System univariant

In this section we consider systems in which the number of degrees of freedom, /, is one. These systems are termed univariant and represent states of coexistence of r + 1 phases. 

In a univariant system, the Gibbs-Duhem equations take a particularly simple form. We may choose dT as the free variation and rewrite Eq. (9-8) in the form 

Equations (9-10) are a set of r + 1 linear algebraic equations for the r + variables dpjdT, dpijdT,. . . , dpJdT. The solution for dpjdT is 

1 mole from phase 1 to phase 2 in equilibrium with it and is 

The phase diagram for a typical one-component system is illustrated in Fig. 9-1. The solid, liquid, and vapor regions are one-phase systems for which / = 2. The curves, whose slopes are given by Eq. (9-14), represent states of coexistence of pairs of phases and are univariant. At the triple point, three phases coexist, and the system then is invariant. 

Take as an example the direct passage of magnesium from solid state into vapor state  

The number of independent coirqronents n - r = 1, and the variance v is 1 therefore, if temperature is fixed, the vapor pressme of magnesimn is also fixed, as is shown by the law of mass action  

For example, in the case of plaster, the transformation of solnble anhydrite (also called anhydrite y), which is a solid solution of calcium sulfate and water, into 

As in section 2.5.1, this representation is incorrect because of the intervention of variable x. Also, it is more accurate to write the double equilibrium of phase transformation and transfer of water in the form  

It is easier to assume that n r = 2 (calcium sulfate and water) and therefore V = 1, and, for example, if the vapor pressure of water is fixed, temperature and the value of X, that is, the water content of the solid solution, are also fixed. 

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Corollary 1.—Every spontaneous isopiestic change in a univariant system evolves heat if it takes place at a temperature... 

Corollary 2.—If there are two opposite isopiestic transformations possible for a univariant system at two different temperatures, the one occurring at a lower temperature will give rise to an evolution, that at the higher temperature to an absorption of heat. 

In traditional method validation, assessment of the calibration has been discussed in terms of linear calibration models for univariate systems, with an emphasis on the range of concentrations that conform to a linear model (linearity and the linear range). With modern methods of analysis that may use nonlinear models or may be multivariate, it is better to look at the wider picture of calibration and decide what needs to be validated. Of course, if the analysis uses a method that does conform to a linear calibration model and is univariate, then describing the linearity and linear range is entirely appropriate. Below I describe the linear case, as this is still the most prevalent mode of calibration, but where different approaches are required this is indicated. 

If the temp, is above the f.p., and ice is accordingly excluded, nine univariant systems, with three solid phases, are theoretically possible. W. Meyerhoffer and A. P. Saunders have studied this system in more detail, and the transition temp, was found to be 4 4°, not 3 7° they worked at 0°, 4" 4°, 16°, and 25° W. C. Blasdale followed up the work at 50°, 75°, and 100°. W. C. Blasdale s experimental data for 0°, 25°, and 50° expressed in eq. mols. of the various salts per 1000 mols. of water, are plotted in Figs. 54 to 56, with respect to four axes representing the four component salts—... 

Univariant Systems (/= 1 p = c + 1) For this case, Gibbs discovered the explicit determinantal solution [J. W. Gibbs. Collected Works (Longmans Green, New York, 1928), Vol. I, pp. 98ff cf. also J. G. Kirkwood and I. Oppenheim. Chemical Thermodynamics (McGraw-Hill, New York, 1961), p. 118] ... 

As indicated in connection with iodine tetroxide, N. A. E. Millon observed the formation of what he considered to be nitroso-iodic acid, and H. Kammorer what he considered to be iodine dinitrosyl tetroxide, I204(N0)8, or maybe O I.O.NO, by treating iodine with nitric acid, or nitrosylsulphuric acid. The yellow powder was considered by H. Kappeler to be iodine nitrate, T.NOj, vide 2. 19, 9. E. Briner said that mixtures of nitric and hydriodie acids do not form a univariant system like stabilized aqua regia, for the iodine produced as a result of the primary reaction is oxidized to iodic acid, with the evolution of nitric oxide. 

Univariant systems containing two components are exemplified by the equilibrium at a eutectic or peritectic. In each case a liquid phase is in equilibrium with two solid phases. Since such systems are univariant, the temperature is a function of the pressure, or the pressure is a function of... 

In this section we consider the heat capacity of a complete system rather than that of a single phase. Equation (9.2) continues to be the basic equation with the condition that d W = 0. The development of the appropriate equations requires expressions for the differential of the enthalpy with a sufficient number of conditions that the heat capacity is defined completely. There are two general cases univariant systems and multivariant systems. 

