Condition of mechanical stability

The YBG equation is a two point boundary value problem requiring the equilibrium liquid and vapor densities which in the canonical ensemble are uniquely defined by the number of atoms, N, volume, V, and temperature, T. If we accept the applicability of macroscopic thermodynamics to droplets of molecular dimensions, then these densities are dependent upon the interfacial contribution to the free energy, through the condition of mechanical stability, and consequently, the droplet size dependence of the surface tension must be obtained. 

The coefficient of compressibility is thus always positive for all stable phases (stable or metastable). This is the condition of mechanical stability. 

The condition of mechanical stability, - (92>/9F)y >0, thus means that... 

For binary systems we must in addition take account of the condition of stability with respect to diffusion processes (15.105). It is in fact this condition which here determines the stability of the system. In 8 we shall show why it is that the condition of mechanical stability plays no part in determining the equilibrium in a binary system. 

Let us now investigate the part played by the condition of mechanical stability in binary systems. This condition can be written... 

The condition of mechanical stability hence flays no part in determining the boundary which is fixed by the fact that at this surface the condition of stability of diffusion ceases to be satisfied. This justifies the method which we have employed in 6 and 7, in which we have made use of only the condition for stability with respect to diffusion. 

Now let us see how the condition of mechanical stability arises in the case of a pure substance. For this purpose we write (16.43) in the equivalent form ... 

We note that the increments (SF), and (Sp), are always of opposite sign in a stable phase, since the condition of mechanical stability (15.41) may be written... 

This inequality, called the condition of mechanical stability, implies that if at a constant temperature the pressure in a system increases, then its volume decreases. It follows from Fig. 37 that along the section BC the van der Waals isotherm does not satisfy the condition (3.15a). This signifies that the isotherms obtaind from equation (3.14) are, at least partially, non-physical, and the corresponding states are physically unattainable. Thus, equation (3.14) is only an approximate state equation. 

Note again that the total compressibility must be positive, from the general condition of mechanical stability of the system. In this particular model, the total compressibility is identified with the relaxation term for the conversion between the two species. 

The convolution analysis is based on the use of convolution data and further manipulation to obtain information on the ET mechanism, standard potentials, intrinsic barriers, and also to detect mechanism transitions. It is worth noting that the general outlines of the methodology were first introduced in the study of the kinetics of reduction of terf-nitrobutane in dipolar aprotic solvents, under conditions of chemical stability of the generated anion radical. For the study of concerted dissociative ET processes, linear scan voltammetry is the most useful electrochemical technique. 

The condition of phase stability for such a system is closely related to the behavior of the Helmholtz free energy, by stating that the isothermal compressibility yT > 0. The positiveness of yT expresses the condition of the mechanical stability of the system. The binodal line at each temperature and densities of coexisting liquid and gas determined by equating the chemical potential of the two phases. The conditions expressed by Eq. (115) simply say that the gas-liquid phase transition occurs when the P — pex surface from the gas... 

Thus Eq. (4) and (5) provide two criticahties based on a purely thermodjmamic analysis of the adsorption in the cylindrical pore. During adsorption or desorption, although the thermodynamic stabihty criteria are satisfied, the condition for mechanical stability of the meniscus has to be satisfied. The mechanical stabihty criteria for the cylindrical meniscus (during adsorption) and hemispherical meniscus (during desorption) are given by [4,6,7,12]... 

These are the higher order conditions for mechanical stability, which, as we shall see in chap. XVI, are satisfied at the critical point of a substance. 

Relation between the Conditions of Mechanical and Diffusion Stability in Binary Systems. 

Remember from Sec. 7.3 that while the the condition dP/dV)j = 0 on the van der Waals loop of an equation of state gave the conditions of mechanical stabiliQ, it did not give the vapor-liquid equilibrium points (that is, the vapor pressure). That had to be determined from the equality of species fugacities in each phase. The situation is much the same here in that the limit of stability from Eq. H.2-9 is not the equilibrium compositions found from the equality of species fugacities in the coexisting liquid phases. 

In the PEFC, the membrane, together with the electrodes, forms the basic electrochemical unit, the membrane electrode assembly (MEA). The first and foremost function of the electrolyte membrane is the transport of protons from anode to cathode. On one hand, the electrodes host the electrochemical reactions within the catalyst layer and provide electronic conductivity, and, on the other hand, they provide pathways for reactant supply to the catalyst and removal of products from the catalyst. The components of the MEA need to be chemically stable for several thousands of hours in the fuel ceU under the prevailing operating and transient conditions. PEFC electrodes are wet-proofed fibrous carbon sheet materials of a few 100 ttm thickness. The functionality of the proton exchange membrane (PEM) extends to requirements of mechanical stability to also ensure effective separation of anode and... 

All packing materials are subject to the development of backpressure under flow conditions. The mechanical stability of the gel determines its maximum allowable flow rate in operation. The pressure and flow characteristics, as illustrated... 

This theoretical study gives rise to some practical conclusions. Firstly, to produce a stable source, the condition of initial stability of Eq. (5.64) must be achieved. Consequently, the parameter, determined from Eq. (5.58), should be as small as possible. All parameters in Eq. (5.58), except for S, are rigidly prescribed either by physico-chemical properties of the system or by standards for production and operatiOTi cmiditions. As to the area of the defect, where the cmitact of the oxidiser with lithium occurs (5 ), it depends, in the first place, oti the presence of impurities in Uthium, including mechanical ones. Therefore, the more pure the anode material is, the lesser is the probability of formation of the local source. Secondly, if the conditirm of Eq. (5.64) is satisfied, then, to get in the stationary stability region. Fig. 5.12, the K constant, determined from Eq. (5.55), should be as large as possible. An increase in piuity and the absence of mechanical impurities in lithium also promote this. Besides, the conductivity of local elements, determined as the product of k and the area of the separator pore section, should be maximum. 

Here A is the Helmholtz free energy and V the volume. The first condition indicates that the limit of mechanical stability is reached, the second that the Helmholtz free energy remains convex. 

Thus, an intensive mechanical treatment of aluminum oxides during an adsorbent production is unfavorable, since it can lead to the decrease of thermal stability. Nevertheless, there are definite conditions of mechanical treatment after which the initial y- and x- aluminas became stable at higher temperatures. 

---