Duhem equation constant pressure

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. 

B. Constant-Pressure Activity Coefficients and the Gibbs-Duhem Equation.. 158... 

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... 

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

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 ... 

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,... 

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). 

The derivatives (dP/dT)S3t and (dxt/dT)sat may be determined experimentally or by solution of the set of Gibbs-Duhem equations applicable to each phase, provided we have sufficient knowledge of the system. If the system is multivariant, a sufficient number of intensive variables—the pressure or mole fractions of the components in one or more phases—must be held constant to make the system univariant. Thus, for a divariant system either the pressure or one mole fraction of one of the phases must be held constant. When the pressure is constant, Equation (9.9) becomes... 

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... 

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... 

At constant pressure VfiiT = V/rJ is the concentration-dependent part of the chemical potential gradient. Through the Gibbs-Duhem equation, we can relate the chemical potentials of heavy h and light 1 components in the gas phase as follows ... 

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... 

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. 

There are three sources of error in the calculated vapor composition when these are calculated from boiling point data random error in each experimental observation systematic error in one or more of the observations and the model is imperfect (this is particularly true for isobaric data because use is made of the Gibbs-Duhem equation which was derived for constant temperature and pressure). In the present work we shall assume that the only error in the data is caused by randomness. 

At constant temperature and pressure, the concentration-dependent activity coefficient can be determined from the free excess enthalpy by differentiation through the mole fraction. These equations are the basis for the methods of Wilson and Prausnitz to calculate the activity coefficient [19, 20], The Gibbs-Duhem equation is again a convenient method for checking the obtained equilibrium data ... 

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