The Statistical Thermodynamics of Liquids

Statistical thermodynamics uses statistical arguments to develop a connection between the properties of individual molecules in a system and its bulk thermodynamic properties. For instance, consider a mole of water molecules at 25° C and standard pressure (1 bar). The thermodynamic state of the system has been defined on the basis of the number of molecules, the temperature, and the pressure. In order to relate the macroscopic thermodynamic properties such as U, G, H and A to the properties of the individual molecules, one would have to solve the Schrodinger wave equation (SWE) for a system composed of 6 x 10 interacting water molecules. This is an impossible task at present but if it were possible, one would obtain a wave function, I y, and an energy, 6)-, for the system. Moreover, 

Estimation of the probability for a given quantum state leads to the definition of the canonical partition function of the macrosystem  

On the basis of the definition of the partition function, the probability that the microsystem is in quantum state 6) is given by 

In the same way, one may estimate the system s pressure P by noting that 

An expression for the entropy S of the ensemble can be found by relating the temperature derivative of the entropy to the heat capacity at constant volume using equation (1.3.9). Accordingly, 

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The contribution of short-range forces to the activity coefficient can be described much better and in greater detail by the methods of the statistical thermodynamics of liquids, which has already created several models of electrolyte solutions. However, the procedures employed in the statistical... 

Three other important equations in the statistical thermodynamics of liquids involve the direct correlation function and provide a connection between the intermolecular potential u(r) for two molecules and the potential of mean force W(r). One way of deriving these equations is by writing an approximate expression for the indirect contribution to the pair correlation function g(r). Keeping in mind that W(r) is defined as — lng(r)/p, and defining gind( ) as the contribution to g(r) from indirect interactions, an approximate expression for c(r) is... 

Equations (1.14) and (1.15) are useful for computing the probability of drawing the queen of hearts in a card game, once you have seen the seven of clubs and the ace of spades. It is also useful for describing the statistical thermodynamics of liquid crystals, and ligand binding to DNA (see page 552). 

A rams, D. S., and J. M. Prausnitz, "Statistical Thermodynamics of Liquid Mixtures A New Expression for the Excess Gibbs Energy of Partly or Completely Miscible Systems," AIChE J., 1975, 21, 116. 

Abrams, D.S. and Prausnitz, J.M., Statistical thermodynamics of liquid mixtures a new expression for the excess Gibbs energy of partly or completely miscible systems, A. I. Chem. E. /., 21 (1975) 116-128. 

Asthagiri, D., Ashbaugh, H. S., and Pratt, L. R., A fresh attack on the statistical thermodynamics of molecular liquids. Technical Report LA-UR-03-5483, Los Alamos National Laboratory (2003fl). 

Experimental determination of excess molar quantities such as excess molar enthalpy and excess molar volume is very important for the discussion of solution properties of binary liquids. Recently, calculation of these thermodynamic quantities becomes possible by computer simulation of molecular dynamics (MD) and Monte Carlo (MC) methods. On the other hand, the integral equation theory has played an essential role in the statistical thermodynamics of solution. The simulation and the integral equation theory may be complementary but the integral equation theory has the great advantage over simulation that it is computationally easier to handle and it permits us to estimate the differential thermodynamic quantities. 

M.M. Telo da Gama, B.S. Almeida, The Uquid State, Set. Progr. Oxford 72 (1988) 75. (Review with some emphasis on the statistical thermodynamics of pure liquid surfaces.)... 

David W. Oxtoby is a physical chemist who studies the statistical mechanics of liquids, including nucleation, phase transitions, and liquid-state reaction and relaxation. He received his B.A. (Chemistry and Physics) from Harvard University and his Ph.D. (Chemistry) from the University of California at Berkeley. After a postdoctoral position at the University of Paris, he joined the faculty at The University of Chicago, where he taught general chemistry, thermodynamics, and statistical mechanics and served as Dean of Physical Sciences. Since 2003 he has been President and Professor of Chemistry at Pomona College in Claremont, California. 

