Gibbs residual

Evaluation of the integrals requires an empirical expression for the temperature dependence of the ideal gas heat capacity, (3p (8). The residual Gibbs energy is related to and by equation 138 ... 

According to equation 132, is the residual Gibbs energy, G. The dimensionless mXio fjP is a new property called the fugacity coefficient, ( ). ... 

Analogous to the defining equation for the residual Gibbs energy is the definition of a partial molar residual Gibbs energy (eq. 161) ... 

The residual Gibbs energy and the fugacity coefficient are useful where experimental PVT data can be adequately correlated by equations of state. Indeed, if convenient treatment or all fluids by means of equations of state were possible, the thermodynamic-property relations already presented would suffice. However, liquid solutions are often more easily dealt with through properties that measure their deviations from ideal solution behavior, not from ideal gas behavior. Thus, the mathematical formahsm of excess properties is analogous to that of the residual properties. 

All eight parameters depend on composition moreover, parameters Co, b, and y are for some applications treated as functions of X By Eq. (4-171), the residual Gibbs energy is... 

The right-hand side of this equation is exactly the quantity that Eq. (4-289), the Gibbs/Duhem equation, requires to be zero for consistent data. The residual on the left is therefore a direct measure of deviations from the Gibbs/Diihem equation. The extent to which values of this residual fail to scatter about zero measures the departure of the data from consistency with respect to this equation. 

If the experimental values P and w are closely reproduced by the correlating equation for g, then these residues, evaluated at the experimental values of X, scatter about zero. This is the result obtained when the data are thermodynamically consistent. When they are not, these residuals do not scatter about zero, and the correlation for g does not properly reproduce the experimental values P and y . Such a correlation is, in fact, unnecessarily divergent. An alternative is to process just the P-X data this is possible because the P-x -y data set includes more information than necessary. Assuming that the correlating equation is appropriate to the data, one merely searches for values of the parameters Ot, b, and so on, that yield pressures by Eq. (4-295) that are as close as possible to the measured values. The usual procedure is to minimize the sum of squares of the residuals 6P. Known as Barkers method Austral. ]. Chem., 6, pp. 207-210 [1953]), it provides the best possible fit of the experimental pressures. When the experimental data do not satisfy the Gibbs/Duhem equation, it cannot precisely represent the experimental y values however, it provides a better fit than does the procedure that minimizes the sum of the squares of the 6g residuals. 

It is a consequence of the Gibbs-Konowalow rule that the compositions of liquid and vapour (i.e., the residue and distillate, respectively) alter in the sense of falling and rising parts of the curve, respectively. One may imagine (Ostwald, loc. cit.) the composition of the liquid to be represented by a heavy mobile... 

In order to obtain an expression for the excess Gibbs energy, we first define g(a), g(c, and gW as the residual Gibbs energies per mole of cells of central anion, central cation and central solvent molecule, respectively. These Gibbs energies are related to the local mole fractions as follows ... 

The molar excess Gibbs energy may now be derived by summing the changes in residual Gibbs energy resulting when xm moles of solvent are transfered from the solvent reference state to their cells in the mixture, and when xa moles of anions and xc moles of cations are transfered from the electrolyte reference state to their respective cells in the mixture. The expression is ... 

Fio. 6. The Gibbs energy difference of the native and denatured states of myoglobin and ribonuclease A calculated per mole of amino acid residues under the same conditions as indicated in Fig. 4. The dot-and-dash lines represent functions obtained in the assumption that the denaturation heat capacity increment is temperature independent. 

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