Duhem equation constant temperature

The pressure at which standard-state fugacities are most conveniently evaluated is suggested by considerations based on the Gibbs-Duhem equation which says that at constant temperature and pressure... 

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 ... 

This result, known as the Gibbs-Duhem equation, imposes a constraint on how the partial molar properties of any phase may vary with temperature, pressure, and composition. In particular, at constant T and P it represents a simple relation among the Af/ to which measured values of partial properties must conform. 

If temperature and pressure are constant, equation 130 reduces to equation 131 (constant T,P) which is a common form of the Gibbs-Duhem equation. 

The advantages of constant-pressure activity coefficients also become clear when we try to relate to one another the activity coefficients of all the components in a mixture through the Gibbs-Duhem equation (P6, P7). For variable-pressure activity coefficients at constant temperature we obtain... 

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... 

In equation (5.27), we used the Gibbs-Duhem equation to relate changes in the chemical potentials of the two components in a binary system as the composition is changed at constant temperature and pressure. The relationship is... 

The Gibbs-Duhem equation is one of the most extensively used relations in thermodynamics. It is written in the following equivalent forms for a binary solution at constant temperature and pressure ... 

It is often useful (e.g. for dilute solutions) to express the adsorption of components with respect to a predominant component, e.g. the solvent. The component that prevails over m components is designated by the subscript 0 and the case of constant temperature and pressure is considered. In the bulk of the solution, the Gibbs-Duhem equation, , nt dpt = 0, is valid, so that... 

Most thermodynamic data for solid solutions derived from relatively low-temperature solubility (equilibration) studies have depended on the assumption that equilibrium was experimentally established. Thorstenson and Plummer (10) pointed out that if the experimental data are at equilibrium they are also at stoichiometric saturation. Therefore, through an application of the Gibbs-Duhem equation to the compositional dependence of the equilibrium constant, it is possible to determine independently if equilibrium has been established. No other compositional property of experimental solid solution-aqueous solution equilibria provides an independent test for equilibrium. If equilibrium is demonstrated, the thermodynamic properties of the solid solution are also... 

From the Gibbs-Duhem equation at constant temperature and pressure [Equation (11.34)], we can write... 

Turning now to adsorption equilibrium, let us apply algebraic methods to a two component 1,2 phase system. From the phase rule there will be two degrees of freedom, but we shall reduce this to one by maintaining the temperature constant. Then for the total system there exists a Gibbs-Duhem equation... 

GIBBS-DUHEM EQUATION, In a system of two or more components at constant temperature and pressure, the sum of the changes for the various components, of any partial molar quunlily. each multiplied by the number of moles of the component present, is zero. The special case of two components is ihe basis of the Gibbs-Duheni equation of the form ... 

Since in addition to the chemical potentials also the electrical potential 99, affects the charged species, electrochemical potentials //, must be used. We use the symbol 99 instead of -ip because this is the Galvani potential (see Section 5.5). The Gibbs-Duhem equation for changes of state functions at constant temperature is... 

This equation is very similar to the Gibbs-Duhem equation under the condition that the temperature and pressure are constant. A more general relation can be obtained by differentiating Equation (6.10) and comparing the result with Equation (6.1). The differentiation of Equation (6.10) gives... 

The similarity to the Gibbs-Duhem equation is quite apparent, and indeed this equation is the Gibbs-Duhem equation if X refers to the Gibbs energy. We should note that the differential dXt, the differential that appears in Equations (6.13) and (6.14), depends upon the differential quantities of the temperature, the pressure, and the mole fractions as expressed in Equation (6.7). At constant temperature and pressure Equation (6.12) becomes a special case of Equation (6.14). 

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... 

In the second case the liquid and vapor are at equilibrium in a closed vessel without restrictions. An inert gas is pumped into the vessel at constant temperature in order to increase the total pressure. For the present we assume that the inert gas is not soluble in the liquid. (The system is actually a two-component system, but it is preferable to consider the problem in this section.) The Gibbs-Duhem equations are now... 

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... 

Many other tests, too numerous to discuss individually, have been devised, all of which are based on the Gibbs-Duhem equation. Only one such test, given by Redlich, is discussed here and is applicable to the case in which both components are volatile and in which experimental studies can be made over the entire range of composition. The reference states are chosen to be the pure liquid at the experimental temperature and a constant arbitrary pressure P0. The values of A/iE[T, P0, x] and A f[T, P0, x] will have been calculated from the experimental data. The molar excess Gibbs energy is given by Equation (10.62), from which we conclude that AGE = 0 when Xj = 0 and when xt = 1. Therefore,... 

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. 

For AlPO s the natural choice of mole is AlPO. For a two component solution at constant temperature the Gibbs-Duhem equation is... 

At constant temperature, the Gibbs-Duhem equation can be rearranged to give the approximate equality... 

It is clear from the Gibbs-Duhem equation that not all the forces V(/ik/T) are independent. For example, for a two-substance system at constant pressure and temperature, we have... 

Using the forces and flows identified in Eq. (7.1), and the Gibbs-Duhem equation for an n-component system at constant temperature and pressure, we obtain... 

A relation between the surface tension y and the surface stress T can be directly derived from the Gibbs-Duhem equation, Eq. (7). At constant temperature, chemical potentials, and electric potential we have... 

Adsorption is described by the Gibbs-Duhem equation. Considering only a single vapor at constant temperature, electric potential, and elastic strain, the change in surface tension is given by... 

The Wilson equation is an imperfect model. In the isobaric case the effect of neglecting the temperature dependence of A21 and Ai2 and in using the Gibbs-Duhem equation, which was derived for constant temperature and pressure, add to the inherent imperfection. 

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