Second virial coefficient determination

Selected entries from Methods in Enzymology [vol, page(s)] Association constant determination, 259, 444-445 buoyant mass determination, 259, 432-433, 438, 441, 443, 444 cell handling, 259, 436-437 centerpiece selection, 259, 433-434, 436 centrifuge operation, 259, 437-438 concentration distribution, 259, 431 equilibration time, estimation, 259, 438-439 molecular weight calculation, 259, 431-432, 444 nonlinear least-squares analysis of primary data, 259, 449-451 oligomerization state of proteins [determination, 259, 439-441, 443 heterogeneous association, 259, 447-448 reversibility of association, 259, 445-447] optical systems, 259, 434-435 protein denaturants, 259, 439-440 retroviral protease, analysis, 241, 123-124 sample preparation, 259, 435-436 second virial coefficient [determination, 259, 443, 448-449 nonideality contribution, 259, 448-449] sensitivity, 259, 427 stoichiometry of reaction, determination, 259, 444-445 terms and symbols, 259, 429-431 thermodynamic parameter determination, 259, 427, 443-444, 449-451. 

Nagasawa and Takahashi, 1972). More specifically, the value of the second virial coefficient determines the excess chemical potential, juE (also known as the excess partial molar Gibbs free energy), which characterizes the formation of biopolymer-solvent and biopolymer-biopolymer pair contacts ... 

It is important to note that the values of the second virial coefficients determined by light scattering are weight-average quantities. The general expression for the second virial coefficient of a polydisperse polymer is as follows (Casassa, 1962) ... 

Our 2nd and 3rd law analysis of four sets of vapor pressure data produces the following results. The second virial coefficient determined by Osborne et al. (5) is used to convert all four vapor pressure data sets to ideal gas fugacity. 

This technique has been considered separately becanse of recent significant developments which have led to p-V-T measurements of the highest accuracy. Historically, accurate gas density measurements were made at different pressnres to remove the effects of molecular interaction by extrapolation to zero pressure and hence to determine atomic weights. Second virial coefficients were derived from these data but the results for organic vaponrs had large uncertainties dne to adsorption. Gas balances have been nsed specifically for second virial coefficient determination [82-zam/ste]. 

Answer by Author Analytical calculations were made with and without quadrupole contributions on the second virial coefficients determined from the Lennard-Jones 6-12 and Kihara potentials. However, the correlation between the experimental and analytical results were superior when the quadrupole contributions were neglected, except for the Kihara potential at 190 K. Therefore, the results presented here do not include these contributions. 

In the next section we shall describe the use of Eq. (8.83) to determine the number average molecular weight of a polymer, and in subsequent sections we shall examine models which offer interpretations of the second virial coefficient. 

Binary interaction parameters are determined for each pq pair p q) from experimental data. Note that = k and k = k = 0. Since the quantity on the left-hand side of Eq. (4-305) represents the second virial coefficient as predicted by Eq. (4-231), the basis for Eq. (4-305) lies in Eq. (4-183), which expresses the quadratic dependence of the mixture second virial coefficient on mole fraction. 

Here eK, gk are the force constants for the pure solute K, which can be determined from measurements of its second virial coefficient, and q, oq are similar, but as yet unknown, constants characteristic for the / -hydroquinone lattice. 

Theta temperature (Flory temperature or ideal temperature) is the temperature at which, for a given polymer-solvent pair, the polymer exists in its unperturbed dimensions. The theta temperature, , can be determined by colligative property measurements, by determining the second virial coefficient. At theta temperature the second virial coefficient becomes zero. More rapid methods use turbidity and cloud point temperature measurements. In this method, the linearity of the reciprocal cloud point temperature (l/Tcp) against the logarithm of the polymer volume fraction (( )) is observed. Extrapolation to log ( ) = 0 gives the reciprocal theta temperature (Guner and Kara 1998). 

Theta temperature is one of the most important thermodynamic parameters of polymer solutions. At theta temperature, the long-range interactions vanish, segmental interactions become more effective and the polymer chains assume their unperturbed dimensions. It can be determined by light scattering and osmotic pressure measurements. These techniques are based on the fact that the second virial coefficient, A2, becomes zero at the theta conditions. 

