Second virial coefficients units

The matter of units also complicates the second virial coefficient, which we consider next. 

The potential energy function prohibits double occupancy of any site on the 2nnd lattice. In the initial formulation, which was designed for the simulation of infinitely dilute chains in a structureless medium that behaves as a solvent, the remaining part of the potential energy function contains a finite repulsion for sites that are one lattice unit apart, and a finite attraction for sites that are two lattice units apart [153]. The finite interaction energies for these two types of sites were obtained by generalizing the lattice formulation of the second virial coefficient, B2, described by Post and Zimm as [159] ... 

The osmotic second virial coefficient A2 is another interesting solution property, whose value should be zero at the theta point. It can be directly related with the molecular second virial coefficient, expressed as B2=A2M /N2 (in volume units). For an EV chain in a good solvent, the second virial coefficient should be proportional to the chain volume and therefore scales proportionally to the cube of the mean size [ 16]. It can, therefore, be expressed in terms of a dimensionless interpenetration factor that is defined as... 

In earlier experiments the effect of branching on the second virial coefficient was not seriously considered because the accuracy of measurements were not sufficient at that time. With the refinements of modern instruments a much higher precision has now been achieved. Thus A2 can also now be measured with good accuracy and compared with theoretical expectations. The second virial coefficient results from the total volume exclusion of two macromolecules in contact [3,81]. Furthermore, this total excluded volume of a macromolecule can be expressed in terms of the excluded volume of the individual monomeric units. In the limit of good solvent behavior this concept leads to the expression [6,27] as shown in Eq. (24) ... 

Tables VI and VII give results corresponding to two series of lignin fractions obtained with a flow-through reactor (3). (The units for dn/dc and A2 are respectively ml.g-1 and mole.ml.g-2). These results show that LALLS allows the determination of low Mw values. The dn/dc values differ from sample to sample but vary only slightly for a given set of fractions. The second virial coefficient exhibits no definite trend. Negative values indicate perhaps some association effects but light scattering alone is not sufficient to ascertain this point.

Here, p and m are the standard chemical potential and concentration (molal scale) of the /-component (z = 1 for solvent, z = 2 for biopolymer) A2 is the second virial coefficient (in molal scale units of cm /mol, i.e., taking the polymer molar mass into account) and m° is the standard-state molality for the polymer. 

Here, the quantities jn ° and ji are, respectively, the chemical potentials of pure solvent and of the solvent at a certain biopolymer concentration V is the molar volume of the solvent and n is the biopolymer number density, defined as n C/M, where C is the biopolymer concentration (% wt/wt) and M is the number-averaged molar weight of the biopolymer. The second virial coefficient has (weight-scale) units of cm mol g. Hence, the more positive the second virial coefficient, the larger is the osmotic pressure in the bulk of the biopolymer solution. This has consequences for the fluctuations in the biopolymer concentration in solution, which affects the solubility of the biopolymer in the solvent, and also the stability of colloidal systems, as will be discussed later on in this chapter. 

Here, m is the molal concentration of the /-component in mol/1000 g water ct is the weight concentration of the /-component in g/ml Vi and vz (with bars) are the specific volumes of water and biopolymer in ml/g and Mf is the molar weight of the biopolymer in g/mol (Da). In turn, the relationship between the second virial coefficients expressed in the different units (molal, At weight, ) is as follows (Wells, 1984) ... 

Figure 6.10 Effect of CITREM on the molecular and thermodynamic parameters of maltodextrin SA-2 (DE = 2) in aqueous medium (phosphate buffer, pH = 7.2, ionic strength = 0.05 M 20 °C) (a) weight average molar mass, Mw (b) radius of gyration, Ra (c) structure sensitive parameter, p, characterizing die architecture of maltodextrin associates (d) second virial coefficient, A2 or A2, on the basis of the weight ( ) and molal (A) scales, respectively. The parameter R is defined as the molar ratio of surfactant to glucose monomer units in the polysaccharide. The indicated cmc value refers to the cmc of the pure CITREM solution. Reproduced from Anokhina et al. (2007) with permission.

Equations of this type are known as virial equations, and the constants they contain are called the virial coefficients. It is the second virial coefficient B that describes the earliest deviations from ideality. It should be noted that B would have different but related values in Equations (26) and (27), even though the same symbol is used in both cases. One must be especially attentive to the form of the equation involved, particularly with respect to units, when using literature values of quantities such as B. The virial coefficients are temperature dependent and vary from gas to gas. Clearly, Equations (26) and (27) reduce to the ideal gas law as p - 0 or as n/V - 0. Finally, it might be recalled that the second virial coefficient in Equation (27) is related to the van der Waals constants a and b as follows ... 

