**6. Concluding remarks**

In the foregoing, we started with a brief discussion of the liquid state, compared with the gaseous and solid states. Among other things, we mentioned that one of the most successes of the microscopic theory of liquids has been to emphasize the importance of the *pair potential u*(*r*) to describe a wide variety of physical properties of liquids. Then, we learned a rudimentary knowledge of the structure of liquids. The latter is essentially described by the static *structure factor S*(*q*), which can be measured directly by elastic scattering of neutrons or X-rays. But *S*(*q*), or more precisely the *pair correlation function g*(*r*), can also be determined by numerical simulations (known as virtual experiments) and with models of the microscopic theory of liquids once the nature of the pair potential *u*(*r*) is known. The comparison between simulation results and those of analytical models essentially allows us to test the models, whereas the comparison of simulation results with experimental results is the ultimate test to judge the efficiency of the pair potential. Subsequently, we described in minute detail the calculations of the thermodynamic properties in terms of the pair potential and pair correlation function. In particular, with the pressure *p* and the chemical potential *μ* to apply equilibrium conditions, it has been shown that we are in a position to determine theoretical estimates for the liquid-vapor coexistence curve. Finally, we got down to basics of the thermodynamic perturbation theory, before presenting the liquid-vapor coexistence curve for the HCY fluid.

In a 1990 time warp, it became apparent that the usefulness of thermodynamic perturbation theory was on the decline compared to the more powerful simulation methods. This was primarily due to the rapid increase in the power of computers. At the same time, the integral equation theory enjoyed renewed popularity with the extensively employed concept of thermodynamic consistency (25), but this aspect goes beyond the scope of this short review. Incidentally, we recall that the thermodynamic consistency consists to adjusting the isothermal compressibility obtained by two different routes. Nevertheless, the thermodynamic perturbation theory remains undoubtedly the most tractable approach to predict the thermodynamic properties of liquids. Let us just quote few articles containing investigations of the HCY fluid with the integral equation theory (26).

As one would expect, the thermodynamic perturbation theory described previously is also important for mixtures. The thermodynamic properties established for studying pure liquids can be applied to binary mixtures, with only few modifications. Specifically, analytic expressions for the hard-sphere free energy *FHS*(*d*1, *d*2) and subsequent thermodynamic quantities are readily generalized for a mixture of hard spheres of different diameters *d*<sup>1</sup> and *d*2. Therefore, if one estimates that a real binary mixture behaves like binary hard-sphere fluid, the variational method consists of minimizing *FHS*(*d*1, *d*2) versus the two hard-sphere diameters, and by taking the resulting minimum upper bound as an approximation to the free energy. The variational method is not limited neither to hard-sphere reference system nor to a specific fluid. It should be stressed that systems like pure liquid metals composed of ions embedded in an electron gas can not be treated as a binary mixture, in the strict sense of the word, but must be suitably reduced to a one-component system of pseudoions. In contrast, a liquid metal made up of two species of pseudoions forms a binary mixture, for which the approach originally developed for simple liquids can be applied. What is it that determines whether or not two metals will mix to form an alloy is a crucial issue to be answered by thermodynamic perturbation theory (27).

I would like to express my acknowledgement to Jean-Marc Bomont for its stimulating discussions.

## **7. References**

30 Thermodynamics book 1

correlation functions, inherent in the calculations of the pressure and the chemical potential, do not represent correctly the growing correlation lengths in approaching the critical point. Looking at the evolution of the binodal lines as a function of the rate of decay *λ* of the Yukawa potential, we remark that higher the critical temperature is, lower *λ* is. At the same time, the domain below the binodal line shrinks. In contrast, when *λ* increases (i. e., the attraction range of the potential becomes shorter), the critical temperature decreases, and the liquid and gaseous phases become indistinguishable. For the hard-sphere potential, as a border case (*λ* → ∞), there is no longer gas-liquid phase transition. On the other hand, one can see that the phase diagrams obtained with the perturbation theory agree with simulation data more favorably for the vapor branch than for the liquid branch. This is not surprising in the extent that the perturbation theory works better for low densities. Lastly, it should be mentioned that the structure and thermodynamic properties of the HCY potential have been extensively studied for the two last decades, as much by computer simulations and integral equation theory as by means of perturbation theory. Note that many other studies of the HCY potential

