Thermodynamics scaling

L. Schafer, T. A. Witten. Renormalization field theory of polymer solutions. I. Scaling laws. J Chem Phys 66 2121-2130, 1977 A. Knoll, L. Schafer, T. A. Witten. The thermodynamic scaling function of polymer solution. J Physique 42 161-m, 1981. 

It was Lord Kelvin who recognized that Carnot s hypothetical engine was of fundamental importance, and used it to define a thermodynamic scale of temperature that has become known as the Kelvin temperature. He set the thermodynamic temperature T of the reservoirs proportional to the amount of heat exchanged at each that is. 

The ITS-90 scale is designed to give temperatures T90 that do not differ from the Kelvin Thermodynamic Scale by more than the uncertainties associated with the measurement of the fixed points on the date of adoption of ITS-90 (January 1, 1990), to extend the low-temperature range previously covered by EPT-76, and to replace the high-temperature thermocouple measurements of IPTS-68 with platinum resistance thermometry. The result is a scale that has better agreement with thermodynamic temperatures, and much better continuity, reproducibility, and accuracy than all previous international scales. 

Since we are interested in this chapter in analyzing the T- and P-dependences of polymer viscoelasticity, our emphasis is on dielectric relaxation results. We focus on the means to extrapolate data measured at low strain rates and ambient pressures to higher rates and pressures. The usual practice is to invoke the time-temperature superposition principle with a similar approach for extrapolation to elevated pressures [22]. The limitations of conventional t-T superpositioning will be discussed. A newly developed thermodynamic scaling procedure, based on consideration of the intermolecular repulsive potential, is presented. Applications and limitations of this scaling procedure are described. 

Oneil KT, Degrade WF (1990) A thermodynamic scale for the hehx-forming tendencies of the commonly occurring amino-acids. Science 250 646-651... 

Sufficient stability of the hydrocarbon ions, as the salt or in the solution, is an obvious prerequisite for these procedures, and, in practice, selecting or designing the stable ions and choosing a proper solvent are tasks of primary importance. As an ordinary stability index for the ions, thermodynamic scales referred to the water molecule, i.e. p CR+ and pKa values, are chosen for the carbocation and carbanion, respectively. 

Definitions - Activity versus Concentration Thermodynamic Scales... 

All equilibrium constants in the present discussion are based on the concentration (not activity) scale. This is a perfectly acceptable thermodynamic scale, provided the ionic strength of the solvent medium is kept fked at a reference level (therefore, sufficiently higher than the concentration of the species assayed). This is known as the constant ionic medium thermodynamic state. Most modern results are determined at 25 °C in a 0.15 M KCl solution. If the ionic strength is changed, the ionization constant may be affected. For example, at 25 °C and 0.0 M ionic strength, the pXj of acetic acid is 4.76, but at ionic strength 0.15 M, the value is 4.55 [24]. 

Valleau, J. P., Thermodynamic scaling methods in Monte Carlo and their application to phase equilibria, Monte Carlo Methods Chem. Phys. 1999,105, 369-404... 

Kiyohara, K. Spyriouni, T. Gubbins, K. E. Panagiotopoulos, A. Z., Thermodynamic scaling Gibbs ensemble Monte Carlo a new method for determination of phase coexistence properties of fluid, Mol. Phys. 19%, 89, 965-974. 

The XVIII Conference Generale des Poids et Mesures adopts the version of the International Temperature Scale ITS-90 The ITS-90 provides die best to-date practical approximation of die thermodynamic scale and offers a reproducibility that is better than the thermodynamic scale. 

Although the two scales are identical numerically, their conceptual bases are different. The ideal gas scale is based on the properties of gases in the limit of zero pressure, whereas the thermodynamic scale is based on the properties of heat engines in the limit of reversible operation. That we can relate them so satisfactorily is an illustration of the usefulness of the concepts so far defined. 

