Ideal mixture, definition

Equilibrium constants do not have units because in the strict thermodynamic definition of the equilibrium constant, the activity of a component is used, not its concentration. The activity of a species in an ideal mixture is the ratio of its concentration or partial pressure to a standard concentration (1 M) or pressure (1 atm). Because activity is a ratio, it is unitless and the equilibrium constant involving activities is also unitless. 

Of course, it is uncommon for the free energy/ to obey (1). In particular, the entropy of an ideal mixture (or, for polymers, the Flory-Huggins entropy term) is definitely not of this form. On the other hand, in very many thermodynamic (especially mean field) models the excess (i.e., nonideal) part of the free energy does have the simple form (1). In other words, if we decompose the free energy as (setting kn = 1)... 

Starting from the definition 5.22 we now establish several important properties of thermodynamic potentials (partial molar quantities of thermodynamic energy functions) for an ideal system of mixture. Differentiating G-H-TS with respect to n, with Tand p constant, we have pt = ht- Tsl and furthermore [d(jWf IT) / dT pn = (1 IT) (dp, / dT) - (p, / T1) = - [(r s, + pt) / T2] = -h,l T2. From this equation we obtain Eq. 5.34 for the partial molar enthalpy hf of a constituent i in an ideal mixture ... 

For an ideal mixture, the enthalpy of mixing is zero and so a measured molar enthalpy of mixing is the excess value, HE. The literature concerning HE -values is more extensive than for GE-values because calorimetric measurements are more readily made. The dependence of HE on temperature yields the excess molar heat capacity, while combination of HE and GE values yields SE, the molar excess entropy of mixing. The dependences of GE, HE and T- SE on composition are conveniently summarized in the same diagram. The definition of an ideal mixture also requires that the molar volume is given by the sum, Xj V + x2 V2, so that the molar volume of a real mixture can be expressed in terms of an excess molar volume VE (Battino, 1971). 

Equation (169) must hold for arbitrary 5 / satisfying E8rij = 0, and clearly the only possibility is that/ (X/) = 1/X/, that is,fiX,) = InX/. The logarithmic form is the only one that satisfies the stated conditions, and hence the discrete equivalent of Eq. (21) simply follows from the definitions. Clearly, the argument can be generalized to a continuous description [one only needs to apply the Gibbs-Duhem equation in its continuous formulation to a 6 (a ) satisfying <8n(y)> = 0], and hence Eq. (21) is identified as simply the definition of a continuous ideal mixture. 

Let us consider an ideal ternary mixture. According to the definition of an ideal mixture (18), the activities of the components ( i) are equal to their mol fractions (xi) and their partial molar volumes are equal to those of the pure components (V = Tf). 

One can also prove that (10.42) is equivalent to the more traditional ideal-gas mixture definition ... 

There are attempts to motivate the definition of ideal mixture by a simpler way, e.g. it is possible to show [149, 150] that if the chemical potential of each constituent depends (besides temperature and pressure) only on the molar fraction of that constituent then this dependence is logarithmic as in (4.437) (it is assumed also that the partial internal energy and volume of at least one constituent depends on temperature and pressure only and that the number of constituents must be 3 as a minimum). 

The alternative motivation of definition (4.437) for (real) gas mixtures comes from a statement that a mixture is ideal if Amagat s law (4.440) is valid at any T, P. Indeed, Amagat s laws means Va = v and then by (4.454), (4.458) below, for fugacity coefficients also = v therefore by (4.463), this is an ideal mixture. 

It is convenient to start with the description of the properties of a real mixture with the ideal mixture as a first approximation. The deviation from ideal behavior is then taken into account by the addition of further terms. Therefore, the definition of an ideal mixture should be explained, before the excess properties are introduced. Starting point for the definition of an ideal mixture are the equations for a mixture of ideal gases. A mixture of ideal gases is characterized by the following behavior ... 

For an ideal mixture the excess part is equal to zero, since the activity coefficient is Yi = 1. From this definition, the connection between the excess Gibbs energy and the activity coefficient can be seen clearly ... 

For condensed matter, Kquation 7 (for each component) can be thought of as a definition of an ideal mixture. Having represented it as... 

The integration constant g , the so-called chemical standard potential, is the partial molar free energy at a pressure P = 1 bar, temperature T and at an activity of unity. In an ideal mixture the activity coefficient is unity, so that the aforementioned definition holds for both real and ideal mixtures. 

By definition, component i experiences in an ideal mixture the same inter-molecular forces as in the reference state and therefore all differences between pjd mix. pj fare caused by differences in the concentration (i.e., dilution) only ... 

The definition of the fugacity of a species in solution is parallel to the definition of the pure-species fugacity. An equation analogous to the ideal gas expression, Eq. (4-73), is written for species i in a flmd mixture ... 

When the gas chromatograph is attached to a mass spectrometer, a very powerful analytical tool (gas chromatography-mass spectrometry, GC-MS) is produced. Vapour gas chromatography allows the analyses of mixtures but does not allow the definitive identification of unknown substances whereas mass spectrometry is good for the identification of a single compound but is less than ideal for the identification of mixtures of... 

From the definition of an ideal gas mixture, we shall have for the free energy of the mixture of i gases in the volume Y the expression m j/ = = XtiiMfy/ri... 

The definition of an Ideal Gas Mixture given in 122, although it leads to results in entire accord with those established by... 

Such a rough comparison of real mixtures with ideal solutions is definitely not perfect but it allows the authors of [230] to proceed using conventional theory. The general conclusion following this comparison is that the quantum. /-diffusion model just slightly differs from its... 

It should be noted that the derivative is negative, so that at certain conditions the denominator of Eq. (15-51) can be zero, resulting in an infinite pressure gradient. This condition corresponds to the speed of sound, i.e., choked flow. For a nonflashing liquid and an ideal gas mixture, the corresponding maximum (choked) mass flux G follows directly from the definition of the speed of sound ... 

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