Duhems 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 kinetics of combustion reactions turn out to be quite complicated they do not satisfy the classical law of mass action and its kinetic formulation. Neither did Duhem s formal conceptions of the existence of regions of false equilibria and of a special chemical friction, which ignores the molecular mechanism of chemical reactions, correspond to reality. 

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,... 

Rational thermodynamics is formulated based on the following hypotheses (i) absolute temperature and entropy are not limited to near-equilibrium situations, (ii) it is assumed that systems have memories, their behavior at a given instant of time is determined by the history of the variables, and (iii) the second law of thermodynamics is expressed in mathematical terms by means of the Clausius-Duhem inequality. The balance equations were combined with the Clausius-Duhem inequality by means of arbitrary source terms, or by an approach based on Lagrange multipliers. 

Studies on thermodynamic restrictions on turbulence modeling show that the kinetic energy equation in a turbulent flow is a direct consequence of the first law of thermodynamics, and the turbulent dissipation rate is a thermodynamic internal variable. The principle of entropy generation, expressed in terms of the Clausius-Duhem and the Clausius-Planck inequalities, imposes restrictions on turbulence modeling. On the other hand, the turbulent dissipation rate as a thermodynamic internal variable ensmes that the mean internal dissipation will be positive and the thermodynamic modeling will be meaningful. 

On the basis of the Duhem-Margules equation, prove that if one component of a binary mixture exhibits positive (negative) deviations from Raoult s Law, the second must do likewise. (See S. Glasstone, "Thermodynamics for Chemists", D. Van Nostrand, New York, 1947, Chapter 14.)... 

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