Note that the value of the intercept, the value of r/RTc at infinite dilution, obeys the van t Hoff equation, Equation (25). At infinite dilution even nonideal solutions reduce to this limit. The value of the slope is called the second virial coefficient by analogy with Equation (27). Note that the second virial coefficient is the composite of two factors, B and (1/2) Vx/M. The factor B describes the first deviation from ideality in a solution it equals unity in an ideal solution. The second cluster of constants in B arises from the conversion of practical concentration units to mole fractions. Although it is the nonideality correction in which we are primarily interested, we discuss it in terms of B rather than B since the former is the quantity that is measured directly. We return to an interpretation of the second virial coefficient in Section 3.4. 

All that remains to be done to complete our derivation of the second virial coefficient in terms of the Flory-Huggins theory is convert volume fractions into practical concentration units. First, we can express the volume fraction of the solute in terms of partial molar volumes ... 

TABLE 1 Second virial coefficient of argon" (B in units cm mol )... 

Second virial coefficient Third virial coefficient Fourth virial coefficient Debye (unit of dipole moment equal to 10 e.s.u.-cm)... 

The second virial coefficient has the dimensions of a volume per mole. The unit often chosen is the Amagat unit which is by definition the molar volume of the gas at 0 °C and 1 atm. The exact value of this unit depends upon the gas considered but is approximately equal to 2-24 x 10 cm. /mole. Alternatively, it is now becoming customary to give B directly in terms of cm. /mole. 

We have already in this and the previous sections made a number of comparisons between the various theories of fluids and the machine computations for the hard sphere system. Unfortunately, many recent developments in theory have been evaluated numerically only to the extent that the fourth and fifth virial coefficients can be compared. The table below lists the values of the fourth and fifth virial coefficients for the three-dimensional hard sphere fluid in units of the second virial coefficient b [cf. Eq. (33)]. The bases of calculation have been identified already in Section III except for the older "netted-chain approximation of Rushbrooke and Scoins. ... 

A theta solvent is one where the second virial coefficient (the coefficient, V in Eq. (7.25)) — which balances the hard-core repulsion and the polymer-polymer attraction — vanishes. This occurs for many solvents only at a particular value of the temperature. (When the second virial coefficient becomes negative, the solution is unstable to phase separation and eventually, to the collapse of the individual chains into compact objects.) The free energy per unit area of a brush in a theta solvent therefore does not have any terms quadratic in the concentration and is written ... 

For uncharged coronal blocks A, the short-ranged (van der Waals) interactions between monomer units are described in terms of a virial expansion. The latter accounts for the monomer-monomer binary (pair) interactions, with second virial coefficient VaO, or the ternary interactions with third virial coefficient waa . We assume that the monomer unit length, a, is the same for both blocks A and B. In the following, we use a as a unit length to make all lengths dimensionless and eliminate a in further equations. We also assume that the (dimensionless) second virial coefficient Va > 0 and that the third virial coefficient Wa 1. 

When the degree of ionization of monomer units in the corona is relatively small, oc << afe < 1, the effect of short-ranged interactions is not negligible. Therefore, in a weakly dissociating corona, both ionic and nonelectrostatic binary interactions (specified by the second virial coefficient Va) should be taken into account. In this case, the corresponding expressions for H ooa and / corona can be represented as [22, 97] ... 

Measurements of viscosity and light scattering have proven that maximum coil dimensions of PAAm/AAcNa in salt solutions are achieved at about 67 mole ) AAcNa. The maximum behaviour was researched by measuring the radius of gyration, second virial coefficient, molecular weight, preferential solvation, viscosity and absorption bands of the copolymer series. Thus, an increase in the possible ways of arranging AAcNa-AAm-AAcNa units (via H-bonds) will lead to extended coil dimensions and therefore influence the viscosity level. In addition, it can... 

To measure the importance of the screening effect, the main parameter of interest is the second virial coefficient A2S of the solvent-solvent interaction. If we define a solvent fraction (the average number of solvent units per site) and a solvent osmotic pressure IIs, the latter has an expression of the form... 

Another practical definition of 0 can be proposed. We may call 0 the temperature at which the second virial coefficient between two very large coils vanishes. Fortunately, these two definitions coincide. When we are on the dividing line, the parameter u (at the m-th iteration) gives (in dimensionless units) the virial coefficient between two subunits. Since the dividing line ends at 0, where u = 0, this coefficient vanishes when the subunits are large enough. The distinction between 0 ai 0 is essentially absent from the polymer literature (which has been written mainly on the mean field level). 

In the vicinity of the 0 temperature, both x(T) and v(T) vary linearly as a function of relative deviation z= (T-0)/T5 1 from the 0 temperature. The second virial coefficient v can be approximated as v Vqt, where Vo is geometrical (bare) exduded-volume parameter, v—>Vo at T>0. Assuming that the effective chain thickness is on the order of the monomer unit length a, one finds Vo = a. For semiflexible polymers, the second virial coefficient for a statistical segment can be estimated at p = f/fl S 1 as Vp s zi a=za p. ... 

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