Equilibrium condition Gibbs-Duhem equation

Solution of the PB equation with these boundary condition enables one to evaluate the spatial distribution of electric potential, field, and ion concentration. Knowing these quantities, one can evaluate the force and energy of interactions between colloid particles or between particles and interfaces. This can be achieved via the thermodynamic equilibrium condition (Gibbs-Duhem relationship) which can be formulated as [13]... 

As usual, our goal is to find the minimum of G( ) in order to determine the equilibrium position ( = eq) of the chemical reaction at constant T and P. From (6.10c) [cf. the Gibbs -Duhem equation (6.36a)], the differential dG under these conditions is simply... 

If one or more chemical reactions are at equilibrium within the system, we can still set up the set of Gibbs-Duhem equations in terms of the components. On the other hand, we can write them in terms of the species present in each phase. In this case the mole numbers of the species are not all independent, but are subject to the condition of mass balance and to the condition that , vtpt must be equal to zero for each independent chemical reaction. When these conditions are substituted into the Gibbs-Duhem equations in terms of species, the resultant equations are the Gibbs-Duhem equations in terms of components. Again, from a study of such sets of equations we can easily determine the number of degrees of freedom and can determine the mathematical relationships between these degrees of freedom. 

The common characteristics of phase transitions are that the Gibbs energy is continuous. Although the conditions of equilibrium and the continuity of the Gibbs energy demand that the chemical potential must be the same in the two phases at a transition point, the molar entropies and the molar volumes are not. If, then, we have two such phases in equilibrium, we have a set of two Gibbs-Duhem equations, the solution of which gives the Clapeyron equation (Eq. (5.73))... 

B) We have pointed out that experimental studies are usually arranged so that the system is univariant. The experimental measurements then involve the determination of the values of the dependent intensive variables for chosen values of the one independent variable. Actually, the values of only one dependent variable need be determined, because of the condition that the Gibbs-Duhem equations, applicable to the system at equilibrium, must be... 

Equations 5.33 and 5.36 constitute four differential equations in the three dependent variables, fj, f2, f3 (the mole fractions and the conditional equilibrium constants are assumed to be known through measurement). If there are more than three exchanging ions, for each term added to the Gibbs-Duhem equation, there will be additional constraints like Eq. 5.36, so that enough equations always will be generated to express the activity coefficients in terms of the conditional equilibrium constants, as in Eq. 5.25. 

Related Calculations. When vapor-liquid equilibrium data are taken under isobaric rather than isothermal conditions, as is often the case, the right-hand side of the preceding Gibbs-Duhem equation cannot as readily be taken to approximate zero. Instead, the equation should be taken as... 

Stability criteria are discussed within the framework of equilibrium thermodynamics. Preliminary information about state functions, Legendre transformations, natural variables for the appropriate thermodynamic potentials, Euler s integral theorem for homogeneous functions, the Gibbs-Duhem equation, and the method of Jacobians is required to make this chapter self-contained. Thermal, mechanical, and chemical stability constitute complete thermodynamic stability. Each type of stability is discussed empirically in terms of a unique thermodynamic state function. The rigorous approach to stability, which invokes energy minimization, confirms the empirical results and reveals that r - -1 conditions must be satisfied if an r-component mixture is homogeneous and does not separate into more than one phase. 

The tv o-dimensional spreading pressure jr is an intensive property of the adsorbed phase. Considering the adsorption of just a single (superscript 0) component i and respecting the Gibbs-Duhem equation provides for constant temperature and pressure and equilibrium conditions the vell-kno vn Gibbs adsorption isotherm (Myers and Prausnitz, 1965) ... 

The second equation is Young-Laplace equation for a bubble and P e is the vapor pressure of an embryo in equilibrium with the liquid. Using the above equation along with Gibbs-Duhem equation it can be shown that the radius of an embryo at equilibrium conditions are ... 

We further impose an experimentally accessible condition that the clathrate hydrate is in equilibrium with a fluid mixture of guest and water. This is realized by requiring that the chemical potentials of water and guest in the fluid phase are equal to those in the clathrate and the pressure in the fluid is equal to p in Eq. (7). Then, A fig in Eq. (8) is replaced in terms of dp and dJfrom the Gibbs-Duhem equations as... 

The temperature on the coexistence curves corresponds to the temperature at which the transition from one phase to another takes place at a given pressure. Thus, if we obtain an explicit relation between the pressure and the temperature that defines the coexistence curve, we can know how the boiling point or freezing point changes with pressure. Using the condition for equilibrium (7.1.2), we can arrive at a more explicit expression for the coexistence curve. Let us consider two phases denoted by 1 and 2. Using the Gibbs-Duhem equation, dyL = -Sjn dT + Vm dp, one can derive a differential relation between p and Tof the system as follows. From (7.1.3) it is clear that for a component k, d i[ = dpi- Therefore we have the equality... 

If the partial pressure on the solid-matter system is increased by dp(s), the partial pressure of water vapour changes by a certain quantity dp g) to maintain the equilibrium in the system. According to the Gibbs-Duhem equation (5.14), equilibrium can be maintained under isothermal conditions dT = 0), provided that... 

The investigation above is due initially to Gibbs (Scient. Papers, I., 43—46 100—134), although in many parts we have followed the exposition of P. Saurel Joum. Phys. diem., 1902, 6, 474—491). It is chiefly noteworthy on account of the ease with which it permits of the deduction, from purely thermodynamic considerations, of all the principal properties of the critical point, many of which were rediscovered by van der Waals on the basis of molecular hypotheses. A different treatment is given by Duhem (Traite de Mecanique chimique, II., 129—191), who makes use of the thermodynamic potential. Although this has been introduced in equation (11) a the condition for equilibrium, we could have deduced the second part of that equation directly from the properties of the tangent plane, as was done by Gibbs (cf. 53). 

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