Entropy of an ideal mixture

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)... 

Here spr is the projected entropy of an ideal mixture. The first term appearing in it, p0 = J dop a), is the zeroth moment, which is identical to the overall particle density p defined previously. If this is among the moment densities used for the projection (or more generally, if it is a linear combination of them), then the term — Tp0 is simply a linear contribution to the projected free energy/pr(p,) and can be dropped because it does not affect phase equilibrium calculations. Otherwise, p0 needs to be expressed—via the A —as a function of the pit and its contribution cannot be ignored. We will see an example of this in Section V. 

To end this section, let us state in full the analogue of Eqs. (29) and (32) for the case of several moment densities, restoring the notation used in the previous sections. The square bracket on the right-hand side of Eq. (32) is the moment expression for the entropy of an ideal mixture. If, as in Section II. A, we measure this entropy per unit volume (rather than per particle, as previously in the current section) and generalize to several moment densities, we find by the combinatorial approach the following moment free energy ... 

This relation is called Dalton s law. With the equation of state above, for example, the entropy of an ideal mixture can be calculated. According to Table 2.1, the total differential of the entropy for a pure ideal gas is... 

Examples.—(1) Prove that the entropy of an ideal gas mixture is the sum of the entropies of the components at the same temperature, each occupying the whole volume of the mixture. 

Figure 7.1 Entropy, enthalpy, and Gibbs free energy changes at T= 298.15 K for forming one mole of an ideal mixture from the components,...

This expression accounts for the configurational entropy of an ideal binary mixture with identical molecular sizes, but not for that of a polymer solution, since polymer chains are large and flexible. For that case, more contributions arise from the chain conformational entropy, first considered by Meyer [19] and then derived by Huggins [20] and Flory [21]. In analogy with a nonreversing random walk on a lattice, the conformational contribution of polymer chains to the partition function is given by... 

Partial molar availability, 24 692 Partial molar entropy, of an ideal gas mixture, 24 673—674 Partial molar Gibbs energy, 24 672, 678 Partial molar properties, of mixtures, 24 667-668... 

Deviation from ideality is entirely due to enthalpic terms. The entropy of mixing is that of 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). 

The free energy of a phase is thus expressed in terms of the amounts of its various components and, through the [xs, in terms of its composition. Pa . .. are called the chemical potentials of A, B,. .. in the phase, at the given temperature and pressure. They generally depend on composition, but sometimes this dependence is relatively simple. Thus for a mixture of ideal gases, U, S, F and G are all sums of terms, one for each gas regarded as though it alone were present. Thus P = — [dFjdV) is a sum of partial pressures (P, Pj, . ..), in accord with Dalton s law, where P F = n RT etc. Now the entropy of an ideal gas has already been found in the form [constant —R log P + Cp log 7] and, since G = H — TS, then, for one mole at partial pressure P the dependence on pressure can be written... 

The entropy of an ideal-gas mixture is obtained by combining the internal energy of Equation (4.261) and the Helmholtz energy of Equation (4.265) ... 

For a so-called ideal mixture, = 0. The entropy of mixing at T = 0 K is also zero. Hence at that very low temperature, ideal mixtures have a straight Gp (x) line between the G-values of the components P x = 1) and Q x = 0). Now, if T > 0, the second term b (which is less than 0) starts to contribute and the Gpg(x) curve, even of an ideal mixture, is no longer linear and touches the vertical axes at x = 0 and X = 1. In a nonideal mixture at OK, 0 and the first term a is no longer linear but parabolic in x. The value of can be positive or negative, depending... 

Since In xt is negative in a mixture, the partial molar entropy of a constituent of an ideal mixture is greater than the molar entropy of the pure substance at the same T and p. 

From Eqs. 11.1.24 and 11.1.25, and the fact that the entropy of a mixture is given by the additivity rule S = Y ,i ntSi, we conclude that the entropy of an ideal gas mixture equals the sum of the entropies of the unmixed pure ideal gases, each pure gas having the same temperature and occupying the same volume as in the mixture. 

It is important to note that all component gas properties are evaluated at the mixture temperature, T, and component partial pressure, P. For the ideal gas mixture, however, enthalpy and internal energy are a fimction of temperature, and hence component gas enthalpy and internal energy are estimated as a function of mixture temperature only as given by Equations 3.47a and 3.47b, respectively. However, entropy of an ideal gas is a function of temperature and pressure, and so the component gas entropy is estimated as a function of partial pressure of the component in the mixture and the gas mixture temperature as given by Equation 3.53. 

The entropy change to form an ideal mixture from the pure components is obtained by differentiating equation (7.7) with respect to T. Since x, is independent of T, the result is... 

Expression (37) is the same result that holds for the ideal gas, where the entropy of mixing results entirely from the increase in randomness of the mixture. It is necessarily positive because all x, < 1 and ln x, < 0. All of the thermodynamic properties of mixing of an ideal solution result from this randomness there are no energy effects. 

Generally, the chemical potential of a constituent substance i in a mixture consists of a unitary part, which is inherent to the pure substance i and independent of its concentration, and a communal part, which depends on the concentration of constituent i [Ref. 3.]. The communal part of the chemical potential of a constituent i in a mixture arises from the entropy of mixing of i For an ideal mixture the molar entropy of mixing of i, s,M, is given from Eq. 3.51 by = -j ln x, and hence the communal part of the chemical potential is expressed by p 4 = -TsM = RT nx, at constant temperature, where x, is the molar fraction of... 

However, even for an ideal mixture, there is an effect on the Gibbs free energy from the entropy of mixing, namely. 

It is of interest to note that since the mole fraction n of any gas in a mixture must be less than unity, its logarithm is negative hence ASm as defined by equation (19.32) is always positive. In other words, the mixing of two or more gases, e.g., by diffusion, is accompanied by an increase of entropy. Although equation (19.32) has been derived here for a mixture of ideal gases, it can be shown that it applies equally to an ideal mixture of liquids or an ideal solid solution. 

Using the energy, volume, and entropy changes on mixing given here, one can easily compute the other thermodynamic properties of an ideal gas mixture (Problem 9.1). The results are given in Table 9.1-1. Of particular interest are the expressions for... 

Appendix A9.1 A Statistical Mechanical Interpretation of the Entropy of Mixing in an Ideal Mixture... 

Gibbs paradox arises from the unsound assumption that the entropy of each gas in an ideal mixture is independent of the presence of the other gases ... 

Figure 10.3. The AG/x plots of a few imaginary mixtures to show the contributions of enthalpy (a) and entropy (b) to binary phase diagrams. Top left an ideal mixture (AH = 0) at some temperature (T > 0). Top right a nonideal mixture (AH > 0) also with an entropy term. Bottom right an ideal mixture at nonzero temperature, where A and B have a different crystal structure. ...

As the mole fractions yi vary between 0 and 1, the entropy change of mixing for an ideal mixture is always different from zero and positive. 

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