The ITS is an artifact scale, designed to relate temperature measurements made with practicable instruments as closely as possible to the thermodynamic scale. The scale is established and controlled by the International Committee of Weights and Measures (BIPM) through its Consultative Committee on Thermometry, which was established in 1937. The BIPM itself is established to maintain and implement the Treaty of the Meter, to which most nations of the wodd subscribe thus the ITS has not only scientific but legal status in most nations. Within nations, the Temperature Scale is maintained by national standards establishments, eg, in the United States the National Institute for Standards and Technology (NIST), in England the National Physical Laboratory (NPL), and in Germany the Physikalisch-Technische Bundesanstalt (PTB). 

The relation between the Centigrade and Kelvin thermodynamic scales is determined by... 

Table HI compiles MC results obtained over the years for the critical temperature and critical density of the RPM. Table in includes also results from the cluster calculations of Pitzer and Schreiber [141]. In a critical assessment of earlier work [40, 141, 179-181, 246], Fisher deduced in 1994 that T = 0.052-0.056 and p = 0.023-0.035 represent the best values [15]. Since then, however, the situation has substantially changed. Caillol et al. [53,247] performed simulations of ions on the surface of a four-dimensional hypersphere and applied finite-size corrections. Valleau [248] used his thermodynamic-scaling MC for systems with varying particle numbers to extract the infinite-size critical parameters. Orkoulas and Panagiotopoulos [52] performed grand canonical simulations in conjunction with a histogram technique. All studies indicate an insufficient treatment of finite-size effects in earlier work. While their results do not agree perfectly, they are sufficiently close to estimate T = 0.048-0.05 and p = 0.07-0.08, as already quoted in Eq. (6). Critical points of some real Coulombic systems match quite well to these figures [5]. The coexistence curve derived by Orkoulas and Panagiotopoulos [52] is displayed in Fig. 9.

Fig. 14.3. The thermodynamic scaling function in the excluded volume limit V (s), divided by the excluded volume overlap s. The broken line gives the asymptotic power law...

Kelvin (degree Kelvin) K Defined in the thermodynamic scale by assigning 273.16 K to the triple point of water (freezing point, 273.15 K = 0°C)... 

Krylov, D., Mikhailenko, I., and Vinson, C. (1994). A thermodynamic scale for leucine zipper stability and dimerization specificity e and g interhelical interactions. EMBOJ. 13, 2849-2861. 

The misconception rests on a confusion between the thermodynamic scale of activities (which measures the tendency of an element to react) and the concentration scale for free, hydrated, ions. Consider for example the not uncommon situation where the free... 

Computer simulation is invariably conducted on a model system whose size is small on the thermodynamic scale one typically has in mind when one refers to phase diagrams. Any simulation-based study of phase behavior thus necessarily requires careful consideration of finite-size effects. The nature of these effects is significantly different according to whether one is concerned with behavior close to or remote from a critical point. The distinction reflects the relative sizes of the linear dimension L of the system—the edge of the simulation cube, and the correlation length —the distance over which the local configurational variables are correlated. By noncritical we mean a system for which L E, by critical we mean one for which L [Pg.46]

The Kelvin scale is thus defined in terms of an ideal reversible heat engine. At first such a scale does not appear to be practical, because all natural processes are irreversible. In a few cases, particularly at very low temperatures, a reversible process can be approximated and a temperature actually measured. However, in most cases this method of measuring temperatures is extremely inconvenient. Fortunately, as is proved in Section 3.7, the Kelvin scale is identical to the ideal gas temperature scale. In actual practice we use the International Practical Temperature Scale, which is defined to be as identical as possible to the ideal gas scale. Thus, the thermodynamic scale, the ideal gas scale, and the International Practical Temperature Scale are all consistent scales. Henceforth, we use the symbol T for each of these three scales and reserve the symbol 9 for any other thermodynamic scale. 

This equation is identical to Equation (3.11). Therefore, the Kelvin thermodynamic scale and the ideal gas scale become identical when the temperature of the triple point of water is assigned the value of 273.16 K. 

Thermal conductivity is expressed in several different internationally recognized ways. One method of expressing thermal conductivity (A) is in terms of the heat flux under steady conditions per square meter for one meter of thickness of one degree Kelvin difference in temperature. Kelvin is a thermodynamic scale and is centigrade starting at absolute zero. 

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