Theorem of Gibbs

Two theorems of Gibbs and of Konovalow —Under what circumstances shall we observe such a state of indifferent equilibrium Two important theorems, discovered by J. Willard Gibbs, found anew by D. Konovalow, give us this information. Here are these two theorems ... 

First Theorem of Gibbs and Konovalow.—Under a canr slant pressure cause the composition of the liquid mixture to vary in a wdl-defined way the boiling-point of this mixture changes] if, for a certain composition of the liquid mixture, the boUing-point passes through a maximum or minimum, this liquid mixture gives off a saturated vapor of the same composition, and reciprocally. 

Application of the second theorem of Gibbs and of Kon cvalow to mixtures of volatile liquids.— The study of the tensions of saturated vapors of a mixture whose composition X is varied at a constant temperature T offers remarkable similarities in all points to those we have made on the subject of boiling-points imder a given pressure. 

Second Case Between the points and (Fig. 65) the curve D has a point I, OF ORDNATE P SMALLER THAN ALL THE others.—According to the second theorem of Gibbs and of Konovalow, this point is an indifferent point when the liquid and the saturated... 

G. Bruni has very well shown this opinion to be inadmissible. We may, in fact, apply to the systems we are studying the theorems of Gibbs and of Konovalow (Art. 194), and particularly the first. It suffices to substitute for the words mixed liquid, mixed vapor, the words mixed crystals, mixed liquid. 

The curve C, formed necessarily by two branches both S3mi-metrical with respect to the line X, has, for the abscissa a point of maximum ordinate. From the first theorem of Gibbs and Konovalow (Art. I94 which may be applied to the double mixture formed by the mixed crystals and the mixed liquid, this point belongs also o the line c, for which it is also a point of maximum or minimum ordinate. At this indifferent point I the mixed hquid, which is inactive by compensation, must give, on freezing, mixed holoedral crystals of composition x=i. 

Each of the points on this line may be regarded, if so wished, as a point of maximum ordinate the theorem of Gibbs and Konovalow may be applied to each of these points whatever the composition of the mixed liquid, it deposits mixed crystals of the same composition. 

Computing thermodynamic properties is the most important validation of simulations of solutions and biophysical materials. The potential distribution theorem (PDT) presents a partition function to be evaluated for the excess chemical potential of a molecular component which is part of a general thermodynamic system. The excess chemical potential of a component a is that part of the chemical potential of Gibbs which would vanish if the intermolecular interactions were to vanish. Therefore, it is just the part of that chemical potential that is interesting for consideration of a complex solution from a molecular basis. Since the excess chemical potential is measurable, it also serves the purpose of validating molecular simulations. 

The most broadly recognized theorem of chemical thermodynamics is probably the phase rule derived by Gibbs in 1875 (see Guggenheim, 1967 Denbigh, 1971). Gibbs phase rule defines the number of pieces of information needed to determine the state, but not the extent, of a chemical system at equilibrium. The result is the number of degrees of freedom Np possessed by the system. 

The experimental verification of Gibbs theorem. Since the osmotic pressure of a solution is generally difficult to measure, it is simplest to choose a case such that Raoult s law holds good and the concentration of the solution may be used in place of osmotic pressure. The solution should therefore be dilute and should be a true solution the solute, that is, must be dispersed as simple molecules and not as molecular aggregates like soap micelles. These conditions were obtained by Donnan and Barker Proc. 

The derivation of the phase rule is based upon an elementary theorem of algebra. This theorem states that the number of variables to which arbitrary values can be assigned for any set of variables related by a set of simultaneous, independent equations is equal to the difference between the number of variables and the number of equations. Consider a heterogenous system having P phases and composed of C components. We have one Gibbs-Duhem equation of each phase, so we have the set of equations... 

An ideal gas is a model gas comprised of imaginary molecules of zero volu that do not interact. Each chemical species in an ideal-gas mixture therefore its own private properties, uninfluenced by the presence of other species. This the basis of Gibbs s theorem ... 

