Equilibrium Invariant

It should be noted that we have here considered the system at constant pressure. If we are not considering the system at isobaric conditions, the invariant equilibrium becomes univariant, and a univariant equilibrium becomes divariant, etc. A... 

It is sometimes convenient to fix the pressure and decrease the degrees of freedom by one in dealing with condensed phases such as substances with low vapour pressure. The Gibbs phase rule for a ternary system at isobaric conditions is Ph + F = C + 1=4, and there are four phases present in an invariant equilibrium, three in univariant equilibria and two in divariant phase fields. Finally, three dimensions are needed to describe the stability field for the single phases e.g. temperature and two compositional terms. It is most convenient to measure composition in terms of mole fractions also for ternary systems. The sum of the mole fractions is unity thus, in a ternary system A-B-C ... 

The phase rule(s) can be used to distinguish different types of equilibria based on the number of degrees of freedom. For example, in a unary system, an invariant equilibrium (/ = 0) exists between the liquid, solid, and vapor phases at the triple point, where there can be no changes to temperature or pressure without reducing the number of phases in equilibrium. Because / must equal zero or a positive integer, the condensed phase rule (/ = c — p + 1) limits the possible number of phases that can coexist in equilibrium within one-component condensed systems to one or two, which means that other than melting, only allotropic phase transformations are possible. Similarly, in two-component condensed systems, the condensed phase rule restricts the maximum number of phases that can coexist to three, which also corresponds to an invariant equilibrium. However, several invariant reactions are possible, each of which maintains the number of equilibrium phases at three and keeps / equal to zero (L represents a liquid and S, a solid) ... 

Similarly, in two-component condensed systems, the condensed-phase rule restricts the maximum number of phases that can co-exist to three, which also corresponds to an invariant equilibrium. However, several invariant reactions are possible (Table 11.2), each of which maintain the number of equilibrium phases at three, and keep/equal to zero. The same terms given in Table 11.2 ate also applied to the structures of the phase mixmres. 

What has been done so far is to consider steady-state diffusion in which neither the flux nor the concentration of diffusing particles in various regions changes with time. In other words, the whole transport process is time independent. What happens if a concentration gradient is suddenly produced in an electrolyte initially in a time-invariant equilibrium condition Diffusion starts of course, but it will not immediately reach a steady state that does not change with time. For example, the distance variation of concentration, which is zero at equilibrium, will not instantaneously hit the final steady-state pattern. How does the concentration vary with time ... 

There is a point of invariant equilibrium of siderite + fayalite + magnetite + quartz + graphite + fluid which fixes T — 600°C and <= 6-8 kbar. 

The time-dependent distribution function f (f) from (4a) can be used to show that a translationally invariant equilibrium distribution function leads to a trans-lationally invariant steady state distribution f, even though the SO in (3b) is not translationally invariant itself. To show this, a point in coordinate space (n,..., r y) shall be denoted by r, and shall be shifted, F —> F, with r, = r, -f a for all i a is an arbitrary constant vector. This gives... 

At 3.5 to 5 M HBr, a significant increase in the 100 to 300 cm band area was registered, and this was postulated to be due to the formation of a species SeOBrj". Again, the molar intensity of the band was varied to yield an invariant equilibrium constant and... 

The invariant reactions of the binary Cr-P system between die liquid and die phases (Cr), Cr3P, Cr2P, CrP form monovariant three-phase equiUbria in die ternary decreasing smoothly to the corresponding invariant equilibria of the binary Fe-P system [1939Vog], No invariant equilibrium was found in the ternary system in the region Cr-CrP-FeP-Fe. 

The liquidus surface projection of the partial Fe-Fc3C-WC-W system consfructed by [1968Jel] contradicts to the constitution of die currently accepted binary systems. In particular, die W2C phase is presented by one modification the Fc3C phase is shown as stable invariant equilibrium widi the participation of the liquid, (6Fe) and the p phase in die Fe-W system is presented as the eutectic one taking place at 1520°C, and the corresponding invariant point is shifted towards the timgsten side. Stracture formation of die C-Fe-W alloys... 

Two invariant three-phase equilibria involving the liquid phase were found by thermodynamic calculation by [2006Pal], A maximum for the incongraent monovariant L + (6Fe) — y lies at 1502°C and a minimum for the incongraent monovariant L + y — (Cu) at 1096°C. They are shown in Fig. 1 and the compositions of the phases are given in Table 3. A four-phase eutectoid type invariant equilibrium involving the (eCo), y, a and (Cu) phases lies close to the Co-Cu side of the system according to flic calculation. Experimental confirmation has not yet been obtained. 

At any point along the liquidus curves TA-eAB and Te-eAB, there exist two phases-either solid A -i- melt, or solid B -i- melt Consequently, the degree of freedom F is 1 that is, either the temperature or the composition can be altered without changing the number of phases present (univariant equilibrium). At the eutectic point Oab the two solid phases A and B are in equilibrium with the melt Thus, the number of phases is P = 3, and F = 0, since any variation of the temperature or the composition will invariably displace the system from point Oab at which two solubility (liquidus) curves intersect (invariant equilibrium). 

Equation (1.72) is the unary Gibbs phase rule. It indicates that the maximum number of phases which can coexist in a unary system is 3 and this results in an invariant equilibrium (f = 0). Note that the equilibria in each type of phase diagram in Figure 1.4 satisfy this condition. 

On the other hand, Szekely and Levine and coworkers " made modifications to the original Washburn approach by introducing the appropriate momentum balance equations and considering the deviation from Poiseuille flow, taking into account entrance effects. However, their equations considered the existence of a single and invariable equilibrium contact angle during the entire rate of capillary rise and made no account of inertial forces. 

Tliere have been subtle debates discussing whether negative degrees of freedom are possible. The answer seems to be yes Starting from an invariant equilibrium, one may calculate which properties an additional phase must have in order to fit in with the equilibrium conditions (Oonk,... 

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