Gibbs-Duhem law

Using the Gibbs-Duhem law (m djU + m2djU2 = 0) (Prigogine and Defay, 1954), the equilibrium osmotic pressure in the biopolymer solution can... 

But Langmuir s isotherm for the solute entails the generalized form of Raoult s law (Eq. 13) as a necessary thermodynamic consequence. This can best be seen from the Gibbs-Duhem equation,... 

In summary, in the limit as x2 —> 0 and xi — 1, /i —>.V /f and f2 —> x2A h..x-It can be shown from the Gibbs-Duhem equation that when the solute obeys Henry s law, the solvent obeys Raoult s law, To prove this, we start with the Gibbs-Duhem equation relating the chemical potentials... 

Chapter 4 presents the Third Law, demonstrates its usefulness in generating absolute entropies, and describes its implications and limitations in real systems. Chapter 5 develops the concept of the chemical potential and its importance as a criterion for equilibrium. Partial molar properties are defined and described, and their relationship through the Gibbs-Duhem equation is presented. 

The behaviour of most metallurgically important solutions could be described by certain simple laws. These laws and several other pertinent aspects of solution behaviour are described in this section. The laws of Raoult, Henry and Sievert are presented first. Next, certain parameters such as activity, activity coefficient, chemical potential, and relative partial and integral molar free energies, which are essential for thermodynamic detailing of solution behaviour, are defined. This is followed by a discussion on the Gibbs-Duhem equation and ideal and nonideal solutions. The special case of nonideal solutions, termed as a regular solution, is then presented wherein the concept of excess thermodynamic functions has been used. 

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... 

We can show that if the solute obeys Henry s law in very dilute solutions, the solvent follows Raoult s law in the same solutions. Let us start from the Gibbs-Duhem Equation (9.34), which relates changes in the chemical potential of the solute to changes in the chemical potential of the solvent that is, for a two-component system... 

We can also show that Raoult s law implies Henry s law by applying the Gibbs-Duhem equation to Raoult s law. From Equation (14.6) and Equation (14.7), we conclude that [compare with Equation (15.21)]... 

The use of the Gibbs-Duhem equation to derive the limiting laws for coUigative properties is based on the work of W. Bloch. 

In Chapters 16 and 17, we developed procedures for defining standard states for nonelectrolyte solutes and for determining the numeric values of the corresponding activities and activity coefficients from experimental measurements. The activity of the solute is defined by Equation (16.1) and by either Equation (16.3) or Equation (16.4) for the hypothetical unit mole fraction standard state (X2° = 1) or the hypothetical 1-molal standard state (m = 1), respectively. The activity of the solute is obtained from the activity of the solvent by use of the Gibbs-Duhem equation, as in Section 17.5. When the solute activity is plotted against the appropriate composition variable, the portion of the resulting curve in the dilute region in which the solute follows Henry s law is extrapolated to X2 = 1 or (m2/m°) = 1 to find the standard state. 

The set of basic equations is completed by the Gibbs-Duhem (the local formulation of the second law of thermodynamics) and the Gibbs relation (which connects the pressure P with the other thermodynamic quantities), which we will use in the following form ... 

In a binary solution, if the solute follows Henry s law, the solvent follows Raoult s law. (One may prove this using Gibbs-Duhem equation.)... 

The Gibbs/ Duhem equation provides a relation between the Lewis/Randall rule and Henry s law. Substituting dGt from Eq. (11.28) for dAft in Eq. (11.8) gives, for a binary solution at constant T and P,... 

By using Raoult s Law for component 1 and the Gibbs-Duhem relation, show that component 2 must satisfy Henry s Law over the composition range x2 - 1 — xx for which Raoult s Law applies for component 1. (See S. Glasstone, "Thermodynamics for Chemists", D. Van Nostrand, New York, 1947, Chapter 14.)... 

Hemy s law is related to the Lewis/Randall rale tlirough the Gibbs/Duhem equation. Writing Eq. (11.14) for a binary solution and replacing M, by G,- = /i gives ... 

Henry s law applies to a species as it approaches infinite dilution in a binary solution, and the Gibbs/Duhem equation insures validity of the Lewis/Randall rule for the other species as it approaches purity. 

The formalism of the statistical mechanics agrees with the requirements of the equilibrium thermodynamics if the thermodynamic potential, which contains all information about the physical system, in the thermodynamic limit is a homogeneous function of the first order with respect to the extensive variables of state of the system [14, 6-7]. It was proved that for the Tsallis and Boltzmann-Gibbs statistics [6, 7], the Renyi statistics [10], and the incomplete nonextensive statistics [12], this property of thermodynamic potential provides the zeroth law of thermodynamics, the principle of additivity, the Euler theorem, and the Gibbs-Duhem relation if the entropic index z is an extensive variable of state. The scaling properties of the entropic index z and its relation to the thermodynamic limit for the Tsallis statistics were first discussed in the papers [16,17],... 

It is understood that these relations are derived adopting several relations from irreversible thermodynamics, e.g., the second law of thermodynamics, the Gibbs-Duhem relation, the linear law and the Onsager reciprocal relations [39, 22, 62, 18, 5]. 

This is the form of the Gibbs-Duhem equation needed to relate the activity of component B in solution to that of component A. Choosing the Raoult law activity for the solvent A, and the Henry law activity for the solute B, equation (1.13.4) may be rewritten as... 

There was justification as well for utilization of the osmotic coefficient data for the linear polyelectrolyte analogue in the Gibbs Duhem equation to compute resin-phase activity coefficients for the exchanging ions. However, since the trend, with dilution, of osmotic coefficient data for fully dissociated polyelectrolytes cannot be deduced beyond the lowest measurable concentj ation as it can with simple electrolytes, where the Debye-Hilckel limiting law applies, the computation with this equation of mean molal activity coefficients meaningfully related to a value of unity for the polyelectrolyte at infinite dilution was impossible. It was necessary to use the equation as shown below to compute mean molal activity coefficient values, y, as a function of counterion concentration, m, relative to 5n indeterminate mean molal activity coefficient,, at the low... 

Using the Gibbs-Duhem equation ((A2.1.27) with dT =0,dp = 0), one can show that the solvent must obey Raoult s law over the same concentration range where Henry s law is valid for the solute (or solutes) ... 

Differentiating both sides with respect to a and using the differential form of the First Law, dU=T dS-P,dV + p dN, one obtains the Gibbs-Duhem equation ... 

This approximation is usually valid when the mole fraction of a component is near one. In a two-component mixture, one can establish from the Gibbs-Duhem relations that if the first component obeys the Meal mixture, then the second component follows Henry s law ... 

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