Virial coefficients estimation

It is interesting to note that three of the approximate theories, including the scaled particle theory, yield the same numerical value for the fourth virial coefficient estimate. The significance... 

It was discovered, that the expression (1) for the description of n into diluted and semi-diluted solutions required of different values of virial coefficient A. In particular, for the estimation of A in a field of diluted solutions it would be better to accept the whole... 

Further development of the Flory-Huggins method in direction of taking into account the effects of far interaction, swelling of polymeric ball in good solvents [4, 5], difference of free volumes of polymer and solvent [6, 7] leaded to complication of expression for virial coefficient A and to growth of number of parameters needed for its numerical estimation, but weakly reflected on the possibility of equation (1) to describe the osmotic pressure of polymeric solutions in a wide range of concentrations. 

A very severe test of these virial-coefficient equations for the sea-water-related Na-K-Mg-Ca-Cl-S0,-H 0 system has been made by Harvie and Weare (37) who calculated tne solubility relationships for most of the solids which can arise from this complex system. There are 13 invariant points with four solids present in the system Na-K-Mg-Cl-SO - O and the predicted solution compositions in all 13 cases agree with the experimental values of Braitsch (38) substantially within the estimated error of measurement. In particular, Harvie and Weare found that fourth virial coefficients were not required even in the most concentrated solutions. They did make a few small adjustments in third virial coefficients which had not previously been measured accurately, but otherwise they used the previously published parameters. 

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. 

The unconventional applications of SEC usually produce estimated values of various characteristics, which are valuable for further analyses. These embrace assessment of theta conditions for given polymer (mixed solvent-eluent composition and temperature Section 16.2.2), second virial coefficients A2 [109], coefficients of preferential solvation of macromolecules in mixed solvents (eluents) [40], as well as estimation of pore size distribution within porous bodies (inverse SEC) [136-140] and rates of diffusion of macromolecules within porous bodies. Some semiquantitative information on polymer samples can be obtained from the SEC results indirectly, for example, the assessment of the polymer stereoregularity from the stability of macromolecular aggregates (PVC [140]), of the segment lengths in polymer crystallites after their controlled partial degradation [141], and of the enthalpic interactions between unlike polymers in solution (in eluent) [142], as well as between polymer and column packing [123,143]. 

Another important application of experimentally determined values of the osmotic second virial coefficient is in the estimation of the corresponding values of the Flory-Huggins interaction parameters x 12, X14 and X24. In practice, these parameters are commonly used within the framework of the Flory-Huggins lattice model approach to the thermodynamic description of solutions of polymer + solvent or polymer] + polymer2 + solvent (Flory, 1942 Huggins, 1942 Tanford, 1961 Zeman and Patterson, 1972 Hsu and Prausnitz, 1974 Johansson et al., 2000) ... 

Amur, K.S., Harlapur, S.F., Aminabhavi, T.M. (1997). A novel analytical method to estimate molar mass and virial coefficients of polymer from osmometry. Polymer, 38, 6417-6420. 

Use these data to estimate 9 and k for this system. Does AS agree with the expected entropy contribution to the second virial coefficient ... 

The calculations were carried out for various values of the parameters, the aspect ratio of segment p, and the number ratio of ionizable groups in the chain f. The other parameters were estimated for NIPA gel. All the values of parameters used are summarized in Table 3. The value of v0 was determined by taking the intermediate value between water and NIPA [20]. The parameters Ca, Cb and Cc for the hydrophobic interaction were determined from the values of isobutyl substituents of amino acids, determined by Nemethy and Scheraga [19]. Since there are no data for the 6 temperature and the virial coefficients of this system, we assumed Te to be 273.15 K, and estimated the virial coefficients... 

Rudzinski and coworkers (35,36) first used the second and third gas-solid virial coefficients obtained from GC data to estimate surface areas. The surface area of silica gel determined using virial expansion data was greater than that obtained using the BET method (Section 11.1). The discrepancy was explained by noting the BET method does not take the lateral interactions into account. These interactions have an effect of decreasing the effective area of the adsorbent, thus making the calculated BET area less than it should be (Table 11.7). 

Here BLS is the second virial coefficient of the polymeric solute in the original solution before ultracentrifugation. BLs is a quantity which can be obtained in light-scattering experiments (17, 25, 30) or in Archibald experiments (31), provided it is calculated from a plot of l/MWapp° vs. c. Here 1/Mw pp° is obtained from values of Mw pp (at rm or rb) that have been extrapolated to zero time. The reason for using Equation 75 is that it leads to a simple method of estimating the MWD in nonideal solutions. 

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. 

To have an idea about the range of the repulsion required to provide such a high virial coefficient, it should be noted that, if the hard-core repulsion, infinite in magnitude, is extended with 15 A (above the 2a separation), 2 increases from 4 to only 5.6. If the range of the hard-core repulsion is extended with 30 A, 2 increases to 7.55, while 60 A leads to 12.8. From these simple estimations one can infer that the repulsion needed to explain the measured second virial coefficient for apoferritin molecules should have a much longer range than that typically observed for the traditional hydration force. 

The estimation of the jamming coverage for the RSA of monodisperse disks is not an important issue, because its value is already accurately known from Monte Carlo simulations [12], However, it is of interest to develop a procedure that can predict the available area and the jamming coverage for a mixture of disks, for which much Less information is available. Even at equilibrium, for which reasonable accurate equations of state for binary mixtures of hard disks are known for low densities [ 19,20], the available area vanishes only for the unphysical total coverage 9 = 9 +0p = 1 (where the subscripts S and L stand for small and large disk radii, respectively), hence there is no jamming . Exact analytical expressions are known only for the first three virial coefficients of a binary mixture of disks [21], The fourth and fifth coefficients were computed numerically for some diameter ratios and molar fractions for an equilibrium gas [22], However, there are no such calculations for the RSA model. 

Fig. 36a and b. Second virial coefficient A2 of a) cellulose nitrate (CN) (Nc = 13.9%) and b) CN (Nc = 12.9%), in acetone78 79> O experimental data 2). Lines are calculated by using the penetration function j/ from short and long range interaction parameters A and B, which are estimated by methods 2C (full line), 2D (broken line), 2E (dotted line), and 2G (chain line), together with experimental [Pg.41]

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