Duhem’s law

Experience has shown that to describe equilibrium systems, or to distinguish between one equilibrium state and another, it is not necessary to enumerate all the properties of a system. It has been observed that for each equilibrium system, whenever a certain number of properties have been specified, all the others have fixed and unchanging values. For example, for any sample of any gas, whenever the pressure and temperature have been fixed, the volume will be observed to take on a fixed value as well. If the temperature and volume are fixed, the pressure will be observed to have a fixed value also. The gas has a number of other properties too, such as heat capacity, viscosity and so on, and it is possible in principle, if not always in practice, to fix all the properties of the system by fixing any two. (We will describe this notion more formally in a later chapter ( 5.4.1) as Duhem s Law.)... 

This is referred to as Duhem s law or principle (and incidentally is the origin of the 2 in the phase rule). For example, all of the properties of a system at equilibrium will be fixed at a stated temperature and pressure, or temperature and volume, etc. If the system is capable of changing composition, then of course there are additional compositional variables. The law refers to systems of fixed composition and only the usual thermodynamic variables are included (ignoring electrical and magnetic fields, etc.). The word maximum occurs to take care of those cases where, due to the existence of more than one phase, the system is univariant or invariant, as we will discuss in connection with the phase rule. 

Duhem s law, which relates to all the intensive and extensive variables, and is applicable only to closed systems. 

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... 

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)]... 

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. 

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,... 

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.)... 

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 - xi for... 

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. 

How could we ever explain This is a question that has bothered philosophers of science for at least six decades. As we can see from the quote above taken from the third book of Nietzsche s The Gay Science, it is a question that has also been of great metaphysical importance. What do we require for an explanation to be valid And what distinguishes explanation frommere description The quote from Nietzsche, strangely enough, could as well stem from the opus of one of the founders of modern philosophy of science and a catholic physicist, Pierre Duhem. Yet, whereas Nietzsche s aim was to discredit metaphysics for its incapability to deliver valid explanations, Duhem s objective was just the opposite i.e., to exclude explanatory claims from science and to leave them completely to metaphysics. Because if the aim of physical theories is to explain experimental laws, theoretical physics is not an autonomous science, it is subordinate to metaphysics (Duhem 1991, 19). 

Alternatively, for the gas phase obeying Dalton s law, y can be obtained directly (i.e., without calculating 7hp and 7 ) using the Duhem s equation [6] ... 

The activity coefficients and the equilibrium gas-phase composition have been calculated by using relationships based on Dalton s law. This also applies to Duhem s equation in the form of Eq. (8.22). For the water HP system this approach is approximately valid at T < 520 K. Comparison of activity coefficients and gas-phase composition obtained within the Redlich-Kister approximations and by numerical integration of Eq. (8.22) indicates that the modified approach of Eq. (8.23) agrees well with the solution of Eq. (8.22). With this modification, Eqs. (8.8), (8.9), (8.17), (8.18), and (8.23) provide the explicit and fairly accurate dependencies of activity coefficients on solution composition and temperature. Based on the activity coefficients, gas-phase composition can be readily obtained by using the approximations for the total pressure and Eqs. (8.3) and (8.4). Application of these formulae for estimating activity coefficients and gas-phase concentrations at higher temperatures [T > 520 K) should be considered as extrapolation. The accuracy of this extrapolation can be worse as compared to total pressure calculations. 

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) ... 

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 ... 

Van t Hoff proved by thermodynamics that Raoult s law of vapour pressure lowering and the formula for the molecular depression of freezing-point follow from the osmotic pressure equation. L. G. Gouy and G. Chaperon, Duhem, and Arrhenius, also showed by thermodynamics that the osmotic pressure and vapour pressure lowering are connected, and hence osmotic pressure and vapour pressure and freezing-point lowerings. 

The general principles established for ideal solutions, such as Raoult s law in its various forms, are of course applicable to solutions of any number of components. Similarly, the Gibbs-Duhem equation is applicable to nonideal solutions of any number of components, and as in the case of binary mixtures various relationships can be worked out relating the activity coefficients for ternary mixtures. This problem has now been attacked from several points of view, a most excellent summary of which is presented by Wolil (35). His most important results pertinent to the problem at hand are summarized here. 

The Glbba-Duhem equation in relation to Raoult s and Henry s laws... 

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