The Second Virial Coefficient

Figure 1 shows second virial coefficients for four pure fluids as a function of temperature. Second virial coefficients for typical fluids are negative and increasingly so as the temperature falls only at the Boyle point, when the temperature is about 2.5 times the critical, does the second virial coefficient become positive. At a given temperature below the Boyle point, the magnitude of the second virial coefficient increases with... 

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

As discussed in Chapter 3, the virial equation is suitable for describing vapor-phase nonidealities of nonassociating (or weakly associating) fluids at moderate densities. Equation (1) gives the second virial coefficient which is used directly in Equation (3-lOb) to calculate the fugacity coefficients. 

Equilibrium constants,, for all possible dimerization reactions are calculated from the metastable, bound, and chemical contributions to the second virial coefficients, B , as given by Equations (6) and (7). The equilibrium constants, K calculated using Equation (3-15). 

VPLQFT is a computer program for correlating binary vapor-liquid equilibrium (VLE) data at low to moderate pressures. For such binary mixtures, the truncated virial equation of state is used to correct for vapor-phase nonidealities, except for mixtures containing organic acids where the "chemical" theory is used. The Hayden-0 Connell (1975) correlation gives either the second virial coefficients or the dimerization equilibrium constants, as required. 

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 FREE CONTRIBUTION TO THE SECOND VIRIAL COEFFICIENT,... 

IF BINARY SYSTEM CONTAINS NO ORGANIC ACIDS. THE SECOND VIRTAL coefficients ARE USED IN A VOLUME EXPLICIT EQUATION OF STATE TO CALCULATE THE FUGACITY COEFFICIENTS. FOR ORGANIC ACIDS FUGACITY COEFFICIENTS ARE PREDICTED FROM THE CHEMICAL THEORY FOR NQN-IOEALITY WITH EQUILIBRIUM CONSTANTS OBTAINED from METASTABLE. BOUND. ANO CHEMICAL CONTRIBUTIONS TO THE 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... 

Corner J 1948 The second virial coefficient of a gas of non-spherical molecules Proc. R. Soc. A 192 275... 

Figure A2.1.7. The second virial coefficient 5 as a fiinction of temperature T/T. (Calculated for a gas satisfying the Leimard-Jones potential [8].)...

Gas mixtures are subject to the same degree of non-ideality as the one-component ( pure ) gases that were discussed in the previous section. In particular, the second virial coefficient for a gas mixture can be written as a quadratic average... 

The same result can also be obtained directly from the virial equation of state given above and the low-density fonn of g(r). B2(T) is called the second virial coefficient and the expansion of P in powers of is known as the virial expansion, of which the leading non-ideal temi is deduced above. The higher-order temis in the virial expansion for P and in the density expansion of g(r) can be obtained using the methods of cluster expansion and cumulant expansion. 

A direct and transparent derivation of the second virial coefficient follows from the canonical ensemble. To make the notation and argument simpler, we first assume pairwise additivity of the total potential with no angular contribution. The extension to angularly-mdependent non-pairwise additive potentials is straightforward. The total potential... 

In the thennodynamic limit (N x, F -> oo withA7F= p), this is just the virial expansion for the pressure, with 7,(7) identified as the second virial coefficient... 

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... 

Figure A2.3.6 illustrates the corresponding states principle for the reduced vapour pressure P and the second virial coefficient as fiinctions of the reduced temperature showmg that the law of corresponding states is obeyed approximately by the substances indicated in the figures. The useflilness of the law also lies in its predictive value.

The principle ideas and main results of tlie theory at the level of the second virial coefficient are presented below. The Mayer/-function for the solute pair potential can be written as the sum of temis ... 

A graphical method, proposed by Zimm (thus tenned the Zinnn plot), can be used to perfomi this double extrapolation to detemiine the molecular weight, the radius of gyration and the second virial coefficient. An example of a Zinnn plot is shown in figure Bl.9.6 where the light scattering data from a solution of poly... 

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. 

In the next sections we shall discuss the second virial coefficient in terms of several different models. Before turning to these models, however, it is important to summarize the foregoing conclusions ... 

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

Equation (8.97) shows that the second virial coefficient is a measure of the excluded volume of the solute according to the model we have considered. From the assumption that solute molecules come into surface contact in defining the excluded volume, it is apparent that this concept is easier to apply to, say, compact protein molecules in which hydrogen bonding and disulfide bridges maintain the tertiary structure (see Sec. 1.4) than to random coils. We shall return to the latter presently, but for now let us consider the application of Eq. (8.97) to a globular protein. This is the objective of the following example. 

The above argument shows that complete overlap of coil domains is improbable for large n and hence gives plausibility to the excluded volume concept as applied to random coils. More importantly, however, it introduces the notion that coil interpenetration must be discussed in terms of probability. For hard spheres the probability of interpenetration is zero, but for random coils the boundaries of the domain are softer and the probability for interpenetration must be analyzed in more detail. One method for doing this will be discussed in the next section. Before turning to this, however, we note that the Flory-Huggins theory can also be used to yield a value for the second virial coefficient. 

To use the Flory-Huggins theory as a source for understanding the second virial coefficient, we return to Eq. (8.53), which gives an expression for jui -jui°. Combining this result with Eq. (8.79) gives... 

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. 

The parameter a which we introduced in Sec. 1.11 to measure the expansion which arises from solvent being imbibed into the coil domain can also be used to describe the second virial coefficient and excluded volume. We shall see in Sec. 9.7 that the difference 1/2 - x is proportional to. When the fully... 

It is interesting to note that for a van der Waals gas, the second virial coefficient equals b - a/RT, and this equals zero at the Boyle temperature. This shows that the excluded volume (the van der Waals b term) and the intermolecular attractions (the a term) cancel out at the Boyle temperature. This kind of compensation is also typical of 0 conditions. 

Many presentations of the second virial coefficient of polymer solutions contain different expressions for the quantities we have discussed. The difference lies in the fact that the factor p( - 0/T) appears in place of 1/2 - x-There are several attitudes we can take toward this difference. For one thing, we can regard the discrepancy as nothing more than different notation ... 

Krigbaumf measured the second virial coefficient of polystyrene in cyclohexane at several different temperatures. The observed values of B as well as some pertinent volumes at those temperatures are listed below ... 

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