Systems, closed univariant

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... 

States of closed systems for which it is necessary to assign values to at least one extensive variable in addition to the mole numbers are called indifferent states. Thus, all nonvariant and univariant states are indifferent states. These states are discussed in greater detail in Section 5.13. 

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... 

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. 

The system has thus been reduced to a univariate problem and conventional analysis may be employed for subsequent quantification if required. In a closely related study of the curing of a cyanate ester rather than an epoxy resin (Cooper, 1999) a similar profile to Figure 3.46 was obtained after PCA (Figure 3.47), and a comparison with the univariate analysis of single peak intensities that are the major peaks in the first principal component after mean centring was made. 

The vertical section of the Nd-Fe-B ternary system which passes through the Fe comer and the phase Nd2Fe14B is shown in fig. 4a. Schneider et al. emphasize that this is not a pseudo-binary section. The dominant feature of this vertical section is the peritectic reaction L + Fe —> at 1180° C. It also follows from the results shown in fig. 4a that cooling of a liquid whose composition corresponds to leads to the formation of primary crystallized Fe. The concentration limit beyond which no primary Fe crystals are formed is at 77 at.% Fe. This is very close to the overal composition of commercial magnets, as will be discussed in more detail in section 3.2. Schneider et al. note that the vertical section of fig. 4a represents the stable situation which applies only to melts that were kept near the liquidus temperatures for a sufficiently long time. For superheated alloys the vertical section is quite different and corresponds to a metastable situation (fig. 4b). A comparison of the two vertical sections reveals that the liquidus temperatures, and the temperature at which the univariant reaction L - + tj begins, are unaltered, but that the temperature at which the 4> phase forms is lower in fig. 4b than in fig. 4a. Furthermore, one notices a new phase in fig. 4b (x) which is formed peritectically at 1130 °C. The latter temperature is below the temperature of the stable reaction L + Fe - 4> (1180 °C). Schneider et al. note that the primary crystallization of is suppressed in the metastable sequence (fig. 4b), in favour of Fe. In the microstructure one now observes that primary Fe is surrounded by a shell of Fe + which is the decomposition product of x- The x phase was identified by Grieb et al. (1987), as a compound of the 2 17 structure type. 

When the supernatant phase is multicomponent, the system is no longer univariant. Although the conditions of Eq. (8.68) must still be satisfied, this does not ensure that the composition of the amorphous phase will remain fixed with changes in A. At constant pressure the equilibrium force need no longer depend solely on the temperature. Consequently, total melting does not have to occur at constant force, in analogy to the behavior of a closed system. 

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