Second virial coefficients, dependence

The second virial coefficient depends on the excluded volume u. The macromolecules arrange themselves with little mutual interference since the total excluded volume Ni u is much smaller than the total volume V. The total number of possible ways of arranging these Ni macromolecules is calculated from the partition function O,... 

Since the second virial coefficient depends only on temperature and integration along an isotherm, the result is... 

The second virial coefficient depends on molar mass. For flexible, nonassociating polymers, a simple scaling relation can be applied ... 

The values of second virial coefficients depend on the chosen polymer-solvent pair, on temperature and pressure, and on molar mass. In good solvents, a scaling relation can be applied for the molar mass dependence ... 

As well as controlling chain dimensions, solvent quality affects the thermodynamics of dilute polymer solutions. This is because interactions between polymer chains are modified by the presence of solvent molecules. In particular, solvent molecules will change the excluded volume for a polymer coil, i.e. how much volume it takes up and prevents neighbouring chains from occupying. In a theta solvent, the excluded volume is zero (this holds for the excluded volume for a polymer segment or the whole coil). The solution is said to he ideal if the excluded volume vanishes. Deviations from ideality for polymer solutions are described in terms of a virial equation, just as deviations from ideal gas behaviour are. The virial equation for a polymer solution in terms of polymer concentration is given by Eq. (2.9). The second virial coefficient depends on interactions between pairs of molecules in particular it is proportional to the excluded volume. Therefore, in a theta solvent, = 0. If the solvent is good then Ai > 0, but if it is poor Ai < 0. If the solvent quality varies as a function of temperature and theta (0) conditions are attained, this occurs at the theta temperature. 

SAPT may compare less favorably with experiment since this potential is purely pairwise additive (the second virial coefficient depends exclusively on pairwise interactions). A cross-section of the water trimer potential shown in Figure 3 includes two-body and three-body SAPT interaction energies as well as those computed from several popular empirical potentials. This comparison shows that while the nonadditive contribution is of the order of 1 kcal mol" or 10% at the trimer level, it is small compared to the differences between the SAPT and empirical potentials which at some angles approach 20 kcal mor. These large differences suggest that the nonadditive contribution is significantly amplified in the bulk water compared to the trimer, as it is the case for solid argon, see below. 

How will the slope of the straight-line plot in Figure 8.5 change as the solvent becomes a progressively better solvent In other words, how does the second virial coefficient depend on solvent quahty ... 

Individual contributions to the second virial coefficient are calculated from temperature-dependent correlations ... 

PARAMETER USED TO CALCULATE PART OF CHEMICAL CONTRIBUTION TO THE SECOND VIRIAL COEFFICIENT. CALCULATED ONE OF TWO WAYS DEPENDING ON THE VALUE OF ETA(IJ). 

CALCULATE THE TEMPERATURE DEPENDENT SECOND VIRIAL COEFFICIENTS. 

The viscosity, themial conductivity and diffusion coefficient of a monatomic gas at low pressure depend only on the pair potential but through a more involved sequence of integrations than the second virial coefficient. The transport properties can be expressed in temis of collision integrals defined [111] by... 

Finally, the assumed spherical synnnetry of the interactions implies that the volume element r 2 is dri2- For angularly-dependent potentials, the second virial coefficient... 

Going beyond die limiting law it is found that the modified (or renonnalized) virial coefficients in Mayer s theory of electrolytes are fiinctions of the concentration through their dependence on k. The ionic second virial coefficient is given by [62]... 

Our primary interest in the Flory-Krigbaum theory is in the conclusion that the second virial coefficient and the excluded volume depend on solvent-solute interactions and not exclusively on the size of the polymer molecule itself. It is entirely reasonable that this should be the case in light of the discussion in Sec. 1.11 on the expansion or contraction of the coil depending on the solvent. The present discussion incorporates these ideas into a consideration of solution nonideality. 

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. 

The two values kp and k are usually not very different, and kp is not strongly composition dependent. Nevertheless, the quadratic dependence of Z — a/RT) on composition indicated by Eq. (4-305) is not exactly preserved. Since this quantity is not a true second virial coefficient, only a value predicted by a cubic equation of state, a strict quadratic dependence is not required. Moreover, the composition-dependent kp leads to better results than does use of a constant value. 

This expression is called the virial equation. The coefficients B, C,. . . are called the second virial coefficient, third virial coefficient, and so on. The virial coefficients, which depend on the temperature, are found by fitting experimental data to the virial equation. 

The second virial coefficient B in Eq. 17 refers to the static case. In the ultracentrifuge the measured value can show a speed dependence [39], an effect which can be minimized by using low speeds and short solution columns. If present it will not affect the value of after extrapolation to zero concentration. 

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. 

To overcome the problem of non-ideality the work be carried out at the Q temperature because in nonideal solutions the apparent Molecular weight is a linear function of concentration at temperatures near Q and the slope depending primarily on the second virial coefficient. 

Other dilute solution properties depend also on LCB. For example, the second virial coefficient (A2) is reduced due to LCB. However, near the Flory 0 temperature, where A2 = 0 for linear polymers, branched polymers are observed to have apparent positive values of A2 [35]. This is now understood to be due to a more important contribution of the third virial coefficient near the 0 point in branched than in linear polymers. As a consequence, the experimental 0 temperature, defined as the temperature where A2 = 0 is lower in branched than in linear polymers [36, 37]. Branched polymers have also been found to have a wider miscibility range than linear polymers [38], As a consequence, high MW highly branched polymers will tend to coprecipitate with lower MW more lightly branched or linear polymers in solvent/non-solvent fractionation experiments. This makes fractionation according to the extent of branching less effective. 

The higher the pressure, the larger the number of terms we have to consider in equations 2.17 and 2.18. Let us assume that the pressure is such that only the second term needs to be considered. Then because the virial coefficients depend only on the temperature, we have ... 

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. 

At moderate pressures, the virial equation of state, truncated after the second virial coefficient, can be used to describe the vapor phase. As suggested by Hirschfelder, et. al. (1 3) the temperature dependence of the virial coefficients is expressed... 

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