Thermodynamics small systems

The time is perhaps not yet ripe, however, for introducing this kind of correction into calculations of pore size distribution the analyses, whether based on classical thermodynamics or statistical mechanics are being applied to systems containing relatively small numbers of molecules where, as stressed by Everett and Haynes, the properties of matter must exhibit wide fluctuations. A fuller quantitative assessment of the situation in very fine capillaries must await the development of a thermodynamics of small systems. Meanwhile, enough is known to justify the conclusion that, at the lower end of the mesopore range, the calculated value of r is almost certain to be too low by many per cent. 

T. L. Hill. Thermodynamics of Small Systems. Minneola Dover, 1994. 

The determination of the character and location of phase transitions has been an active area of research from the early days of computer simulation, all the way back to the 1953 Metropolis et al. [59] MC paper. Within a two-phase coexistence region, small systems simulated under periodic boundary conditions show regions of apparent thermodynamic instability [60] simulations in the presence of an explicit interface eliminate this at some cost in system size and equilibration time. The determination of precise coexistence boundaries was usually done indirectly, through the... 

For obvious reasons, we need to introduce surface contributions in the thermodynamic framework. Typically, in interface thermodynamics, the area in the system, e.g. the area of an air-water interface, is a state variable that can be adjusted by the observer while keeping the intensive variables (such as the temperature, pressure and chemical potentials) fixed. The unique feature in selfassembling systems is that the observer cannot adjust the area of a membrane in the same way, unless the membrane is put in a frame. Systems that have self-assembly characteristics are conveniently handled in a setting of thermodynamics of small systems, developed by Hill [12], and applied to surfactant self-assembly by Hall and Pethica [13]. In this approach, it is not necessary to make assumptions about the structure of the aggregates in order to define exactly the equilibrium conditions. However, for the present purpose, it is convenient to take the bilayer as an example. 

D. H. E. Gross, Microcanonical Thermodynamics, Phase Transitions in Small Systems, World Scientific Lecture Notes in Physics, World Scientific, Singapore, 2001. 

C. Bustamante, J. Liphardt, and F. Ritort, The nonequilibrium thermodynamics of small systems. Phys. Today 58, 43 8 (2005). 

It is simplistic to suppose that only an infinite system warrants macroscopic designation. We suspect that certain quite small systems (e.g., 1 mm3 of He gas) might be satisfactorily macroscopic for thermodynamic purposes. Let us frame the definition of macroscopic in more precise and operational terms that allow a realistic finite limit to be established. 

The oldest, most well-established chemical separation technique is precipitation. Because the amount of the radionuclide present may be very small, carriers are frequently used. The carrier is added in macroscopic quantities and ensures the radioactive species will be part of a kinetic and thermodynamic equilibrium system. Recovery of the carrier also serves as a measure of the yield of the separation. It is important that there is an isotopic exchange between the carrier and the radionuclide. There is the related phenomenon of co-precipitation wherein the radionuclide is incorporated into or adsorbed on the surface of a precipitate that does not involve an isotope of the radionuclide or isomorphously replaces one of the elements in the precipitate. Examples of this behavior are the sorption of radionuclides by Fe(OH)3 or the co-precipitation of the actinides with LaF3. Separation by precipitation is largely restricted to laboratory procedures and apart from the bismuth phosphate process used in World War II to purify Pu, has little commercial application. 

The micellization of surfactants has been described as a single kinetic equilibrium (10) or as a phase separation (11). A general statistical mechanical treatment (12) showed the similarities of the two approaches. Multiple kinetic equilibria (13) or the small system thermodynamics by Hill (14) have been frequently applied in the thermodynamics of micellization (15, 16, 17). Even the experimental determination of the factors governing the aggregation conditions of micellization in water is still a matter of considerable interest (18, 19) and dispute (20). 

Oversaturation — A thermodynamically unstable system that contains more of the dissolved material than would be dissolved by that solvent at equilibrium. It can also refer to a vapor of a compound whose partial pressure is higher than the vapor pressure of that compound at that temperature. Small particles can trigger the precipitation of dissolved material or the condensation of vapor in over saturated systems since they provide a suitable interface to start the formation of the new phase [i]. 

If the surfactant concentration in a macroemulsion is greatly increased, or if the monomer concentration is greatly reduced, a microemulsion results. Microemulsions are thermodynamically stable systems in which all of the monomer resides within the micelles. At high surfactant concentration, the micelles may form a bicontinuous network, rather than discrete micelles. Polymerization (with water- or oil-soluble initiator) of the monomer within a microemulsion is referred to as microemulsion polymerization. The particles produced in this way are extremely small, ranging from 10 to 100 nm. 

Thermodynamic properties of small systems. Phys. Rev. 124, 1673 (I96lb). 

A prediction of AE /AEq to within 0.1 kcal/mol may produce a AG /AGo, accurate to maybe 0.2 kcal/mol. This corresponds to a factor of 1.4 error (at T = 300 K) in the rate/equilibrium constant, which is poor compared to what is routinely obtained by experimental techniques. Calculating AG /AGq to within 1 kcal/mol is still only possible for fairly small systems. This corresponds to predicting the absolute rate constant, or the equilibrium distribution, to within a factor of References 1. H. Eyring, J. Chem. Phys., 3 (1934), 107. 2.1. N. Leyine, Physical Chemistry, McGraw-Hill, 1983 K. Lucas, Applied Statistical Thermodynamics, Springer-Verlag, 1991. ... 

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