The condition of stability

The variation of Gibbs energy with pressure The variation of Gibbs energy with temperature 

Checklist of key concepts 128 Checklist of key equations 129 Further information 3.1 The phase rule 129 

Further information 3.2 Measures of concentration 130 Discussion questions Exercises Projects 

The thermodynamics of phase changes of pure materials is also important because it prepares us first for the study of mixtures and then for the study of chemical equilibria (Chapter 4). Some of the thermodynamic concepts developed in this chapter also form the basis of important experimental techniques in biochemistry, such as the measurement of molar masses of proteins and nucleic acids and the investigation of the binding of small molecules to proteins. 

Because the Gibbs energy, G = H-TS, provides a signpost of spontaneous change when the pressure and temperature are constant, and we need to know the conditions under which a transition from one state to another becomes spontaneous, it is at the centre of all that follows. In particular, we need to know how G depends on the pressure and temperature. As we work out these dependencies, we shall acquire deep insight into the thermodynamic properties of biologically important substances and the transitions they can undergo. 

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It is necessary to show that the subharmonic resonance whose existence we have just ascertained is stable. Here the condition of stability is very simple, since in the stroboscopic method we deal with the stability of the singular point (and not of the stationary motion). 

From (82) we see that the fractions in (31) are all positive, and since the conditions of stability at points 1 and 2 require that ... 

In other papers by the same group, the effects of sulfur adsorbed or segregated on the Ni surface on corrosion or passivation were described, including the sulfur-induced enhancement of dissolution and the blocking of passivation. It was shown how the conditions of stability of adsorbed sulfur monolayers could be predicted on thermodynamical grounds and this was illustrated by a potential-pH diagram for adsorbed sulfur on nickel in water at 25 °C. (See Refs. 22, 25-29 and papers cited therein.)... 

Equilibrium models are powerful tools for describing the conditions of stability of redox components in natural water systems. More extended quantitative inferences must be made with great caution because the systems are generally dynamic rather than equilibrated. 

Thus, the conditions to remain in this cyclic trajectory are mnp. However, one can take advantage of the fact that variables m, , and p are not independent to simplify the expression of the conditions of stability of... 

BaAg2(N02)4.H20, from a soln. of the component salts. The formula is here due to M. Oswald, who examined the three nitrites of the alkaline earths, and found only barium forms a double salt. The conditions of stability are represented by AB, Fig. 85. The corresponding diagrams for strontium and calcium nitrites do not show any evidence of the formation of silver strontium nitrite, or of a silver calcium nitrite nor is any strontium barium nitrite formed, for the two salts give a series of mixed crystals, (Ba,Sr)(N02)2.H20. 

In certain instances of poisoning, especially in the case of base metal catalysts, the deactivation can be simply explained by the formation of new bulk solid phases between the base metal and the poison. Examples are the formation of lead vanadates (14) in vanadia catalysts, or of sulfates in copper-chromite and other base metal catalysts (81). These catalyti-cally inactive phases are identifiable by X-ray diffraction. Often, the conditions under which deactivation occurs coincide with the conditions of stability of these inert phases. Thus, a base metal catalyst, deactivated as a sulfate, can be reactivated by bringing it to conditions where the sulfate becomes thermodynamically unstable (45). In noble metal catalysts the interaction is assumed to be, in general, confined to the surface, although bulk interactions have also been postulated. 

We have established the conditions that must be satisfied at equilibrium, but we have not discussed the conditions that determine whether a single-phase system is stable, metastable, or unstable. In order to do so, we consider the incremental variation of the energy of a system, AE, rather than the differential variation of the energy, SE, for continuous virtual variations of the system. Higher-order terms must then be included. The condition of stability is that... 

We can now consider the conditions of stability for pure substances, binary systems, and ternary systems based on Equations (5.122), (5.132), and (5.134), respectively. In order to satisfy the conditions, the coefficient of each term (except the last term in each applicable equation, which is zero) must be positive. If any one of the terms is negative for a hypothetical homogenous system, that system is unstable and cannot exist. 

For ternary systems the conditions of stability in addition to those given by Equations (5.135) and (5.136) are... 

The determination of the conditions of stability for systems containing more than three components requires only the continuation of the methods that have been discussed. Additional conditions of stability of a homogenous system in an electrical or magnetic field can be obtained by use of the same methods discussed here with the introduction of the appropriate variables. 

