Gibbs-Duhem equation liquid phase

Many reactions encountered in extractive metallurgy involve dilute solutions of one or a number of impurities in the metal, and sometimes the slag phase. Dilute solutions of less than a few atomic per cent content of the impurity usually conform to Henry s law, according to which the activity coefficient of the solute can be taken as constant. However in the complex solutions which usually occur in these reactions, the interactions of the solutes with one another and with the solvent metal change the values of the solute activity coefficients. There are some approximate procedures to make the interaction coefficients in multicomponent liquids calculable using data drawn from binary data. The simplest form of this procedure is the use of the equation deduced by Darken (1950), as a solution of the ternary Gibbs-Duhem equation for a regular ternary solution, A-B-S, where A-B is the binary solvent... 

During each phase transition of the type illustrated here, both of the intensive parameters P and T remain constant. Because of the difference in density however, when a certain mass of liquid is converted into vapour, the total volume (extensive parameter) expands. From the Gibbs-Duhem equation (8.8) for one mole of each pure phase,... 

A review is presented of techniques for the correlation and prediction of vapor-liquid equilibrium data in systems consisting of two volatile components and a salt dissolved in the liquid phase, and for the testing of such data for thermodynamic consistency. The complex interactions comprising salt effect in systems which in effect consist of a concentrated electrolyte in a mixed solvent composed of two liquid components, one or both of which may be polar, are discussed. The difficulties inherent in their characterization and quantitative treatment are described. Attempts to correlate, predict, and test data for thermodynamic consistency in such systems are reviewed under the following headings correlation at fixed liquid composition, extension to entire liquid composition range, prediction from pure-component properties, use of correlations based on the Gibbs-Duhem equation, and the recent special binary approach. 

PI4.1 An azeotrope is a constant-boiling solution in which evaporation causes no change in the composition of the liquid. In other words, the composition of the liquid and gaseous phases must be identical. If the vapors may be assumed to be perfect gases, then the ratio of the two partial pressures is equal to the ratio of the mole fractions in the liquid. Use the Gibbs-Duhem equation to show that, at the azeotrope,... 

Two cases arise. The simpler case is one in which we imagine that the liquid is confined in a piston-and-cylinder arrangement with a rigid membrane that is permeable to the vapor but not to the liquid, as indicated in Figure 10.1. Pressure may then be exerted on the liquid independently of the pressure of the vapor. The temperatures of the two phases are equal and are held constant. The Gibbs-Duhem equation for the vapor phase is... 

First we consider the binary systems when no inert gas is used. When only one of the components is volatile, the intensive variables of the system are the temperature, the pressure, and the mole fraction of one of the components in the liquid phase. When the temperature has been chosen, the pressure must be determined as a function of the mole fraction. When both components are volatile, the mole fraction of one of the components in the gas phase is an additional variable. At constant temperature the relation between two of the three variables Pu x1 and yt must be determined experimentally the values of the third variable might then be calculated by use of the Gibbs-Duhem equations. The particular equations for this case are... 

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... 

Consider the A-B binary liquid system in equilibrium with the vapour phase at a constant temperature. Is the composition of the vapour the same as that of the liquid Not necessarily. Let s apply the Gibbs-Duhem equation to the liquid phase. 

These data can be studied in two ways. The first is to use the Gibbs-Duhem equation and numerical integration methods to calculate the vapor-phase mole fractions, as considered in Problem 10.2-6. A second method is to choose a liquid-phase activity coefficient model and determine the values of the parameters in the model that give the best fit of the experimental data. We have, from Eq. 10.2-2b, that at the jth experimental point... 

Eq. (1.26) can be simplified (with use of Gibbs-Duhem equation) for a liquid phase containing two components only, as ... 

A liquid mixture in equilibrium with this solid phase contains c thermodynamic components i, where c > c. The Gibbs-Duhem equation for the liquid mixture is ... 

Throughout the above discussion it has been supposed that the total pressure is held constant by use of an additional component of the vapour phase which is not soluble in the liquid. If this is not the case, the correct form of the Gibbs-Duhem equation is equation (7 39)... 

Under isothermal conditions, the Gibbs-Duhem equation for gas and liquid phases can be written as... 

The Gibbs-Duhem equations of the liquid phase (/) and the outer phase o) read... 

Instead of serving as a basis for the testing of redundant data, the Gibbs-Duhem equation may be used in quite a different manner, one which aids the experimenter in the design of a VLE experiment of minimal complexity. For purposes of discussion, we assume ideal-gas behavior for the vapor phase, and pressure-independence of liquid-phase properties. In this case, Eq. (10) reduces to... 

If solubility of alcohol in the aqueous phase is small or its influence neglected (i.e. the aqueous phase is treated as a two component mixture) then the Gibbs-Duhem equations for both liquid phases are... 

The Gibbs-Duhem equation limits the possible mathematical forms we may choose to represent liquid-phase (or vapor- or solid-phase) activity coefficients. 

Most widely used liquid-phase activity coefficient equations represent the group g liJiTXaXj as some relatively simple algebraic function of the hquid mol fractions. If we choose g l(RTXaXi,) = some constant, we find the symmetrical equation, which is the simplest activity coefficient equation which is consistent with the Gibbs-Duhem equation. More complex functions are more successful at fitting experimental VLE data. 

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