In the foregoing, we started with a brief discussion of the liquid state, compared with the gaseous and solid states. Among other things, we mentioned that one of the most successes of the microscopic theory of liquids has been to emphasize the importance of the *pair potential u*(*r*) to describe a wide variety of physical properties of liquids. Then, we learned a rudimentary knowledge of the structure of liquids. The latter is essentially described by the static *structure factor S*(*q*), which can be measured directly by elastic scattering of neutrons or X-rays. But *S*(*q*), or more precisely the *pair correlation function g*(*r*), can also be determined by numerical simulations (known as virtual experiments) and with models of the microscopic theory of liquids once the nature of the pair potential *u*(*r*) is known. The comparison between simulation results and those of analytical models essentially allows us to test the models, whereas the comparison of simulation results with experimental results is the ultimate test to judge the efficiency of the pair potential. Subsequently, we described in minute detail the calculations of the thermodynamic properties in terms of the pair potential and pair correlation function. In particular, with the pressure *p* and the chemical potential *μ* to apply equilibrium conditions, it has been shown that we are in a position to determine theoretical estimates for the liquid-vapor coexistence curve. Finally, we got down to basics of the thermodynamic perturbation theory, before presenting the liquid-vapor coexistence curve

In a 1990 time warp, it became apparent that the usefulness of thermodynamic perturbation theory was on the decline compared to the more powerful simulation methods. This was primarily due to the rapid increase in the power of computers. At the same time, the integral equation theory enjoyed renewed popularity with the extensively employed concept of thermodynamic consistency (25), but this aspect goes beyond the scope of this short review. Incidentally, we recall that the thermodynamic consistency consists to adjusting the isothermal compressibility obtained by two different routes. Nevertheless, the thermodynamic perturbation theory remains undoubtedly the most tractable approach to predict the thermodynamic properties of liquids. Let us just quote few articles containing

As one would expect, the thermodynamic perturbation theory described previously is also important for mixtures. The thermodynamic properties established for studying pure liquids can be applied to binary mixtures, with only few modifications. Specifically, analytic

investigations of the HCY fluid with the integral equation theory (26).

with these various methods are available in literature (24).

**6. Concluding remarks**

for the HCY fluid.


**32** 

*U.S.A.* 

**Probing Solution Thermodynamics** 

*Department of Pharmaceutical Sciences, Washington State University* 

Solution microcalorimetry has entrenched itself as a major technique in laboratories concerned with studying the thermodynamics of chemical systems. Recent developments in the calorimeter marketplace will undoubtedly continue to popularize microcalorimeters as mainstream instruments. The technology of microcalorimetry has in turn benefited from this trend in terms of enhanced sensitivity, signal stability, physical footprint and userfriendliness. As the popularity of solution microcalorimeters has grown, so has an impressive body of literature on various aspects of microcalorimetry, particularly with respect to biophysical characterizations. The focus of this chapter is on experimental and analytical aspects of solution microcalorimetry that are novel or represent potential pitfalls. It is hoped that this information will aid bench scientists in the formulation and numerical analysis of models that describe their particular experimental systems. This is a valuable skill, since frustrations often arise from uninformed reliance on turnkey software that accompany contemporary instruments. This chapter will cover both differential scanning calorimetry (DSC) and isothermal titration calorimetry (ITC). It targets physical chemists, biochemists, and chemical engineers who have some experience in calorimetric techniques as well as nonlinear regression (least-square analysis), and are interested in quantitative

DSC measures the heat capacity (Cp) of a sample as the instrument "scans" up or down in temperature. For reversible systems, direct interpretation of the data in terms of thermodynamic parameters requires that chemical equilibrium be re-established much more rapidly than the scan rate. This can be verified by comparing data obtained at different scan rates. For transitions involving a change in molecularity (*e.g.*, self-association/dissociation, ligand binding/unbinding), reversibility can also be confirmed by the lack of hysteresis between heating and cooling experiments. The optimal scan rate is ultimately a compromise between the requirement for reversibility and the desire for reasonable throughput; typically this falls between 0.5 to 1.0 °C/min for most systems in dilute aqueous solutions.

The observed or apparent thermodynamics of solution systems generally include linked contributions from other solutes in addition to the species of interest. They include buffers,

thermodynamic characterizations of noncovalent interactions in solution.

**2. Differential scanning calorimetry** 

**2.1 Experimental conditions for DSC** 

**1. Introduction** 

**by Microcalorimetry** 

Gregory M. K. Poon

