Integration of the Gibbs-Duhem equation

In experimental investigations of thermodynamic properties of solutions, it is common that one obtains the activity of only one of the components. This is in particular the case when one of the components constitutes nearly the complete vapour above a solid or liquid solution. A second example is when the activity of one of the components is measured by an electrochemical method. In these cases we can use the Gibbs-Duhem equation to find the activity of the second component. 

We have already derived the Gibbs-Duhem equation in Chapter 1.4. At constant p and T  

In terms of activity and mole fractions this yields 

We may also integrate the Gibbs-Duhem equation using an Henrian reference state for B  

An alternative method of integrating the Gibbs-Duhem equation was developed by Darken and Gurry [10]. In order to calculate the integral more accurately, a new function, a, defined as 

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An expression for V can be obtained from equation (5.29) by integration of the Gibbs-Duhem equation. Starting with the Gibbs-Duhem equation equation (5.23) applied to volume gives... 

L can also be obtained from Lj by integration of the Gibbs Duhem equation... 

A graphical integration of the Gibbs-Duhem equation is not necessary if an analytical expression for the partial properties of mixing is known. Let us assume that we have a dilute solution that can be described using the activity coefficient at infinite dilution and the self-interaction coefficients introduced in eq. (3.64). 

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... 

Jones liquid. Circles denote molecular dynamics results dashed curves were derived from integration of the Gibbs-Duhem equation using molecular dynamics data. Convenient molecular dimensions were used. Reprinted with permission from Deitrick et al. [3],... 

Possible correlatingequationsfor In yi in a binary liquid system are givenfollowing. For one of these cases determine by integration of the Gibbs/Duhem equation [Eq. (11.96)] the corresponding equation for In yz. Wliat is the corresponding equation for... 

By using activity coefficients obtained from b.m.f. measurements, integration of the Gibbs-Duhem equation permits the evaluation of the activity of the solvent (water) cf., Newton and Tippetts, J. Am. Chem. Soc., 58, 280 (1936). 

Values of partial molar thermodynamic functions of silver and gold in the Ag Au alloy as a function of composition for 500 °C. The values for gold were obtained by integration of the Gibbs-Duhem equation (Section 3.2.4, data for 0.169 were omitted for the integration of the... 

Integrations of the Gibbs-Duhem Equation. For ua moles of component A, the decrease in free energy when transferred from the pure state to solution in B is, according to Eq. (3.10)... 

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). 

As described in 9 6 the immediate result of measuring a fireezing-point depression or an osmotic pressure is the activity coefficient of the solvent. Provided that these results are available over a range of concentrations which extend up to very high dilution it is x>ossible to calculate the activity coefficient of the solvie by integration of the Gibbs-Duhem equation. 

Previous experimental works have attempted to make a connection between liquid porosimetry and gas adsorption by proposing transformations between the respective isotherms based upon macroscopic considerations [31-33], We have shown that the Hamiltonian symmetry contained in our model leads to an exact transformation between gas adsorption and liquid porosimetry curves [20], The integration of the Gibbs-Duhem equation expressed in terms of activity leads to... 

Sometimes the thermodynamic data are expressed in the form of an empirical equation. For example, the activity coefficient of a component in a solution is often expressed as a function of composition in terms of an empirical equation. In such cases, the Gibbs-Ouhem equation can be solved analytically instead of graphically. The following example illustrates the analytical integration of the Gibbs-Duhem equation. 

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... 

For a nonvolatile substance we must find a way to determine its activity coefficient that does not depend on measuring its vapor pressure. We will discuss three different methods. The first is through integration of the Gibbs-Duhem equation. The second is through a theory due to Debye and Hiickel, which can be applied to electrolyte solutes. The third method for electrolyte solutes is an electrochemical method, which we will discuss in Chapter 8. Published data are available for common electrolytes, and some values are included in Table A. 11 in Appendix A. 

The final step, step (iii), consists in setting >. = 1. The scaled particle now is an ordinary particle too and its chemical potential, /r, also is that of an ordinary particle. The ideal part of the chemical potential is given by Eq. (5.59), an equation yet to be derived, plus the excess part given by Eq. (5.29), where X = 1. We obtain the pressure in our hard sphere gas via integration of the Gibbs-Duhem equation, i.e. 

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