Gibbs theorem

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. 

Although absolutely conclusive quantitative evidence for Gibbs theorem is so far lacking, its qualitative truth is apparent from so many distinct sources that we may build upon it with considerable confidence, and in the remainder of this book it will be treated as a valid principle. 

The name surfactant refers to molecules that are surface-active , usually in aqueous solutions. Surface-active molecules adsorb strongly at the water-air interface and, because of this, they substantially reduce its surface energy (Gibbs theorem). This is the opposite behaviour from that observed for most inorganic electrolytes, which are desorbed at the air interface and hence raise the surface energy of water (slightly). Surfactant molecules are amphiphilic, that is, they have both hydrophilic and hydrophobic moieties, and it is for this reason that they adsorb so effectively at interfaces (note that amphi means of both kinds in Greek). 

From Fig.7 the calculated 3 -values of 1i1 CgSOC-CjFNa solution at different surface tension are shown in Table B together with the X g- obtained from the adsorption data by the Gibbs theorem. For the 1i1 mixed solution, assuming ideal (as the solution is dilute), then ... 

In the formation of polymeric foams, a number of the relationships described below are applicable, at least to some extent, when the polymer phase is stUl a liquid. In order to form a stable foam, there must be at least two components, one which is preferentially absorbed at the surface. The Gibbs theorem teaches that the surface tension is dependent upon the type and amount of absorbed solute, as follows ... 

Consider a closed system. Gibbs theorem enables us to determine the Gibbs variance Vq pertaining solely to the intensive physico-chemical variables (external and compositional). Duhem s theorem gives a variance Vd=P pertaining to both the intensive and extensive values of a closed... 

We can show that the above conclusion is true in the case of chemical reactions. These results give full importance to the physical implications of the principle of minimum entropy production. As is well known, the entropy is a maximum at the equilibrium state, and it increases monotonically with time as demonstrated by Boltzmann s and Gibbs / -theorems. The present theorem states that the entropy production is not only a minimum at the stationary state, but also that it decreases monotonically until it attains the stationary value. 

This variance is calculated by Gibbs theorem (or phase rule). If the system contains n constituents connected to each other by r linearly independent chemical reactions and if it consists of cp phases, the variance is as follows ... 

Let us again consider the Gibbs theorem for these systems. In both cases, the number of independent components is 2 (salt and water) and the number of external physical parameters taken into account is also 2 (pressure, temperature). In both systems, there is a gas phase and if cp is the number of solid phases, the Gibbs theorem is written ... 

This influence of grain sizes in fact increases the variance of the system by 1, by increasing the term p of the Gibbs theorem, with the introduction of surface tension. 

Gibbsitic [14762-49-3] Gibbs-Kelvin equation Gibbs phase rule Gibbs s phase rule Gibbs s theorem Gibbs-Thomson equation... 

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. 

Previous to the researches of Ivonowalow, the vapour pressures of mixtures had been investigated theoretically by G. Kirchhoff (Pogg. Ann. (1858), 103, 104 Ostw. Klass. No. 101), and by Gibbs Scientific Papers, Yol. I.). The latter had established the theorem relating to mixtures with stationary vapour pressures. 

Gas, cells, 464, 477, 511 characteristic equation, 131, 239 constant, 133, 134 density, 133 entropy, 149 equilibrium, 324, 353, 355, 497 free energy, 151 ideal, 135, 139, 145 inert, 326 kinetic theory 515 mixtures, 263, 325 molecular weight, 157 potential, 151 temperature, 140 velocity of sound in, 146 Generalised co-ordinates, 107 Gibbs s adsorption formula, 436 criteria of equilibrium and stability, 93, 101 dissociation formula, 340, 499 Helmholtz equation, 456, 460, 476 Kono-walow rule, 384, 416 model, 240 paradox, 274 phase rule, 169, 388 theorem, 220. Graetz vapour-pressure equation, 191... 

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. 

Tbe angel is probably named after the great American physical chemist J. Willar Gibbs, who developed most of the theorems that permit the application of thermodynamics to interfaces. 

Clausius great paper of 1850 can be recognized as a landmark in the development of thermodynamics. As remarked by Thomson in 1851, the merit of first establishing [Carnot s theorem] upon correct principles is entirely due to Clausius. In his 1889 eulogy of Clausius, Gibbs praised the 1850 paper in the following terms ... 

The following three chapters will be devoted to summarizing some leading theorems and results from Gibbs great monograph. 

Why does the Gibbs-Roozeboom triangle work The answer becomes apparent from the following elementary geometrical theorem ... 

Boltzmann s tombstone in Vienna bears the famous formula 5 = k log W (W = Wahrscheinlichkeit—probability) that was a signature of his audacious concepts. The alternative formula (13.69) (which reduces to Boltzmann s in the limit of equal a priori probabilities pa) was ultimately developed by Gibbs, Shannon, and others in a more general and productive way (see Sidebar 13.4), but the key step of employing probability to trump Newtonian determinism was his. Boltzmann was long identified with efforts to establish the //-theorem and Boltzmann equation within the context of classical mechanics, but each such effort to justify the second law (or existence of atoms) in the strict framework of Newtonian dynamics proved futile. Boltzmann s deep intuition to elevate probability to a primary physical principle therefore played a key role in efforts to find improved foundation for atomic and molecular concepts in the pre-quantum era. 

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