Virial coefficients evaluation

If /i = 1 ahn, it is sufficient to retain only the first temi on the right. However, one does not need to know the virial coefficients one may simply use volumetric data to evaluate the integral. 

For dilute, teal gases, where ternary and higher collisions can be neglected, the angle of deflection can be employed to evaluate a number of physical properties. Of course appropriate distributions of the values of g and b must be introduced. The resulting expressions for the virial coefficients and the transport properties (viscosity, diffusion and thermal conductivity) are quite complicated. The interested reader is referred to advanced books on this subject... 

Actually we are dealing here with a system where the difference in size of the two components is appreciable (p > 0.1) and this might well go beyond the limit of validity of the APM. It is, however, interesting to note that an evaluation of eJB and r B for the present system has been made by Thomaes et a/.33 from second virial coefficients. This leads roughly to ... 

Archibald (1947) showed that measurement of c and dc/dr at the cell boundaries permit Molecular weight determination at any stage in the equilibrium process. However, when measurements are made early enough, before the molecular species have time to redistribute in the cell, the weight-average Molecular weight and second virial coefficient can be evaluated. In practice, measurements can be made... 

It can be shown that the virial type of activity coefficient equations and the ionic pairing model are equivalent, provided that the ionic pairing is weak. In these cases, it is in general difficult to distinguish between complex formation and activity coefficient variations unless independent experimental evidence for complex formation is available, e.g., from spectroscopic data, as is the case for the weak uranium(VI) chloride complexes. It should be noted that the ion interaction coefficients evaluated and tabulated by Cia-vatta [10] were obtained from experimental mean activity coefficient data without taking into account complex formation. However, it is known that many of the metal ions listed by Ciavatta form weak complexes with chloride and nitrate ions. This fact is reflected by ion interaction coefficients that are smaller than those for the noncomplexing perchlorate ion (see Table 6.3). This review takes chloride and nitrate complex formation into account when these ions are part of the ionic medium and uses the value of the ion interaction coefficient (m +,cio4) for (M +,ci ) (m +,noj)- Io... 

Because all the above deductions are valid for infinite low polymer concentrations but practical measurements have to be carried out at finite values of c, it is necessary to include the second virial coefficient A2. Thus the equation according to which evaluation of light-scattering experiments can be done is ... 

Nonlinear regression analysis of the dependence of c2(r) upon (r), a transform of the radial distance i leads to evaluation of the reference thermodynamic activity, Mzzz(rf), and the osmotic second virial coefficient, Au/Mi, expressed on a weight basis (litre/g) rather than a molar basis (litre/mol). Furthermore, the values of Miy v/ and ps can be obtained by curve-fitting the sedimentation equilibrium distribution for low biopolymer concentrations (M fr) cf for all r) to the equations (5.36) and (5.37) in order to deduce the quantity [Mj( 1 - v/ps)] from the coefficient of the exponent (Winzor et al., 2001 Deszczynski et al., 2006). 

Schmidt, D.G., Payens, T.A. (1972). The evaluation of positive and negative contributions to the second virial coefficient of some milk proteins. Journal of Colloid and Interface Science, 39, 655-662. 

Since the limiting value of Tr/mPshown in Figure 3.6 applies to these data also, it is possible to evaluate the second virial coefficient at these pH levels. Evaluate B and z, the effective protein charge, at the pH values shown. (Note that a slight variation in NaCl concentrations at these different pH levels should be taken into account for a more accurate determination of z.)... 

The diffuse part of the double layer is of little concern to us at this point. Chapters 11 and 12 explore in detail various models and phenomena associated with the ion atmosphere. At present it is sufficient for us to note that the extension in space of the ion atmosphere may be considerable, decreasing as the electrolyte content of the solution increases. As micelles approach one another in solution, the diffuse parts of their respective double layers make the first contact. This is the origin of part of the nonideality of the micellar dispersion and is reflected in the second virial coefficient B as measured by osmometry or light scattering. It is through this connection that z can be evaluated from experimental B values. 