Rudorff s experiments, compared with the theorems of J. Willard Gibbs, show us that the mixed crystals should be considered not as two phases, but as a single phase these crystals are not therefore, as many writers have supposed simp y mechanical mixtures, a juxtaposition or a mixing of crystalline particles of potassium sulphate and cr3nstalline pa tides of ammonium sulphate in them the two component salts are physically mixed in a manner as intimate as for an aqueous solution every volume, however small which may be cut from one of these crystals, contains a certain quantity of each one of these salts these mixed crystals, formed by two isomorphous bodies, constitute, according to the... 

Nevertheless, whatever be the outcome of this revolution, it seems to me there is injustice in making the glory of Gibbs consist in this alone, by seeing in him merely the author of the phase rule. In his immortal work. On the Equilibrium of Heterogeneous Substances, this rule is not all it is but one theorem, and is accompanied by other propositions whose importance is not less the theorems on... 

An ideal gas (Sec. i. is a irrodel gas comprised of iirragiirary irrolecules of zero volume that do irot iirteract. Thus, propertiesfor each chemical species are iirdepeirdeirt of the preseirce of other species, and each species has its own set of private properties. This is the basis for the following statement of Gibbs s theorem. ... 

In the last chapter we established two powerful general theorems, those of Gibbs and Duhem, relating to heterogeneous systems. We shall now consider in more detail the quantitative behaviour of some simple systems, beginning with a study of the phase changes of a pure substance. A study of more complex heterogeneous systems will follow in later chapters after we have discussed the thermodynamic conditions of stability. 

This book is the first volume of a Treatise on Thermodynamics based on the methods of Gibbs and De Donder. It deals with the following topics fundamental theorems, homogeneous systems, heterogeneous systems, stability and moderation, equilibrium displacements and equilibrium transformations, solutions, azeotropy, and indifferent states. The second volume deals with surface tension and adsorption while the third and last will be concerned with irreversible phenomena. 

Equilibrium segregation and equilibrium adsorption at solid-gas interfaces have often been formally treated as identical phenomena since both obey the Gibbs adsorption theorem. However, Gibbs rigourous results are difficult to apply due to the lack of information about various parameters, especially the composition dependence of the surface tension ). Therefore, a number of alternative approachs, based on experimental results have been attempted to predict and explain surface segregation. 

The macroscopic description of the adsorption on electrodes is characterised by the development of models based on classical thermodynamics and the electrostatic theory. Within the frames of these theories we can distinguish two approaches. The first approach, originated from Frumkin s work on the parallel condensers (PC) model,attempts to determine the dependence of upon the applied potential E based on the Gibbs adsorption equation. From the relationship = g( ), the surface tension y and the differential capacity C can be obtained as a function of E by simple mathematical transformations and they can be further compared with experimental data. The second approach denoted as STE (simple thermodynamic-electrostatic approach) has been developed in our laboratory, and it is based on the determination of analytical expressions for the chemical potentials of the constituents of the adsorbed layer. If these expressions are known, the equilibrium properties of the adsorbed layer are derived from the equilibrium equations among the chemical potentials. Note that the relationship = g( ), between and , is also needed for this approach to express the equilibrium properties in terms of either or E. Flere, this relationship is determined by means of the Gauss theorem of electrostatics. 

The energy that is bound in one mole ofwater is given by its enthalpy of formation. It differs from the reaction Gibbs energy by the product of the thermodynamic temperature and the entropy of reaction. According to the second fundamental theorem of thermodynamics, a part of the enthalpy of reaction can be applied as thermal energy with a maximum of AQr = TASr which is the amount of energy that corresponds to the entropy of reaction ASr at the thermodynamic temperature T(see Eq. (5.15)). 

Thus, the Euler theorem gives for the function of Gibbs energy at a surface that ... 

Theorems I. and V., or II. and VI., lead to the Theorem VII., due to Gibbs (1876) If a small circuit is drawn around the triple point, it cuts alternately stable and unstable branches of the curves of transition meeting at that point. 

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