Condition of equilibrium

For a more detailed illustration, we consider two chambers filled with a gas. The gas may exchange volume energy via a piston. This is shown in Fig. 6.2. The gas with volume V), has a pressure of ph, and the gas with volume V has a pressure of Pi. We presume that the lower pressure pi is smaller than the higher pressure ph. 

Therefore, if the system moves toward equilibrium, its total volume energy will decrease, i.e.. 

However, the statement d(7 0 cannot be true because we have presumed that the total energy should remain constant, as is demanded in an isolated system. In fact, we are dealing here with the experiment of Gay-Lussac. We have forgotten another thermodynamic variable, namely the entropy. The true and complete expression for the energy reads as 

We are dealing here with an isodynamic process, but not with an equilibrium process. Further, since we do not allow the exchange of matter, the mol numbers and thus the corresponding chemical potentials are not relevant in the present consideration. 

In an equilibrium process, d f/ = 0 together with all the constraints in the system. If only the volume may exchange freely, then dSi = 0 and d5/, = 0, but Eq. (6.3) still holds. Thus, we get by adding v(dVtot - dVh + dVi) = 0 to the energy differential equation (6.5) the condition 

Suppose now that we have a system containing a number of phases that may exchange matter in response to changing conditions, but that the system as a whole is of fixed composition. From Chapter 5, we know that if a closed system is constrained to a constant entropy and constant volume, then its energy content will seek a minimum value, and the system will be at equilibrium when dUs,v = 0. In the present case we 

Still want to consider a system that is closed overall, but within which matter is free to move between phases, i.e., in which the phases are open. Still, because the system is closed overall, the same criterion (dUs,v = 0) applies. If we denote the various phases in the system by accents, we can consider that during any increment of change of energy dU the various phases contribute dU, dU , etc., so that 

For the system, S, V, and the quantity of each component are constant, so that  

Equations (14.21) express the conditions for thermal, mechanical, and chemical equilibrium of the system, i.e., that temperature, pressure, and the chemical potential of 

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As in Section III-2A, it is convenient to suppose the two bulk phases, a and /3, to be uniform up to an arbitrary dividing plane S, as illustrated in Fig. Ill-10. We restrict ourselves to plane surfaces so that C and C2 are zero, and the condition of equilibrium does not impose any particular location for S. As before, one computes the various extensive quantities on this basis and compares them with the values for the system as a whole. Any excess or deficiency is then attributed to the surface region. 

To understand the conditions which control sublimation, it is necessary to study the solid - liquid - vapour equilibria. In Fig. 1,19, 1 (compare Fig. 1,10, 1) the curve T IF is the vapour pressure curve of the liquid (i.e., it represents the conditions of equilibrium, temperature and pressure, for a system of liquid and vapour), and TS is the vapour pressure curve of the solid (i.e., the conditions under which the vapour and solid are in equili-hrium). The two curves intersect at T at this point, known as the triple point, solid, liquid and vapour coexist. The curve TV represents the... 

A condition of equilibrium is reached (70-90 per cent, of bisulphite compound with equivalent quantities of the reagents in 1 hour), but by using a large excess of bisulphite almost complete conversion into the... 

Comparable results are not obtained with the less reactive iodine, because the hydrogen iodide formed tends to reduce the iodo compound and a condition of equilibrium is produced ... 

Various constraints may be put on this expression to produce alternative criteria for the directions of irreversible processes and for the condition of equilibrium. For example, it follows immediately that... 

To fully understand the formation of the N13S2 scale under certain gas conditions, a brief description needs to be given on the chemical aspects of the protective (chromium oxide) Ci 203/(nickel oxide) NiO scales that form at elevated temperatures. Under ideal oxidizing conditions, the alloy Waspaloy preferentially forms a protective oxide layer of NiO and Ci 203 The partial pressure of oxygen is such that these scales are thermodynamically stable and a condition of equilibrium is observed between the oxidizing atmosphere and the scale. Even if the scale surface is damaged or removed, the oxidizing condition of the atmosphere would preferentially reform the oxide scales. 

Let the concentration of solvent (B) in equilibrium with the silica gel surface be (c) g/ml. Let a fraction (a) of the surface be covered with a mono-layer of the polar solvent (B) and, of that fraction (a), let a fraction ( 3) be covered by a second layer of the polar solvent (B). The number of molecules striking and adhering to the surface covered with a mono-layer of polar solvent (A) and that covered with a mono-layer of solvent (B) per unit time will be (n ) and be (n") respectively. Furthermore, let the number of molecules of solvent (A) leaving the mono-layer surface and the bi-layer surface per unit time be ni and 2 respectively. Now, under conditions of equilibrium,... 

The changes in the average chain length of a solution of semi-flexible selfassembling chains confined between two hard repulsive walls as the width of the sht T> is varied, have been studied [61] using two different Monte Carlo models for fast equihbration of the system, that of a shthering snake and of the independent monomer states. A polydisperse system of chain molecules in conditions of equilibrium polymerization, confined in a gap which is either closed (with fixed total density) or open and in contact with an external reservoir, has been considered. 

For such a condition of equilibrium to be reached, the atoms must acquire sufficient energy to permit their displacement at an appreciable rate. In the case of metal lattices, this energy can be provided by a suitable rise in temperature. In the application of coatings the diffusion process is arrested at a suitable stage when there is a considerable solute concentration gradient between the surface and the required depth of penetration. 