It should be noted that it is assumed that the intermolecular forces do not affect the internal degrees of freedom so that is independent of whether these forces are present or not. When they are absent (Zf = 0), the integral Z collapses to and equation (2.2.31) becomes the same as equation (2.2.23). The important task of the statistical thermodynamics of imperfect gases and liquids is to evaluate Z. This subject is discussed in detail later in this chapter. However, the nature of the intermolecular forces which give rise to the potential energy U is considered next. 

Fluctuations in thermodynamics automatically imply the existence of an underlying structure that has created them. We know that such structure is comprised of molecules, and that their large number allows statistical studies, which, in turn, allow one to relate various statistical moments to macroscopic thermodynamic quantities. One of the purposes of the statistical theory of liquids (STL) is to provide such relations for liquids (Frisch and Lebowitz 1964 Gray and Gubbins 1984 Hansen and McDonald 2006). In such theories, many macroscopic quantities appear as limits at zero wave number of the Fourier transforms of statistical correlation functions. For example, the Kirkwood-Buff theory allows one to relate integrals of the pair density correlation functions to various thermo-physical properties such as the isothermal compressibility, the partial molar volumes, and the density derivatives of the chemical potentials (Kirkwood and Buff 1951). If one wants a connection between detailed correlations and integrated moments, one may ask about the nature of the wave-number dependence of these quantities. It turns out that the statistical theory of liquids allows an answer to such a question very precisely, which leads to new types of questions. The Ornstein-Zemike equation (Hansen and McDonald 2006), which is an exact equation of the STL, introduces the concept of correlation length which relates to the spatial extension of the density and/or concentration (the latter in the case of mixtures) fluctuations. This quantity cannot be accessed from pure... 

D. A. McQuarrie, Statistical Mechanics, Harper Row, New York, 1976. [This well-known book provides an extensive treatment of the statistical thermodynamics of gases, liquids, and sohds. Chapter 13 provides a comprehensive description of configurational integral equations. Also see the discussion in J. M. Ziman, Models of Disorder The Theoretical Physics of Homogeneously Disordered Systems, Cambridge University Press, Cambridge 1979.]... 

Lumsden s Thermodynamics of Alloys develops the thermodynamics of metals from the fundamental laws up to the applications of statistical mechanics. The concept of entropy is approached via randomness rather than via Carnot cycles. Applications of the theory to the correlation of the properties of pure metals are shown and the theories of solution are developed. Solid solutions, and the statistical mechanics of liquids, liquid solutions, and imperfect crystals, are also considered. 

It is shown that the problem of formulating the statistical thermodynamics of networks can be resolved by the use of ideas from quantum statistical mechanics by appropriate generalization. Examples are given in terms of gaussian and liquid crystal polymer networks. 

Sato H, Fujikake H, Lino Y, Kawakita M, Kikuchi H (2002) Flexible grayscale ferroelectric liquid crystal device containing polymer walls and networks. Jpn J Appl Phys 41 5302-5306 Sato H, Fujikake H, Kikuchi H, Kurita T (2003) Rollable polymer stabilized ferroelectric liquid crystal device using thin plastic substrates. Opt Rev 10(5) 352-356 Schrader DM, Jean YC (1988) Positron and positronium chemistry. Elsevier, Amsterdam Shinkawa K, Takahashi H, Fume H (2008) Ferroelectric liquid crystal cell with phase separated composite organic film. Ferroelectrics 364 107-112 Simha R, Somcynsky T (1969) On the statistical thermodynamics of spherical and chain molecule fluids. Macromolecules 2 342-350... 

The topic of capillarity concerns interfaces that are sufficiently mobile to assume an equilibrium shape. The most common examples are meniscuses, thin films, and drops formed by liquids in air or in another liquid. Since it deals with equilibrium configurations, capillarity occupies a place in the general framework of thermodynamics in the context of the macroscopic and statistical behavior of interfaces rather than the details of their molectdar structure. In this chapter we describe the measurement of surface tension and present some fundamental results. In Chapter III we discuss the thermodynamics of liquid surfaces. 

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