Second virial coefficient, enthalpy

Despite the importance of mixtures containing steam as a component there is a shortage of thermodynamic data for such systems. At low densities the solubility of water in compressed gases has been used (J, 2 to obtain cross term second virial coefficients Bj2- At high densities the phase boundaries of several water + hydrocarbon systems have been determined (3,4). Data which would be of greatest value, pVT measurements, do not exist. Adsorption on the walls of a pVT apparatus causes such large errors that it has been a difficult task to determine the equation of state of pure steam, particularly at low densities. Flow calorimetric measurements, which are free from adsorption errors, offer an alternative route to thermodynamic information. Flow calorimetric measurements of the isothermal enthalpy-pressure coefficient pressure yield the quantity 4>c = B - TdB/dT where B is the second virial coefficient. From values of obtain values of B without recourse to pVT measurements. 

We recall from the transition between Equations (74) and (75) that RT is the enthalpy contribution to the second virial coefficient. In the present context, we designate this A//. ... 

As pointed out in Chapter III, Section 1 some specific diluent effects, or even remnants of the excluded volume effect on chain dimensions, may be present in swollen networks. Flory and Hoeve (88, 89) have stated never to have found such effects, but especially Rijke s experiments on highly swollen poly(methyl methacrylates) do point in this direction. Fig. 15 shows the relation between q0 in a series of diluents (Rijke assumed A = 1) and the second virial coefficient of the uncrosslinked polymer in those solvents. Apparently a relation, which could be interpreted as pointing to an excluded volume effect in q0, exists. A criticism which could be raised against Rijke s work lies in the fact that he determined % in a separate osmotic experiment on the polymer solutions. This introduces an uncertainty because % in the network may be different. More fundamentally incorrect is the use of the Flory-Huggins free enthalpy expression because it implies constant segment density in the swollen network. We have seen that this means that the reference dimensions excluded volume effect. 

Abbreviations A H Huggins coefficient M, molar mass R, radius of gyration RD, core radius p, association number AmcH°, standard enthalpy of micellization, AmlcG°, standard Gibbs energy of micellization A2, second virial coefficient. Ru, hydrodynamic radius. 

For true compatability of solute and solvent, matching of all these partial solubility parameters (i.e., 8, Bp, 8 ) is necessary. The total solubility parameter can be easily calculated [1, p. 307] finm the material s enthalpy of vaporization, vapor pressure as a function of temperature, surface tension, thermal expansion coefficient, critical pressure, and second virial coefficient of its vapor, as well as by calculating its value for the chemical structure of the material. For the calculation of the Hildebrand solubility parameter fi om chemical structure, we use Small s [58] equation ... 

The enthalpy of micelle formation of various mixed sodium dodecylsulfate (NaDDS) and sodium deoxycholate (NaDOC) systems was measured by calorimeter In aqueous systems. The heat of micelle formation, AH, showed a maximum around NaDDS NaDOC molar ratio 1. These data are analyzed In comparison to the aggregation number of mixed micelles and the second virial coefficient, Bg. 

This study is a continuation of our previous investigations, in which the aggregation phenomena of surfactant molecules (amphiphiles) in aqueous media to form micelles above the critical micelle concentration (c.m.c.) has been described based on different physical methods (11-15). In the current literature, the number of studies where mixed micelles have been investigated is scarcer than for pure micelles (i.e., mono-component). Further, in this study we report various themodynamlc data on the mixed micelle system, e.g., ci H25soi4Na (NaDDS) and sodium deoxycholate (NaDOC), enthalpy of micelle formation (by calorimetry), and aggregation number and second virial coefficient (by membrane osmometry) (1 6). 

Hence, if we argue that the alkyl group of NaDDS is more hydrophobic than NaDOC, then we should have expected endothermic increase with addition of NaDOC. However, from Figure 2 we find the reverse. Thus we conclude that, due to the steric hindrance in the packing of NaDDS and NaDOC alkyl parts, the enthalpy of mixed micelles behaves non-ideally. These conclusions are in agreement with the data of second virial coefficient, B2 (Figure 2). 

Derive equations to calculate the enthalpy departure using each of the following methods (a) the ideal gas equation, (b) the virial equation of state truncated after the second virial coefficient, (c) the Soave-Redlich-Kwong equation of state. 

By the mid-1970s, the existence of CH- -X H-bonds had achieved general acceptance [15], with verification coming from a range of different measurements that included vapor pressure, azeotrope formation, second virial coefficient, solubility, freezing-point diagrams, enthalpies of mixing, dipole moments, viscosity, refractive index, and electronic, vibrational, and NMR... 

Equation 25 was developed from an empirical representation of thg second virial coefficient correlation of Pitzer and Curl (I) parameter b was left unchanged at its classical value of 0.0866. Because of the substantial improvement in the prediction of and its temperature derivatives for nonsimple fluids, the Barner modification of the RK equation gave improved estimates of enthalpy deviations for nonpolar vapors and for vapor-phase mixtures of hydrocarbons. However, the new equation was unsuitable for fugacity calculations. 

The term m = 0.74048 Vm°/Vm = 1/6 7rN0cJm/Vm> where Vm° is the close-packed volume, N0 is the Avogadro number, and Vm is the molar volume of the system. V° is a simple function of the temperature (T) (10) with a characteristic value V°° at T = 0 K. The last term in Equation 12 was introduced by Alder et al. (II). Dnm are 24 universal constants common for all substances whose radial and higher distribution functions are the same functions of u/kT and the reduced density p = V°/V. As shown by Chen and Kreglewski (10) and Simnick, Lin, and Chao (12), Equation 12 is the most accurate known equation with four characteristic constants a, V°° (V° at T = 0 K), u°/k, and rj/k (see Equations 13 and 14). They also have shown (10) that in order to obtain agreement with second virial coefficient data of the gas and the internal energy or the enthalpy of the liquid, it is necessary to assume that u(r) is a function of T as required by the theory of noncentral forces between nonspherical molecules (13)... 

