Cross coefficients, second

The coefficient Bij characterizes a bimolecular interaction between molecules i and J, and therefore Bij = Bji. Two lands of second virial coefficient arise Bn and By, wherein the subscripts are the same (i =j) and Bij, wherein they are different (i j). The first is a virial coefficient for a pure species the second is a mixture property, called a cross coefficient. Similarly for the third virial coefficients Cm, Cjjj, and are for the pure species and Qy = Cyi = Cjn, and so on, are cross coefficients. 

Tsonopoulous, C. "Second Virial Cross Coefficients Correlation and Prediction of k..," Adv. in Chem., in press(1979). 

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. 

Previously, we considered the case where heat and mass flows are coupled in a reaction diffusion system with heat effects, in which the cross coefficients Zrq. Zqr. and LlS, LSl have vanished (Demirel, 2006). Here, we consider the other three cases. The first involves the stationary state balance equations. In the second case, there is no coupling between the heat flow and chemical reaction with vanishing coefficients Zrq and Zqr. Finally, in the third, there is no coupling between the mass flow and chemical reaction because of vanishing cross-coefficients of ZrS and LSl. The thermodynamically coupled modeling equations for these cases are derived and discussed briefly in the following examples. 

Two types of virial coefficients have appeared Bh and B22, for which successive subscripts are the same, and B12, for which the two subscripts different. The first type represents the virial coeffident of a pure species second is a mixture property, known as a cross coefficient. Both are functions temperature only. 

The generalexpressionforcalculationof In from second-virial-coefficientdata is given by Eq. (11.61). Values of tire pure-speciesvirial coefficients Bkk, Bu, etc., are foimd from die generalized correlation represented by Eqs. (3.59), (3.61), and (3.62). The cross coefficients Bik, Bij, etc., are found from an extension of the same correlation. For this purpose, Eq. (3.59) is rewrittenin the more general form ... 

Although developed for pure materials, these correlations can be extended to gas or vapor mixtures. Basic to this extension are the mixing rules for the second virial coefficient and its temperature derivative as given by Eqs. (4-60) and (4-62). Values for the cross coefficients Bp with i and their derivatives are provided by Eq. (4-72) written in extended form ... 

The second virial pure component and cross coefficients are next calculated ... 

For gases at low and moderate pressures, it is often preferable to use the virial expansion that provides successive corrections to the ideal-gas law (see Chapter 4, Thermodynamics ). The first correction (called the second virial coefficient, B) has been derived from volumetric data for many pure fluids. Higher virial coefficients are much less well known, as are the cross-coefficients for interactions between unlike molecules. Estimation techniques for second (and to a lesser extent third) virial coefficients exist [15] and work reasonably well for many fluids, especially organic compounds of low polarity. Dymond et al. have compiled extensive experimental data for virial coefficients [32]. 

Second Virial Cross-Coefficients Correlation and Prediction of kij... 

Second virial cross efficients between the polar chemical and the light gases appear in the literature less often than second virial coefficients for pure polar substances. However, these coefficients (B ) can be used to determine AP(TC) of the polar solute. By substituting Equations 5, 12, and 13 in Equation 16 and by using the definition of the second virial cross coefficient (3) ... 

Equations 17 and 14 can be fit to the experimental second virial cross coefficients to find AP(TC). 

Water Solute in Hydrocarbon-Rich Vapor and Liquid. The pure component parameters of water solute AP(TC) and a were determined by using Equations 5 and 16 to fit the gas-phase volumetric properties of steam (5) and the second virial cross coefficients of steam and light gases such as methane, ethane, and nitrogen (6). The least-squares minimization technique was used to find the parameters that gave the minimum deviations between calculated and experimental pressures and second virial cross coefficients. (Table I lists the parameters for pure steam and of other compounds used in this study.)... 

Methanol in Hydrocarbon-Rich Vapor and Liquid. The volumetric properties of methanol gas (12) and the second virial cross coefficients of methanol and light gases (13) were used to determine the pure-component parameters AP(TC) and a for methanol. Table II shows the enthalpy departure of gaseous methanol from ideal gas at three temperatures and several pressures. For comparison, the experimental values (14) and the values calculated by the Soave equation (1) are also shown. Table II indicates that the Won modified equation of state predicts the enthalpy departure of methanol very well at low temperatures and fairly well at high temperatures, but that the original Soave equation considerably underestimates the enthalpy departure at all temperatures and pressures. Since the original Soave equation was meant to be applied only to hydrocarbons, we are not surprised at this result. Comparison of calculated and experimental second virial cross coefficients between methanol and methane (and also C02) is presented elsewhere (15).)... 

As the second term in Eq. (2.153) is non-zero, the chemical potential of the insoluble component does not depend on the adsorption of the soluble component provided that both surface pressure and adsorption of the insoluble component are fixed. In turn, as the surface concentration of the insoluble component is fixed, the requirement for constant activity of this component implies the independence of this activity coefficient of adsorption of the soluble component. Clearly, this requirement is satisfied not only for the trivial case of an ideal monolayer, but also for non-ideal monolayers, provided that the activity cross-coefficients of the components (or intermolecular interaction parameters) vanish. For example, if the equation of state Eq. (2.35) is used for a non-ideal (with respect to the enthalpy) mixed two-component monolayer, it follows from Eq. (2.153) that Eqs. (2.151) and (2.152) are applicable when ai2 = 0. Clearly, the condition of Eq. (2.153) imposes certain restrictions to the applicability of Pethica s model. The generalised Pethica equation (2.151) was thermodynamically analysed in [64, 65]. Moreover, an attempt to verify Eq. (2.151) experimentally was undertaken in [65], which also confirms its validity for mixed monolayers comprised of two non-ionic surfactants, or for mixtures of non-ionic and ionic surfactant, or two ionic surfactants. 

Tessier, P.M., Sandler, S.I., Lenhoff, A.M. Direct measurement of protein osmotic second virial cross coefficients by cross-interaction chromatography. Protein Sci. 13, 1379-1390 (2004)... 

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