Guldberg and Waage (1867) clearly stated the Law of Mass Action (sometimes termed the Law of Chemical Equilibrium) in the form The velocity of a chemical reaction is proportional to the product of the active masses of the reacting substances . Active mass was interpreted as concentration and expressed in moles per litre. By applying the law to homogeneous systems, that is to systems in which all the reactants are present in one phase, for example in solution, we can arrive at a mathematical expression for the condition of equilibrium in a reversible reaction. 

That all actual processes are irreversible does not invalidate the results of thermodynamic reasoning with reversible processes, because the results refer to equilibrium states. This procedure is exactly analogous to the method of applying the principle of Virtual Work in analytical statics, where the conditions of equilibrium are derived from a relation between the elements of work done during virtual i.e., imaginary, displacements of the parts of the system, whereas such displacements are excluded by the condition of equilibrium of the system. 

Criterion (1) is seen to be identical with Horstmann s principle it has been largely employed in the treatment of equilibria by Planck. It is, however, not always convenient in application because the systems which actually occur in practice are not isolated we shall therefore modify the relation so as to make it suitable for non-isolated systems. In this investigation we shall recover the first general method for determining the conditions of equilibrium—the principle of dissipation of energy. 

The magnitude on the left is the heat absorbed in the isothermal change, and of the two expressions on the right the first is dependent only on the initial and final states, and may be called the compensated heat, whilst the second depends on the path, is always negative, except in the limiting case of reversibility, and may be called the uncompensated heat. From (3) we can derive the necessary and sufficient condition of equilibrium in a system at constant temperature. 

The suffix x indicates that besides T, all the variables xu a, . . . during the change of which external work is done, are maintained constant (adynamic condition). Thus, if the only external force is a normal and uniform pressure p, then x — v, the volume of the system, and (11) is the condition of equilibrium at constant temperature and volume. 

Since in the majority of cases we have to deal with constant pressure, it is most useful to express the conditions of equilibrium in terms of the potential . 

If the reaction occurs at constant temperature and volume we have as the condition of equilibrium ... 

The determination of the chemical potentials of the components of a solution, in terms ofp, T and the masses (or concentrations) is, from what precedes, equivalent to finding the conditions of equilibrium.. 

Conditions of equilibrium, 92 Conduction of heat, 48, 84, 454 Configuration, 22, 107 Connodal curve, 243 Conservation of energy, 35 Contact potential differences, 470 Continuity of states, 174 Corresponding states, 228, 237 Creighton. See Southern. 

J.S. Anderson, The Conditions of Equilibrium of Nonstoichiometric Chemical Compounds, Proc. Roy. Soc. (London), A185, 69-89 (1946). 

The temperature at which this condition is satisfied may be referred to as the melting point Tm, which will depend, of course, on the composition of the liquid phase. If a diluent is present in the liquid phase, Tm may be regarded alternatively as the temperature at which the specified composition is that of a saturated solution. If the liquid polymer is pure, /Xn —mS where mS represents the chemical potential in the standard state, which, in accordance with custom in the treatment of solutions, we take to be the pure liquid at the same temperature and pressure. At the melting point T of the pure polymer, therefore, /x2 = /xt- To the extent that the polymer contains impurities (e.g., solvents, or copolymerized units), ixu will be less than juJ. Hence fXu after the addition of a diluent to the polymer at the temperature T will be less than and in order to re-establish the condition of equilibrium = a lower temperature Tm is required. 

The derivation of the quantitative relationship between this equilibrium temperature and the composition of the liquid phase may be carried out according to the well-known thermodynamic procedures for treating the depression of the melting point and for deriving solubility-temperature relations. The condition of equilibrium between crystalline polymer and the polymer unit in the solution may be restated as follows ... 

Turning now to the case of equilibrium with an infinite external solution containing c/ moles of electrolyte Mp Av per liter, we note that Eq. (B-3) is then equivalent to Eq. (45). As a further condition of equilibrium, it is required that the activity of this electrolyte be the same inside and outside the gel. The activity of the electrolyte being equal to the product of the activities of the individual ions into which it dissociates, this condition may be stated as follows... 

Now we focus our attention on the conditions of equilibrium for a fluid spheroid rotating about a constant axis. In this case the mutual position of fluid particles does not change and all of them move with the same angular velocity, a>. As is well known, there is a certain relationship between the density, angular velocity, and eccentricity of an oblate spheroid in equilibrium. In studying this question we will proceed from the equation of equilibrium of a fluid, described in the first section. 

Thus, the pressure has a maximum at the center and the decreases as a parabolic function and it is equal to zero at the pole. Next, consider the distribution of pressure in the channel A, where both the attraction and centrifugal forces act on any particle. Inasmuch as a difference of a pressure at terminal points of both channels is the same and a >, it is natural to assume that the attraction field in the channel A is smaller and suppose that the correction factor is equal to the ratio of axes, bja. Correspondingly, a condition of equilibrium is... 

Next we describe La Coste s invention and demonstrate that it allows one to make a spring with much smaller stiffness. From the condition of equilibrium it follows that... 

Taking into account the condition of equilibrium. Equation (3.161), the last equality is simplified... 

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