Equilibrium Bivariant

By the variance, or number of degrees of freedom of the system, we mean the number of independent variables which must be arbitrarily fixed before the state of equilibrium is completely determined. According to the number of these, we have avariant, univariant, bivariant, trivariant,. . . systems. Thus, a completely heterogeneous system is univariant, because its equilibrium is completely specified by fixing a single variable— the temperature. But a salt solution requires two variables— temperature and composition—to be fixed before the equilibrium is determined, since the vapour-pressure depends on both. 

This means that two thermodynamic parameters can now vary freely while the system still remains in equilibrium, and the system is said to be bivariant. In this case, the composition becomes an important parameter and must be added to the partial pressure and temperature. Thus, the partial pressure over a nonstoichiometric oxide in a sealed tube will no longer depend solely upon the temperature but on the composition as well (Fig. 7.6) (see also Section 6.8.2). 

The hydrates of sodium carbonate and vapour furnish a system with two independent components, and, if one hydrate be present, the system is bivariant, and at a given temp., the hydrate can exist in equilibrium with water vapour at different press. if two hydrates be present the system is univariant, and it is stable only when the press, at a given temp, has one unique value so that if the... 

If there is no constant influx of fluid of a certain composition, decomposition of magnetite ceases. The limiting case is a dry system closed to CO2. By analogy with systems closed to water, in such a system with constant pressure P — Pf = const) the fluid phase disappears entirely, and the Mgt + Sid + Hem association (system Fe-C-O) becomes bivariant and can exist stably below the P-T curve (see Fig. 77) in the stability field of the Sid -1- Hem (+ fluid) association. From these considerations the Mgt -I- Sid + Hem association cannot be used to judge the low-temperature limit of mineral formation the upper limit is fixed quite definitely inasmuch as removal of CO2 begins at P P and the reaction proceeds irreversibly to the right. The extensive occurrence of magnetite in oxide-carbonate iron-formations of low-rank metamorphism apparently indicates the absence of equilibrium or even a deficiency of COj and special dry conditions. 

At higher temperatures and in the presence of a fluid, bivariant grunerite + magnetite equilibrium occurs, which is superseded above 450-510°C by... 

Equation (42) cannot be used if NO concentrations approach their equilibrium values, since the net production rate then depends on the concentration of NO, thereby bringing bivariate probability-density functions into equation (40). Also, if reactions involving nitrogen in fuel molecules are important, then much more involved considerations of chemical kinetics are needed. Processes of soot production similarly introduce complicated chemical kinetics. However, it may be possible to characterize these complex processes in terms of a small number of rate processes, with rates dependent on concentrations of major species and temperature, in such a way that a function w (Z) can be identified for soot production. Rates of soot-particle production in turbulent diffusion flames would then readily be calculable, but in regions where soot-particle growth or burnup is important as well, it would appear that at least a bivariate probability-density function should be considered in attempting to calculate the net rate of change of soot concentration. 

A very simple case of a bivariant system is furnished by a solid salt in presence of an aqueous solution of this salt two independent components, the salt and the water, have two phases, the solid salt and the solution. For every temperature and pressure such a system is in equilibrium the solution is then saturated with the salt the concentration of the saturated solution depends upon the temperature to which it is brought and upon the pressure it supports but it is independent of the masses of salt and water that the S3rstem contains. Also, if to a knowledge of the temperature and pressure we join the knowledge of the total mass of the salt and water in the 83nstem, the masses of the solution and of the undissolved salt are determined. 

Remark on the law of equilibrium of bivariant systems. — Here we must guard against a possible confusion. We have said that, when the temperature and pressure were stated, the concentration of the saturated solution was determined we understand by this that it is impossible, at a given temperature and pressure, to find a series of saturated solutions such that the concentration varies in a continuous manner from one solution to the following but it is not to be understood that the constitution of the saturated solution is determined without amhiquity it may happen, in fact, that to a given temperature and pressure corre-... 

A similar remark may be made relative to the composition of each of the phases of a bivariant system in equilibrium at a given pressure and temperature it is a remark whose importance we shall see while studying in Chap. XI the indifferent states of a bivariant system. 

The points on the intersections of these faces represent a state of equilibrium, under the given pressure, between the mixture and two distinct solid precipitates for these states the system has become bivariant the bounding line between the domain of the precipitate C and that of C represents all the possible states of equilibrium between the liquid mixture, the precipitate C, and the precipitate C. ... 

If the system consists only of solid mercuric oxide and a mixture of oxygen and mercury vapor, it is divided into two phases only and is bivariant for such a system there is no longer question of a curve of transformation tensions the knowledge of the temperature is not sufficient to determine the pressure supported by the system in equilibrium. 

If the temperature attains, then exceeds, the point of aqueous fusion of these crystals of chlorine hydrate, they disappear the system which then contains but two phases, the liquid mixture and the gaseous mixture, is bivariant at a given temperature the tension of the gaseous mixture which remains in equilibrium above the liquid solution may assume an infinity of values to increase it, the addition of chlorine is sufficient to decrease it, removal of a part of the gaseous mixture. 

