Statistical thermodynamics pressure

From the third law of thermodynamics, the entiopy 5 = 0 at 0 K makes it possible to calculate S at any temperature from statistical thermodynamics within the hamionic oscillator approximation (Maczek, 1998). From this, A5 of formation can be found, leading to A/G and the equilibrium constant of any reaction at 298 K for which the algebraic sum of AyG for all of the constituents is known. A detailed knowledge of A5, which we already have, leads to /Gq at any temperature. Variation in pressure on a reacting system can also be handled by classical thermodynamic methods. 

For example, the measured pressure exerted by an enclosed gas can be thought of as a time-averaged manifestation of the individual molecules random motions. When one considers an individual molecule, however, statistical thermodynamics would propose its random motion or pressure could be quite different from that measured by even the most sensitive gauge which acts to average a distribution of individual molecule pressures. The particulate nature of matter is fundamental to statistical thermodynamics as opposed to classical thermodynamics, which assumes matter is continuous. Further, these elementary particles and their complex substmctures exhibit wave properties even though intra- and interparticle energy transfers are quantized, ie, not continuous. Statistical thermodynamics holds that the impression of continuity of properties, and even the soHdity of matter is an effect of scale. 

For a calculation of d. see R- H. Fowler. Statistical Thermodynamics. Second Edition, Cambridge University Press. 1956. p. 127. In Section 1.5a of Chapter 1 we defined the compressibility and cautioned that this compressibility can be applied rigorously only for gases, liquids, and isotropic solids. For anisotropic solids where the effect of pressure on the volume would not be the same in the three perpendicular directions, more sophisticated relationships are required. Poisson s ratio is the ratio of the strain of the transverse contraction to the strain of the parallel elongation when a rod is stretched by forces applied at the end of the rod in parallel with its axis. 

The formal thermodynamic analogy existing between an ideal rubber and an ideal gas carries over to the statistical derivation of the force of retraction of stretched rubber, which we undertake in this section. This derivation parallels so closely the statistical-thermodynamic deduction of the pressure of a perfect gas that it seems worth while to set forth the latter briefly here for the purpose of illustrating clearly the subsequent derivation of the basic relations of rubber elasticity theory. 

Mass discrimination with distillation effects. Let us assume that the isotope composition of an element is being measured by thermal ionization. This method consists in ionizing the sample atoms by evaporation on a metal filament. Statistical thermodynamics (e.g., Denbigh, 1968) tells us that, while vapor pressure is a function... 

In all hydrodynamic methods we have the effect of both the hydrodynamic and thermodynamic interactions and these do not contribute additively but are coupled. This explains why the theoretical treatment of [77] and of the concentration dependence of has been so difficult. So far a satisfactory result could be achieved only for flexible linear chains [3, 73]. Fortunately, the thermodynamic interaction alone can be measured by static scattering techniques (or osmotic pressure measurement) when the scattering intensity is extrapolated to zero scattering angle (forward scattering). Statistical thermodynamics demonstrate that this forward scattering is given by the osmotic compressibility dc/dn as [74,75]... 

Now that we have considered the calculation of entropy from thermal data, we can obtain values of the change in the Gibbs function for chemical reactions from thermal data alone as well as from equilibrium data. From this function, we can calculate equilibrium constants, as in Equations (10.22) and (10.90.). We shall also consider the results of statistical thermodynamic calculations, although the theory is beyond the scope of this work. We restrict our discussion to the Gibbs function since most chemical reactions are carried out at constant temperature and pressure. 

Figure 13.3. A P- V-T surface for a one-component system in which the substance contracts on freezing, such as water. Here Tj represents an isotherm below the triple-point temperature, 72 represents an isotherm between the triple-point temperature and the critical temperature, is the critical temperature, and represents an isotherm above the triple-point temperature. Points g, h, and i represent the molar volumes of sohd, hquid, and vapor, respectively, in equilibrium at the triple-point temperature. Points e and d represent the molar volumes of solid and liquid, respectively, in equihbrium at temperature T2 and the corresponding equilibrium pressure. Points c and b represent the molar volumes of hquid and vapor, respectively, in equilibrium at temperature and the corresponding equihbrium pressure. From F. W. Sears and G. L. Sahnger, Thermodynamics, Kinetic Theory, and Statistical Thermodynamics. 3rd ed., Addison-Wesley, Reading, MA, 1975, p. 31.

Transport properties are often given a short treatment or a treatment too theoretical to be very relevant. The notion that molecules move when driven by some type of concentration gradient is a practical and easily grasped approach. The mathematics can be minimized. Perhaps the most important feature of the kinetic theory of gases is the recognition that macroscopic properties such as pressure and temperature can be derived by suitable averages of the properties of individual molecules. This concept is an important precursor to statistical thermodynamics. Moreover, the notion of a distribution function as a general... 

