Statistical mechanics ideal mixture

Given this experimental result, it is plausible to assume (and is easily shown by statistical mechanics) that the chemical potential of a substance with partial pressure p. in an ideal-gas mixture is equal to that in the one-component ideal gas at pressure p = p. 

It has long been recognized that the validity of the BKW EOS is questionable.12 This is particularly important when designing new materials that may have unusual elemental compositions. Efforts to develop better EOSs have been based largely on the concept of model potentials. With model potentials, molecules interact via idealized spherical pair potentials. Statistical mechanics is then employed to calculate the EOS of the interacting mixture of effective spherical particles. Most often, the exponential-6 (exp-6) potential is used for the pair interactions ... 

The ideal gas free energy functional is defined exactly from statistical mechanics, dropping the temperature-dependent terms that do not affect the fluid structure. Free energy functional contribution due to the excluded volume of the segments is calculated from Rosenfeld s (1989) DFT for a mixture of hard spheres. The functional derivatives of these free energy functional contributions, which are actually required to solve the set of Euler-Lagrange equations, are straightforward. 

In Section A.l, the general laws of thermodynamics are stated. The results of statistical mechanics of ideal gases are summarized in Section A.2. Chemical equilibrium conditions for phase transitions and for reactions in gases (real and ideal) and in condensed phases (real and ideal) are derived in Section A.3, where methods for computing equilibrium compositions are indicated. In Section A.4 heats of reaction are defined, methods for obtaining heats of reaction are outlined, and adiabatic flame-temperature calculations are discussed. In the final section (Section A.5), which is concerned with condensed phases, the phase rule is derived, dependences of the vapor pressure and of the boiling point on composition in binary mixtures are analyzed, and properties related to osmotic pressure are discussed. 

In equation (49), which is the van t Hoff equation, —dH/de may be replaced by — AH, since these two quantities are equal for ideal-gas reactions. Relationships analogous to equation (49) may be derived for each of the equilibrium constants defined in Section A. 3, but for reactions in systems other than ideal-gas mixtures, — AH and — dH/de may not, in general, be equated in these expressions. Heats of reaction can be determined directly either by spectroscopic measurements followed by the application of statistical mechanics (for ideal-gas reactions) or by calorimetric measurements of Q (for arbitrary reactions). Since the measurement of equilibrium compositions may be simpler than either of the above procedures, in practice equation (49) is often used to obtain heats of reaction from experimental values of Kp at neighboring temperatures. 

The statistical mechanical interpretation of Eq. 9.1-8 is that an ideal gas mixture is a completely mixed or random mixture. This is discussed in Appendix A9.1. 

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

At first sight this looks like nothing more than a polynomial expansion of the ideal gas law. However, it turns out to have real physical significance, and the form of the equation follows directly from statistical mechanics. The details can be found in most textbooks on statistical mechanics (see, for example, Mayer and Mayer, 1940 Hill, 1960, Chapter 15). We will outline the underlying theory very briefly here because virial equations (or similar approaches) appear several times in this book—see for example the discussion of the Pitzer equations for the non-ideal properties of salt solutions in Chapter 17 and Chapter 16 on gas mixtures. 

To obtain a physical interpretation for the residual Gibbs energy, we start with an ideal-gas mixture confined to a closed vessel. As the process, we consider the reversible isothermal-isobaric conversion of the ideal-gas molecules into real ones. Although this process is hypothetical, it is a mathematically well-defined operation in statistical mechanics the process amounts to a "turning on" of intermolecular forces. We first want to obtain an expression for the work, but since the process involves a change in molecular identities, we must start with the general energy balance (3.6.3). For a system with no inlets and no outlets, (3.6.3) becomes... 

The equations given for enthalpy and entropy of ideal-gas mixture were given here without proof. They can be proven using the tools of statistical mechanics, but this is beyond the scope of this book. Nonetheless, we can arrive at these equations by qualitative arguments. Since molecules in the ideal-gas state do not interact, the internal energy of the mixture is the same as the total internal energy of the pure components at same pressure and temperature this means = o. And since the volume of the mixture is the sum of the pure component volumes, we conclude the same for enthalpy, or AHm > = o. 

From Eqs. 9.2.46 and 9.2.50, the solute chemical potential is given by /t-b = C/b + pV — TS. In the dilute solution, we assume Ub and Ir are linear functions of xr as explained above. We also assume the dependence of 5b on xr is approximately the same as in an ideal mixture this is a prediction from statistical mechanics for a mixture in which aU molecules have similar sizes and shapes. Thus we expect the deviation of the chemical potential from ideal-dilute behavior, /tr = + RT InxB, can be... 

For some reactions use can also be made of statistical-mechanical equations either (rarely) alone or in combination with some of the quantities alluded to above. These are reactions taking place in systems for which we have a model which is at once realistic enough and mathematically tractable enough to be useful. An example is the calculation of the standard equilibrium constant of a gas reaction (and thence of the yield, but only if the gas mixture is nearly enough perfect) from spectroscopically determined molecular properties. Another example is the use of the Debye-Hiickel theory or its extensions to improve the calculation of the yield of a reaction in a dilute electrolyte solution from the standard equilibrium constant of the reaction when, as is usually so, it is not accurate enough to assume that the solution is ideal-dilute. 

Modification of the theory onto the case of adsorption fi om non-ideal mixtures and introduction of the lateral interactions between the adsorbed molecules are possible, but may lead to rather complicated relations [86,88]. On the other hand, the localized models may easily be generalized onto adsorption on a heterogeneous surface [89,90]. This is related to the fact that the coefficients b (T) may be directly expressed in terms of the adsorption energies by means of the statistical mechanics ... 

These are well-established thermodynamic relations that describe the properties of mixing as a function of the molar fractions of mixing components. Typically, these relations are presented in an ad hoc manner in thermodynamics textbooks. In this section, we have demonstrated that they can be naturally derived from first principles of statistical mechanics. Properties of non-ideal mixtures can also be derived, but, as expected, the derivations, which include particle interactions, quickly become cumbersome. The interested reader is referred to the literature at the end of the chapter for more details. 

---