Quite otherwise will be the succession of phenomena according to the second hypothesis the system, formed only of two phases, ammonia gas and the solid solution, is bivariant at a single temperature T the system may be observed in equilibrium for an infinity of different values for the tension of the ammonia gas the pressure exerted b this gas at the moment of equilibrium will increase if ammonia gas is added to the system, and will diminish if gas is withdrawn. 

While m is less than s, the hydrogenized palladium is composed of a single solid solution Hie system, which includes only two phases, is bivariant it does not have a fixed dissociation tension the tension of the gaseous hydrogen for the system in equilibrium increases with the amount of hydrogen held in the solid solutions the representative point describes a curve Oa which rises from left to right. 

These properties no longer hold when we study systems whose variance exceeds unity or, at least, are only met with in certain particular cases thus in the next chapter we shall see, in studying bivariant S3mtems, that such a system may sometimes have a particular equilibrium state where are found, all the properties which the equilibrium states of a monovariant S3rstem have manifested but even in the case where such a state of equilibrium will be possible it will represent an exception among the cases of equilibrium of the system considered. 

What we have just said regarding a bivariant S3rstem may be repeated for any multivariant system whatever If certain exceptional equilibrium stales which are indifferent are excluded, every state of equilibrium of a bivariant or multivariant system is stable when the temperature and pressure are kept constant. 

Consider at a gi en pressure and temperature a bivariant system whose twb components are a salt and water and whose two phases are the solid salt and an aqueous solution when the system is in equilibrium, the solution has a definite concentration S it is scUwraJted at the given pressure and temperature. 

Under a given pressure, such as the atmospheric pressure, and at the temperature T, there is stable equilibrium in a bivariant system formed by ice in contact with a salt solution s is the concentration of the solution. i... 

In all cases, by the words is determined is not to be understood a determination which excludes all ambiguity it may happen, and does in certain cases which we shall meet in this chapter, that at a given pressure and temperature a bivariant system formed of the same independent components presents two distinct states of equilibrium corresponding to different compositions of the several phases. 

Consequently, when a bivariant system is thus given, it is impossible, except for the particular case we have just mentioned, to vary the masses of the different phases which are held in equilibrium without causing their composition to vary, so that at constant pressure and temperature the system in equilibrium could not undergo any modification without destroying the equilibrium the exceptional case aside, the equilibrium of a bivariant system is not an indifferent equilibrium this distinguishes the bivariant tems sharply from the monovariant systems. 

It may be shown that the state of equUibrium of a bivariant system is stable except for the special case, when it is indifferent this proposition is of considerable importance, for it shows that in general one may apply to bivariant s3rstems the two laws of displacement of equilibrium by variation of pressure and of the displacement of equilibrium by change of temperature in fact, in the preceding chapter we have borrowed several examples of these laws from the study of bivariant systems. 

When the two precipitates coexist in contact with the solution, the system, always formed of two independent components, is divided into three phases it is no longer bivariant, but monovariant in general it cannot be in equilibrium under the pressure considered, except at a particular temperature which we shall denote by d. 

When the representative point is elsewhere than at w, it is impossible for our bivariant system to remain in equilibrium it must be transformed until the complete disappearance of one of its phases takes place. What laws govern these transformations To determine these laws, we must distinguish three cases, as follows ... 

Formation in solution of a racemic compound.—The precipitation within a solution of one of the substances we have just studied leads to the study of the equilibrium of a system no longer bivariant, but trivariant this study is, from the experimental point of view, much less advanced than the preceding it has given rise nevertheless to several interesting researches among this number is the analysis of the conditions of formation of the double racemate of sodium and ammonium, anal3rsis for which we are indebted to Van t Hoff and van Deventer. ... 

At temperatures included between 1130 and 1000° mixed crystals of martensite may be observed in equilibrium with graphite ciystals to each temperature corresponds an equilibrium state of this bivariant system at atmospheric pressure and the composition of each pha e is given for this state of equilibrium therefore at every temperature comprised between 1130° and 1000° the martensite crystals which may coexist with the graphite crystals have a given composition, and the law connecting this composition with the temperature may be represented by a certain curve. 

At this moment the system is divided into three phases the solution, the double salt, the solid carbonate of magnesium it is besides formed of three independent components, bicarbonate of potassium, magnesium carbonate, water it is therefore a bivariant system under atmospheric pressure there should correspond to every temperature a state of true equilibrium defined by a given composition of the solution. The solution containing almost exclusively potassium bicarbonate and water, its composition may be fixed by its concentration a. The equilibrium would then correspond, for each temperature T, to a value S of the concentration at this temperature, in presence of a solution of concentration less than S(sdouble salt would decompose on the contrary, in presence of a solution of concentration greater than S(s>5), the bicarbonate of potassium would combine with the magnesium carbonate. 

---