Equation (7-38) is an expression for the enthalpy of a pure real gas that has been evaluated expUcitly up to terms proportional to p. Evaluation of the enthalpy thus demands information concerning the dependence of the second virial coefficient on temperature and thermal data on the behavior of the heat capacity at constant pressure in the limit of zero pressure. In addition, there is an undetermined constant of integration. We note that the enthalpy of an ideal gas at T and p is given by h T). Thus the zero-pressure limit of the enthalpy of a pure real gas is the same as the enthalpy of a corresponding ideal gas. 

Values for the virial coefficients are derived from experimental measurements which can be conveniently classified as follows low pressnre p-V-T measnrements high pressnre p-V-T measnrements speed of sonnd measurements vaponr pressnre and enthalpy of vaporization measnrements refractive index/dielectric constant measurements and Jonle-Thomson experiments. These will be discussed in Chapter 1.2, and methods of data evalnation described in Chapter 1.5. Much attention has been paid to the correlation of virial coefficient data and the more satisfactory methods are considered in Chapter 1.3, together with a brief discussion of the theoretical calculation of the second virial coefficient from pair potential energy functions which have been derived a priori or by consideration of other dilute gas properties. So far, this calculation is only applicable to molecules with a spherically symmetric intermolecular potential energy function, for which... 

Second virial coefficients for a large number of organic compounds have been calculated from enthalpies of vaporization and vapour pressure data [81-hos/sco] at the former U.S. Bureau of Mines Research Center, Bartlesville by J.P. McCullough, D.W. Scott and G. Waddington. The exact Clapeyron equation... 

The new supplementary volume is again divided into the seven chapters as used before (1) Introduction, (2) Vapor-Liquid Equilibrium (VLE) Data and Gas Solubilities of Copolymer Solutions, (3) Liquid-Liquid Equilibrium (LLE) Data of Copolymer Solutions, (4) High-Pressure Fluid Phase Equilibrium (HPPE) Data of Copolymer Solutions, (5) Enthalpy Changes for Copolymer Solutions, (6) PVT Data of Copolymers and Solutions, and (7) Second Virial Coefficients (A2) of Copolymer Solutions. Finally, appendices quickly route the user to the desired datasets. 

Gas chromatography is primarily an analytical separation technique. However, since the basic process is an equilibration of a solute between two immiscible phases, the chromatographic technique may be used to measure such physical properties as activity coefficients, second virial coefficients of gas mixtures, partition coefficients, adsorption and partition isotherms, and complex formation constants. Other properties which can be measured with less accuracy, from secondary measurements or from temperature variation studies, include surface areas, heats of adsorption, and excess enthalpies and excess entropies of solution. A number of reviews and discussions on these measurements have appeared in the literature. The present work is restricted to a review of activity-coefficient measurements. 

Although in favourable cases S can be determined to 1 to 2 cm mol the uncertainty in B depends on the precision with which the virial coefficients of the pure components are known. On the other hand, for some applications, such as the calculation of enthalpies of mixing or the correction of vapour-liquid equilibria data, is the quantity of interest. Pressure-change measurements are not likely to be of utility for obtaining information about interaction third virial coefficients because the contribution of the second virial coefficients to the... 

W. M. Haynes and R. D. Goodwin, Thermophysical Properties of Normal Butane from 135 to 700 K at Pressures to 70 MPa, U.S. Dept, of Commerce, National Bureau of Standards Monograph 169, 1982, 192 pp. Tabulated data include densities, compressibility factors, internal energies, enthalpies, entropies, heat capacities, fugacities and more. Equations are given for calculating vapor pressures, liquid and vapor densities, ideal gas properties, second virial coefficients, heats of vaporization, liquid specific heats, enthalpies and entropies. 

Some implicit databases are provided within the Polymer Handbook by Schuld and Wolf or by Orwoll and in two papers prepared earlier by Orwoll. These four sources list tables of Flory s %-function and tables where enthalpy, entropy or volume changes, respectively, are given in the literature for a large number of polymer solutions. The tables of second virial coefficients of polymers in solution, which were prepared by Lechner and coworkers (also provided in die Polymer Handbook), are a valuable source for estimating the solvent activity in the dilute polymer solution. Bonner reviewed vapor-liquid equilibria in concentrated polymer solutions and listed tables containing temperature and concentration ranges of a certain number of polymer solutions." Two CRC-handbooks prepared by Barton list a larger number of fliermodynamic data of polymer solutions in form of polymer-solvent interaction or solubility parameters." ... 

The virial equation can not only describe the PvT behavior of gases, but also the different residual functions. Both forms (Leiden form and Berlin form) are well-suited for this purpose, but most of the thermodynamic functions can be derived easier using the Berlin form, because this form is volume explicit. Volume explicit means that the equation can be solved explicitly for vas a function of T and P. As an example, the calculations of the fugacity coefficient and of the residual function of the enthalpy are shown, using the virial equation in the Berlin form truncated after the second virial coefficient B. 

Thus we continued with numerical methods for the fugacity, entropy, and enthalpy functions (13), although we did present an empirical equation for the second virial coefficient (14). This work was done by Bob Curl he did an excellent job but found the almost interminable graphical work very tiresome. Thus I was pleased that the British Institution of Mechanical Engineers Included Curl in the award of their Clayton Prize for this work. A fifth paper with Hultgren (15) treated mixtures on a pseudocritical basis, and a sixth with Danon (16) related Kihara core sizes to the acentric factor. 

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