Molecular weight from osmotic pressure measurements Degree of polymerization and molecular weight distribution Excluded volume from osmotic pressure measurements Theta temperature from second virial coefficient data Evaluation of charges on macroions from osmotic pressures... 

It seems to me that we can scarcely progress in our understanding of the structural and kinetic effects of the H-bond without knowing the AG and AH terms involved, so I intend to discuss some methods of determining them. The references will provide simple examples of the methods mentioned. The most significant AG and AH values are those evaluated from equilibrium measurements in the gas phase—either by classical vapour density measurements, the second virial coefficient [1], or from, spectroscopic, specific heat or thermal conductance [2], or ultrasonic absorptions [3]. All these methods essentially measure departures from the ideal gas laws. The second virial coefficient provides a measure of the equilibrium constant for the formation of collision dimers in the vapour as was emphasized by Dr. Rowlinson in the discussion, this factor is particularly significant as only the monomer-dimer interaction contributes to it. 

The second virial coefficients of a number of n-alkanes (from butane to hexadecane) are evaluated by using the RIS model. It Is shown that a simple site-site potential model is able to reproduce the second virial coefficient of several n-alkanes in a wide range of temperature. 

The virial coefficients B(T), C(T), D(T),... are functions of temperature only. Although these coefficients might be treated simply as empirical fitting parameters, they are actually deeply connected to the theory of intermolecular clustering, as developed by J. E. Mayer (Sidebar 13.5). More specifically, the second virial coefficient B(T) is related to the intermolecular potential for pairs of molecules, the third virial coefficient C(T) to that for triples of molecules, and so forth. For example, knowledge of the intermolecular pair potential V(R) (see Sidebar 2.8) allows B T) to be explicitly evaluated by statistical mechanical methods as... 

All terms except the.last are known from Corner s equation. The last term, corresponding to the 4th virial coefficient, was evaluated by analysis of deton vel for PETN as a function of loading density. It turns out that iIf can be absorbed in the 3rd virial coefficient C... 

The differentiation of the term containing the second virial coefficient may present some difficulty. In a specific case the summation could be expanded and then the differentiation carried out. In the general case we must evaluate... 

Experimentally, A2 can be evaluated by determining the initial slope of n/(RTc) plotted against c. However, the estimation of A3 and the higher virial coefficients is not a simple matter for experimentalists. Available experimental information about the virial coefficients is largely limited to A2. 

Using eqs 7,10,11, and 13, one can determine the charge on an apoferritin molecule at any pH, provided that the values of the dissociation constants Ku and An, are known. Further, using eqs 1 and 2, one can evaluate the second virial coefficient for the interaction between two particles at constant surface charge. 

At pressures up to 40 tons/in2, corresponding to density of 0.35 g/cc, only the first 3 terms in the equation need be kept. Thus the pressure dependence of the thermodynamic props can be evaluated from a knowledge of the 2nd 3rd virial coeffs of the various gaseous products. Tables are presented which cover the range 1600° - 4000° K, and which have found considerable application in internal ballistics. These tables give covolumes of propellants with a systematic error of less than 5%- The basis of Corner s theory is the expression of the 2nd virial coefficient of a gas as a simple function of the parameters of the intermolecular field... 

R = gas constant and T = absolute temperature). The so-called first virial coefficient, Bi, is simply equal to the reciprocal of the molecular weight, M. The evaluation of the second virial coefficient, Bi, has become one of the principal theoretical developments in recent years. In his theory of polymer solutions Flory (1953) has shown that Bi is proportional to a term (1 — 6/T), which vanishes T = 0. Accordingly 0 may be considered as the ideal temperature at which the above equation is reduced to the well-known van t Hoff s law, i.e.,... 

Named functions, HRB(TR,PR,OMEGA) and SRB(TR,PR,OMEGA), forevaluation of Yi iRTc and S /R by the generalized virial-coefficient correlation were described in Sec. 6.7. Similarly, we introduce here a function named PHIB(TR,PR,OMEGA) for evaluation of

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The Debye-Hiickel potential needs modification for high values of ze. The second virial coefficient suitable for this case has been evaluated... 

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