The conditions of stability developed in Section 5.15 suggest a boundary between stable and unstable systems. This boundary is determined by the conditions that one of the quantities that determine the stability of a system becomes zero at the boundary at one side of the boundary the appropriate derivative has a value greater than zero, whereas on the other side its value is less than zero. The derivative is a function of the independent... 

Because of the odd order of the variations, the term could have negative values for appropriately chosen variations of the volume and mole number. However, according to the condition of stability, negative values of AA cannot occur and therefore the third-order term must be zero. The coefficient (d3A/dn3)T V is zero because of the chemical potential. The other coefficients become zero when (d2P/dV2)T and (dP/dV)Tn are zero. This condition is consistent with the horizontal points of inflection at the critical point. 

For this term to be positive, (d3P/dV3)T must be negative. If (d3P/dV3)T n = 0, the entire term becomes zero. Under such circumstances the fifth-order term must be zero and the sixth-order term would have to be studied for its positive character. Thus, the conditions of stability of the critical phase in a one-component system must be... 

The conditions of stability for the critical solution phase can be discussed in terms of the Gibbs energy at constant temperature and pressure in terms of Equation (5.155). The second-order term is... 

When (dpjdx j- P = 0, (32G/3n )TP 2 = 0 according to Equation (5.138). The entire term then becomes zero for the critical solution phase. The third-order term must be zero when the second-order term is zero, because otherwise some variations would result in negative values of the term. The condition to make the third term zero is that (d3G/dn])T P ri2 or the equivalent (82pi/3xi) is zero. If the fourth-order term is positive, then (d4G/dn4)T P is positive. Thus, the conditions of stability for the critical solution phase are... 

An additional problem arises when the stability of the critical phase involving the gas-liquid equilibrium in a binary system is studied. The conditions of stability of a homogenous system at constant pressure are d2A/dV2)T x> 0 and d25/dx )T P > 0 from Equations (5.136) and (5.141), respectively. The question arises of which of the two conditions becomes zero first as the boundary between stable and unstable phases is approached. 

Problems concerning the conditions of stability of homogenous systems for critical phases in ternary systems are very similar to those for the gas-liquid phenomena in binary systems, because of two independent variables at constant temperature and pressure. The conditions for stability are (82G/dnl)T P 2 3>0 and (820, given by Equations (5.146) and (5.147), respectively. Inspection of the condition equivalent to Equation (5.148) given by Equation (5.152) shows that (82(j>/dnl)TP> 2i 3 and, therefore, it is the condition expressed by Equation (5.147) or (5.150) that determines the boundary between stable... 

The existence of a solid itself, the solid surfaces, the phenomena of adsorption and absorption of gases are due to the interactions between different components of a system. The nature of the interaction between the particles of a gas-solid system is quite diverse. It depends on the nature of the solid s atoms and the gas-phase molecules. The theory of particle interactions is studied by quantum chemistry [4,5]. To date, one can consider that the prospective trends in the development of this theory for metals and semiconductors [6,7] and alloys [8] have been formulated. They enable one to describe the thermodynamic characteristics of solids, particularly of phase equilibria, the conditions of stability of systems, and the nature of phase transitions [9,10]. Lately, methods of calculating the interactions of adsorbed particles with a surface and between adsorbed particles have been developing intensively [11-13]. But the practical use of quantum-chemical methods for describing physico-chemical processes is hampered by mathematical difficulties. This makes one employ rougher models of particle interaction - model or empirical potentials. Their choice depends on the problems being considered. 

With respect to the molecular interactions the simplest asymmetric films are these from saturated hydrocarbons on a water surface. Electrostatic interaction is absent in them (or is negligible). Hence, of all possible interactions only the van der Waals molecular attraction forces (molecular component of disjoining pressure) can be considered in the explanation of the stability of these films. For films of thickness less than 15-20 nm, the retardation effect can be neglected and the disjoining pressure can be expressed with Eq. (3.76) where n = 3. When Hamaker s constants are negative the condition of stability is fulfilled within the whole range of thicknesses. 

The conditions of stability of asymmetric films is expressed by the following inequalities [539] and analogous relations [531,540]... 

At A2 > 2Aj, the increment 8P becomes negative. This means a violation of the condition of stability of the stationary state in the given autocatalytic chemical system when the fluctuation of the concentration of Aj appears with respect to its stationary value provided that A2 > 2Ai. The violation of the stability happens because the fluctuation in the driving force under the discussed conditions and the increment of the reaction rate caused by the fluctuation appear to be opposite in sign. 

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