The state of a closed, univariant system is defined by assigning values to the temperature, the volume, and the mole numbers of the components. For a closed system the mole numbers are constant. Then, to define the heat... 

The simplest univariant system is a one-component, two-phase system. The development of the expression for (8H/dT)V n for such a system is discussed in Section 8.3. The next-simplest system is a two-component, three-phase system. The appropriate equations for this system are... 

The state of a multivariant system is defined by assigning values to either the temperature, volume, and mole numbers of the components or the temperature, pressure, and mole numbers. Thus, we define heat capacities at constant volume or heat capacities at constant pressure for such closed systems. The equations and method of calculation are exactly the same as those outlined for univariant systems when the heat capacity at constant volume is desired. For the heat capacity at constant pressure, Equation (9.14) or (9.15) and the set of equations, one for each component, illustrated by Equation (9.18) are still applicable. The method of calculation is the same, with the exception that the volume of the system is a dependent variable... 

The values of An [T, P, x] at different compositions obtained by the case of Equation (10.86) are not isothermal, because in the univariant system the temperature is a function of the mole fractions. Corrections must be made to obtain isothermal values of An. We choose some arbitrary temperature T0. Then, according to Equation (11.120),... 

However, if we eliminate dju2 from Equation (10.236) by the use of Equation (10.234) and then substitute for d/ij by an equation giving dju3 as a function of the temperature and two mole fractions, we obtain one equation involving three variables for a univariant system. This difficulty can be avoided if we first express d/i2 as... 

Metastable Equilibria The change of SR to SM occurs very slowly. If enough time for the change is not allowed and SR is heated rapidly, it is possible to pass well above the transition point (B) without obtaining SM. In that case, the curve AB extends to O. The curve AO is known as metastable vaporisation curve of SR. The phases SR and Sv will be in metastable equilibrium along this curve. It is a univariant system. 

There are other types of equilibria, in addition to the invariant type, which can be deduced from Eq. 2.5. For example, when three phases of a two-component system are in equilibrium, such as with a closed vessel containing hydrogen gas in equilibrium with a metal and the metal hydride, immersed in a water bath, it is possible to change the value of just one variable (temperature or pressure or composition) without changing the number of phases in equilibrium. This is called univariant equilibrium (/ = 1). If the composition is held constant, temperature and pressure will have a fixed relationship in a univariant system. Hence, if the pressure of hydrogen gas in the vessel is increased slightly, the temperature of its contents remains the same as heat escapes through the vessel walls to the water bath. 

Presence of Complex Molecules.— The Phase Rule, we have seen, takes no account of molecular complexity, and so it is found that the system water— vapour or the system acetic acid— vapour behaves as a univariant system of one component, although in the liquid and sometimes also in the vapour different molecular species (simple and associated molecules) are present. Such systems, however, it should be pointed out, can behave as one-component systems only if at each tmperature there exists an equilibrium between the different molecular species [pseudo-components) in each phase separately and as between the two phases and only if these equilibria are established sufficiently rapidly. By this is meant that the time required for establishing equilibrium is short compared with that required for determining the vapour pressure. When these conditions are satisfied, the system will behave as a univariant system of one component. 

Equilibrium between Solid and Liquid. Curve of Fusion.— There is still another univariant system, the existence of which, at definite values of temperature and pressure, the Phase Rule allows us to predict. This is the system solid—liquid. A crystalline solid on being heated to a certain temperature melts and passes into the liquid state and since this system solid— liquid is univariant, there will be... 

Similar changes are produced when the volume of the system is altered. Alteration of volume may take place either while transference of heat to or from the system is cut off (adiabatic change), or while such transference may occur (isothermal change). In the latter case, the temperature of the system will remain constant in the former case, since at the triple point the pressure must be constant so long as the three phases are present, increase of volume must be compensated by the evaporation of liquid. This, however, would cause the temperature to fall (since communication of heat from the outside is supposed to be cut off), and a portion of the liquid must therefore freeze. In this way the latent heat of evaporation is counterbalanced by the latent heat of fusion. As the result of increase of volume, therefore, the process occurs L S V. Diminution of volume, without transference of heat, will bring about the opposite change, S + V -> L. In the former case there is ultimately obtained the univariant system S—Y in the latter case there will be obtained either S—L or L— V, according as the vapour or solid phase disappears first. 

The different areas in Fig. 12, bounded by the curves for the univariant systems, represent the conditions for the stable existence of single phases, as represented in the diagram. From this diagram it is seen that the region of stability of ice III. is completely circumscribed. 

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