Vafaei Shaghayegh, Tomberli Bruno, Gray C G
Department of Physics and Biophysics Interdepartmental Group, University of Guelph, Guelph, Ontario N1G2W1, Canada.
Department of Physics, Capilano University, Vancouver, British Columbia V7J3H5, Canada.
J Chem Phys. 2014 Oct 21;141(15):154501. doi: 10.1063/1.4897980.
McMillan and Mayer (MM) proved two remarkable theorems in their paper on the equilibrium statistical mechanics of liquid solutions. They first showed that the grand canonical partition function for a solution can be reduced to one with an effectively solute-only form, by integrating out the solvent degrees of freedom. The total effective solute potential in the effective solute grand partition function can be decomposed into components which are potentials of mean force for isolated groups of one, two, three, etc., solute molecules. Second, from the first result, now assuming low solute concentration, MM derived an expansion for the osmotic pressure in powers of the solute concentration, in complete analogy with the virial expansion of gas pressure in powers of the density at low density. The molecular expressions found for the osmotic virial coefficients have exactly the same form as the corresponding gas virial coefficients, with potentials of mean force replacing vacuum potentials. In this paper, we restrict ourselves to binary liquid solutions with solute species A and solvent species B and do three things: (a) By working with a semi-grand canonical ensemble (grand with respect to solvent only) instead of the grand canonical ensemble used by MM, and avoiding graphical methods, we have greatly simplified the derivation of the first MM result, (b) by using a simple nongraphical method developed by van Kampen for gases, we have greatly simplified the derivation of the second MM result, i.e., the osmotic pressure virial expansion; as a by-product, we show the precise relation between MM theory and Widom potential distribution theory, and (c) we have extended MM theory by deriving virial expansions for other solution properties such as the enthalpy of mixing. The latter expansion is proving useful in analyzing ongoing isothermal titration calorimetry experiments with which we are involved. For the enthalpy virial expansion, we have also changed independent variables from semi-grand canonical, i.e., fixed {N(A), μ(B), V, T}, to those relevant to the experiment, i.e., fixed {N(A), N(B), p, T}, where μ denotes chemical potential, N the number of molecules, V the volume, p the pressure, and T the temperature.
麦克米兰和迈耶（MM）在其关于液体溶液平衡统计力学的论文中证明了两个引人注目的定理。他们首先表明，通过对溶剂自由度进行积分，溶液的巨正则配分函数可以简化为仅具有有效溶质形式的配分函数。有效溶质巨配分函数中的总有效溶质势可分解为单个、两个、三个等溶质分子孤立基团的平均力势分量。其次，根据第一个结果，现在假设溶质浓度较低，MM推导出了渗透压关于溶质浓度幂次的展开式，这与低密度下气体压力关于密度幂次的维里展开式完全类似。所得到的渗透压维里系数的分子表达式与相应的气体维里系数具有完全相同的形式，只是用平均力势取代了真空势。在本文中，我们将研究局限于溶质为A物种和溶剂为B物种的二元液体溶液，并做三件事：(a) 通过使用半巨正则系综（仅对溶剂为巨正则）而非MM所使用的巨正则系综，并避免使用图形方法，我们极大地简化了MM第一个结果的推导；(b) 通过使用范坎彭为气体开发的一种简单的非图形方法，我们极大地简化了MM第二个结果即渗透压维里展开式的推导；作为一个副产品，我们展示了MM理论与维登势分布理论之间的精确关系；以及(c) 我们通过推导其他溶液性质（如混合焓）的维里展开式扩展了MM理论。后一个展开式在分析我们正在进行的等温滴定量热实验中被证明是有用的。对于焓维里展开式，我们还将自变量从半巨正则的，即固定的{N(A), μ(B), V, T}，改为与实验相关的，即固定的{N(A), N(B), p, T}，其中μ表示化学势，N表示分子数，V表示体积，p表示压力，T表示温度。