Elementary and advanced treatments of such cellular functions are available in specialized monographs and textbooks (Bergethon and Simons 1990 Levitan and Kaczmarek 1991 Nossal and Lecar 1991). One of our objectives in this chapter is to develop the concepts necessary for understanding the Donnan equilibrium and osmotic pressure effects. We define osmotic pressures of charged and uncharged solutes, develop the classical and statistical thermodynamic principles needed to quantify them, discuss some quantitative details of the Donnan equilibrium, and outline some applications. 

As mentioned above, the primary focus of this chapter is on osmotic pressure and its basis in solution thermodynamics. We consider both classical and statistical thermodynamic interpretations of osmotic pressure. The next three sections are devoted to this. The last two sections describe osmotic effects in charged systems and a few applications of osmotic phenomena. 

From basic statistical-thermodynamics arguments [60] the mean molecular speed in a gas can be determined approximately from the relationship between pressure and mass density as... 

It is useful to be able to express the pressure in terms of the partition function. (The result will be used subsequently in the statistical thermodynamic expression derived for the enthalpy.)... 

Transition-state theory is based on the assumption of chemical equilibrium between the reactants and an activated complex, which will only be true in the limit of high pressure. At high pressure there are many collisions available to equilibrate the populations of reactants and the reactive intermediate species, namely, the activated complex. When this assumption is true, CTST uses rigorous statistical thermodynamic expressions derived in Chapter 8 to calculate the rate expression. This theory thus has the correct limiting high-pressure behavior. However, it cannot account for the complex pressure dependence of unimolecular and bimolecular (chemical activation) reactions discussed in Sections 10.4 and 10.5. 

The virial isotherm equation, which can represent experimental isotherm contours well, gives Henry s law at low pressures and provides a basis for obtaining the fundamental constants of sorption equilibria. A further step is to employ statistical and quantum mechanical procedures to calculate equilibrium constants and standard energies and entropies for comparison with those measured. In this direction moderate success has already been achieved in other systems, such as the gas hydrates 25, 26) and several gas-zeolite systems 14, 17, 18, 27). In the present work AS6 for krypton has been interpreted in terms of statistical thermodynamic models. 

Hydrate Nonstoichiometry. The cause of the nonstoichiometric properties of hydrates has been considered. Evidence for the view that only a fraction of the cavities need to be occupied is obtained from both the experimental observations of variation in composition, and the theoretical success of the statistical thermodynamic approach of van der Waals and Platteeuw (1959) in Chapter 5. Typical occupancies of large cavities are greater than 95%, while occupancy of small cavities vary widely depending on the guest composition, temperature, and pressure. 

In this section two prediction techniques are discussed, namely, the gas gravity method and the Kvsi method. While both techniques enable the user to determine the pressure and temperature of hydrate formation from a gas, only the KVSI method allows the hydrate composition calculation. Calculations via the statistical thermodynamics method combined with Gibbs energy minimization (Chapter 5) provide access to the hydrate composition and other hydrate properties, such as the fraction of each cavity filled by various molecule types and the phase amounts. 

Because the two prediction techniques of this chapter were determined over half a century ago, they apply only to si and sH hydrates, without consideration of the more recent sH, which always contains a heavier component. For structure H equilibrium, only the statistical thermodynamics method of Chapter 5 is available for prediction of hydrate pressure, temperature, and composition. 

Section 5.1 presents the fundamental method as the heart of the chapter— the statistical thermodynamics approach to hydrate phase equilibria. The basic statistical thermodynamic equations are developed, and relationships to measurable, macroscopic hydrate properties are given. The parameters for the method are determined from both macroscopic (e.g., temperature and pressure) and microscopic (spectroscopic, diffraction) measurements. A Gibbs free energy calculation algorithm is given for multicomponent, multiphase systems for comparison with the methods described in Chapter 4. Finally, Section 5.1 concludes with ab initio modifications to the method, along with an assessment of method accuracy. 

The Center for Hydrate Research has been conducting hydrate experiments for over 30 years in efforts to improve flow assurance strategies. The first statistical thermodynamic model for hydrates was developed in 1959 and involved many assumptions, including the assumption that volume is constant. This model predicted hydrate formation temperatures and pressures reasonably well at temperatures near the ice point and at low pressures. However, as industry moves to deeper waters, there is a need for a hydrate model that can predict hydrate formation at higher temperatures and pressures. 

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