.7) is used to define an ideal solution of two components, values for the changes in thermodynamic properties resulting from the formation of such a solution follow directly. 

Specific, surface confined reactions not only directly involve catalysis but also the built-up of sdf-assembled multilayers (see Fig. 9.1 (3)) with co-functionalities for more complex (bio-) catalytic systems such as proteins or the directed deposition of active metals. Furthermore, SAM on flat substrates can be used for the study and development of e.g. catalytic systems, but are not useful for large scale applications because they have very limited specific surface. Here, nanoparticle systems covered with 3D-SAMs are the ideal solution of combining the advantages of high surface area, defined surface composition and accessibility of proximal active catalytic centers. 

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