Special care has to be taken if the polymer is only soluble in a solvent mixture or if a certain property, e.g., a definite value of the second virial coefficient, needs to be adjusted by adding another solvent. In this case the analysis is complicated due to the different refractive indices of the solvent components [32]. In case of a binary solvent mixture we find, that formally Equation (42) is still valid. The refractive index increment needs to be replaced by an increment accounting for a complex formation of the polymer and the solvent mixture, when one of the solvents adsorbs preferentially on the polymer. Instead of measuring the true molar mass Mw the apparent molar mass Mapp is measured. How large the difference is depends on the difference between the refractive index increments ([dn/dc) — (dn/dc)A>0. (dn/dc)fl is the increment determined in the mixed solvents in osmotic equilibrium, while (dn/dc)A0 is determined for infinite dilution of the polymer in solvent A. For clarity we omitted the fixed parameters such as temperature, T, and pressure, p. 

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... 

Of the preponderance of small ions, the colligative properties of polyelectrolytes in ionising solvents measure counterion activities rather than Molecular weight. In the presence of added salt, however, correct Molecular weights of polyelectrolytes can be measured by membrane osmometry, since the small ions can move across the membrane. The second virial coefficient differs from that previously defined, since it is determined by both ionic and non-ionic polymer-solvent interactions. 

Several techniques are available for measuring values of interaction second virial coefficients. The primary methods are reduction of mixture virial coefficients determined from PpT data reduction of vapor-liquid equilibrium data the differential pressure technique of Knobler et al.(1959) the Bumett-isochoric method of Hall and Eubank (1973) and reduction of gas chromatography data as originally proposed by Desty et al.(1962). The latter procedure is by far the most rapid, although it is probably the least accurate. 

Young, C.L. Cruickshank, A.J.B. Gainey, B.W. Hicks, C.P. Letcher, T.M. Moody, R.W. Gas-Liquid Chromatographic Determination of Cross-Term Second Virial Coefficients using Glycerol, Trans. Far. Soc., 65, 1014(1969). 

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. 

An important series of papers by Professor Pitzer and colleagues (26, 27, 28, 29), beginning in 1912, has laid the ground work for what appears to be the "most comprehensive and theoretically founded treatment to date. This treatment is based on the ion interaction model using the Debye-Huckel ion distribution and establishes the concept that the effect of short range forces, that is the second virial coefficient, should also depend on the ionic strength. Interaction parameters for a large number of electrolytes have been determined. 

Second virial coefficients represent the first approximation to the system equation of state. Yethiraj and Hall [148] obtained the compressibility factor, i.e., pV/kgTn, for small stars. They found no significant differences with respect to the linear chains in the pressure vs volume behavior. Escobedo and de Pablo [149] performed simulations in the NPT ensemble (constant pressure) with an extended continuum configurational bias algorithm to determine volumetric properties of small branched chains with a squared-well attractive potential... 

The next step consists of the determination of the size of the macromolecules in space. Two equivalent sphere radii can be measured directly by means of static and dynamic LS. Another one can be determined from a combination of the molar mass and the second virial coefficient A2. Similarly, an equivalent sphere radius is obtained from a combination of the molar mass with the intrinsic viscosity. This is outlined in the following sections. 

The MOLWT-II program calculates the molecular weight of species in retention volume v(M(v)), where v is one of 256 equivalent volumes defined by a convenient data acquisition time which spans elution of the sample. I oment of the molecular weight distribution (e.g., Mz. Mw. Mn ) are calculated from summation across the chromatogram. Along with injected mass and chromatographic data, such as the flow rate and LALLS instruments constants, one needs to supply a value for the optical constant K (Equation la), and second virial coefficient Ag (Equation 1). The value of K was calculated for each of the samples after determination of the specific refractive index increment (dn/dc) for the sample in the appropriate solvent. Values of Ag were derived from off-line (static) determinations of Mw. 

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