Gibbs-Duhem equation integration constants

If we vary the composition of a liquid mixture over all possible composition values at constant temperature, the equilibrium pressure does not remain constant. Therefore, if integrated forms of the Gibbs-Duhem equation [Equation (16)] are used to correlate isothermal activity coefficient data, it is necessary that all activity coefficients be evaluated at the same pressure. Unfortunately, however, experimentally obtained isothermal activity coefficients are not all at the same pressure and therefore they must be corrected from the experimental total pressure P to the same (arbitrary) reference pressure designated P. This may be done by the rigorous thermodynamic relation at constant temperature and composition ... 

A consistency test described by Chueh and Muirbrook (C4) extends to isothermal high-pressure data the integral (area) test given by Redlich and Kister (Rl) and Herington (H2) for isothermal low-pressure data. [A similar extension has been given by Thompson and Edmister (T2)]. For a binary system at constant temperature, the Gibbs-Duhem equation is written... 

For the ternary solution, the Gibbs-Duhem equation can be easily integrated to calculate the activity coefficient of water when the expressions for the activity coefficients of the electrolytes are written at constant molality. For Harned s rule, integration of the Gibbs-Duhem equation gives the activity of water as ... 

Table III presents the values of the constants used in the calculations. The t/o data have been obtained from the variation of Pq with Z by numerical integration of the Gibbs-Duhem equation using the Runge-Kutta method (22,23). The comparison of the Pq values of Table I with those obtained by some previous workers (24, 25) shows that our results are at most higher by 0.5-1 Torr.

When the excess chemical potential of the solute in the liquid phase is required as a function of the mole fraction at the constant temperature T0 and pressure P, an integration of the Gibbs-Duhem equation must be used. For this the infinitely dilute solution of the solute in the solvent must be... 

Equation 3.1-1 can be integrated at constant S, V, and to obtain G = -TS + PV+Y, Pj If we write out the total differential of G using this equation and subtract equation 3.1-1, we obtain the Gibbs-Duhem equation. 

Similar to the derivation of the Gibbs-Duhem equation, it is also possible to show the dependence of surface tension on the chemical potentials of the components in the interfacial region. If we integrate Equation (201) between zero and a finite value at constant A, T and nb to allow the internal energy, entropy and mole number to almost from zero to some finite value, this gives... 

In bulk thermodynamics one derives the Gibbs-Duhem equation by integrating equation (AIV.l), keeping the intensive properties constant (dT = 0, p and pt constant) to obtain... 

Mean activity coefficients derived from cryoscopic measurements have been tabulated by C. M. Criss in Appendices 2.4.15-19. These values should be used with caution as considerable uncertainties can arise from (a) the experimental method, fb) the value used for the cryoscopic constant, (c) integration using the Gibbs-Duhem equation (eqn. 2.8.13). 

In bulk thermodynamics, one can derive the Gibbs-Duhem equation by integration of Equation (9.5) while holding the intensive properties T, P, and (Xi constant to give... 

In each of these cases, differentiation of the equations shows that they are true integrations of the Gibbs-Duhem equation, and the values of Aab and Aba are the terminal values of log 7a and log 7b [Eqs. (3.56) and (3.57)]. More complex equations with constants other than the terminal log 7 values are also possible (35). 

Moderate temperature changes result in such minor changes in activity coefficient that constant-pressure data are ordinarily satisfactory for application of the various integrated forms of the Gibbs-Duhem equation. 

Calculate activity coefficients for the system chloroform-acetone at 35.17 C. from vapor-liquid data reported in International Critical Tables (Vol. Ill, p. 286). Fit one of the integrated Gibbs-Duhem equations to the data. With the help of heat of solution data, ibid. Vol. V, pp. 151, 155, 158, estimate the values of the equation constants for 55.1 C., and calculate the activity coefficients for this temperature. Compare with those computed from vapor-liquid data at this temperature, ihid.j Vol. Ill, p. 286. 

As was shown in Chapter 2 (equations (2.4.10) and (2.4.11)), the molar enthalpy of a pure compound Hm T) can be obtained from the tabulated values of Cp T), the heat capacity at constant pressure. For ideal mixtures the Hjok are the same as that of a pure compound. For nonideal mixtures a detailed knowledge of the molar heat capacities of the mixture is needed to obtain the //mi-For a pure compound, knowing p(po> T) at pressurepo and temperature T, the value of p(p, T) at any other pressure p can be obtained using the expression dp = —SjndT+ Vradp which Allows from the Gibbs-Duhem equation (5.2.4), where the molar quantities 5m = S/N and Vm = V/N- Since Tis fixed, dT = 0 and we may integrate this expression with respect to p to obtain... 

In the case of binaiy systems, at constant T, the activities of solvent and solute (components 1 and 2) are determined from integration of the Gibbs-Duhem equation... 

The thermodynamic equilibrium constant IC is then obtained by integrating the appropriate form of the Gibbs-Duhem equation to give as corresponding... 

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