Non-variant systems

In this case, we find polymorphic transformations of solids such as a-sulfur into P-sulfur. The presence of only one independent component with two phases leads to a non-variant system when temperature at the equilibrium between the two forms is fixed. 

We also come across variance zero with stoichiometric transformations between two solids, for example, the synthesis of tricalcium silicate, which involves two independent components with three phases  

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System having degrees of freedom three, two, one or zero are known as trivariant, bivariant, univariant (or monovariant) and non-variant systems, respectively. 

A system for which degree of freedom is zero is called non-variant system. Under such conditions, there can be no change in temperature, pressure and concentration, because if any change is made in either of them, one or more phases may disappear. Such systems are also called self defined systems. 

Enzyme labels are usually associated with solid-phase antibodies in the technique known as enzyme-linked immunosorbent assay (ELISA). There are several variants of this technique employing both competitive and non-competitive systems. However it is best used in combination with two monoclonal antibodies in the two-site format in which an excess of antibody is bound to a solid phase such as a test-tube or microtitre plate the test antigen is then added and is largely sequestered by the antibody (Figure 7.12). After washing... 

For systems that are nearly linear or time-variant, the concept of the impulse (complex frequency) response is still applicable. For weakly non-linear systems the characterization can be extended by including measurements of the non-linearity (noise, distortion, clipping point). For time-variant systems the characterization can be extended by including measurements of the time dependency of the impulse response. Some of the additional measurements incorporate knowledge of the human auditory system which lead to system characterizations that have a direct link to the perceived audio quality (e.g. the perceptually weighted signal to noise ratio). 

So, this point is non-variant, i.e., in order to define the system at O, we have not to mention any variable factor, i.e.. the system is self-defined. 

Triple point is that point where all the three phases in a one component system exist in equilibrium. At this point, both the variables, e.g., temperature and pressure are fixed, i.e., they have definite values. Thus, triple point is a non-variant point, i.e., degree of freedom is zero. It is also clear from phase rule equation i.e.,... 

The crystallization path of a mixture of any composition except of Xm, is similar as in the previous case of the systems with one two-phase region. When cooling the mixture of composition Xm, at temperature Tm the liquid and solid phases are in equilibrium and they have identical composition. The system seems to be univariant, since there are two components, two phases, and thus one degree of freedom. However, in the case of an extreme on the boundary line, the degree of freedom decreases by one. The system in the minimum is thus non-variant and the cooling of the system stops until all the melt completely freezes. 

Fig. 9 Multiblock copolymers consisting of a poly(ethylene glycol) soft block and a tetrapeptide Ala-Gly-Ala-Gly, crystalline hard block in two variants a Templated system in which an aromatic hairpin turn is used to force parallel jS-sheet formation, b Non-templated system in which peptide segments are free to form parallel and/or antiparallel /1-sheets. Reprinted with permission from [43]. Copyright 2001 American Chemical Society...

Solid, liquid, and vapour of a given substance can, as has been seen, coexist in equilibrium only at a flxed temperature and pressure. Such a system is said to be non-variant or to possess no degrees of freedom. SoUd and liquid can coexist over a range of temperatures, provided that for each temperature an appropriate pressure be chosen. Liquid and vapour can be in equilibrium at a continuous series of pressures, each corresponding to a definite temperature. These systems are called monovariant, and are said to possess one degree of fireedom. 

The steady-state optimal Kalman filter can be generalized for time-variant systems or time-invariant systems with non-stationary noise covariance. The time-varying Kalman filter is calculated in two steps, filtering and prediction. For the nonlinear model the state estimate may be relinearized to compensate the inadequacies of the linear model. The resulting filter is referred to the extended Kalman filter. If once a new state estimate is obtained, then a corrected reference state trajectory is determined in the estimation process. In this manner the Filter reduces deviations of the estimated state from the reference state trajectory (Kwon and Wozny, 1999 Vankateswarlu and Avantika, 2001). In the first step the state estimate and its covariance matrix are corrected at time by using new measurement values >[Pg.439]

The systems with degrees of freedom of zero, one, two and three are referred to as the non-variant, monovariant, divariant and trivariant systems, respectively. For example, if we have a system with one species and treat only the pressure, / = 3—/), therefore the condition for coexistence of gas, liquid and solid phases is / = 0, which is referred to as the triple point. ... 

So, this point is non-variant, i.e., in order to define the system at O, we have not to mention any of the variable factors, i.e., the system is self-defined. Saying the system self defined means in simpler words, that we cannot alter the temperature or pressure or both without changing the number of phases. If, for exeimple, vm lower the temperatme keeping the pressure constant, the vapour will then be deposited as ice and water will freeze. 

The two curves AB and BC meet at the common point B. Since, rhombic and vapour sulphm exist in equilibrium along AB while monoclinic and vapour sulphur exist in equilibrium along BC, therefore, at B the three phases that can exist in equilibrium at any one time are rhombic, monoclinic and vapour sulphur. Hence, B is known as the first triple point of the system and is non-variant, i.e., it is a self-defined point. According to phase rule ... 

Curve EF is known as melting point or fusion curve of rhombic sulphur. The two phases co-existing in equilibrium dong EF are rhombic sulphur and liquid sulphur. The system is univariant. As the two curves BE and EF meet at the common point E, so at E, the conditions of temperatme and pressure are such that three phases, viz., rhombic, monoclinic and liquid sulphur exist in equilibriiun. It is known as the third triple point of the system which is non-variant. (F = 1 - 3 + 2 = 0). 

Nd2Te3-Te. The liquidus of the system consists of the 18 fields of primary crystallization, the field NdTe being the largest one. The monovariant curves intersect in 18 non-variant points, 6 of them being eutectic and 12 being peritectic. 

We notice that at each triple point, for the pressure Pi and P2, two equilibriums occur simultaneously and the system becomes non-variant, both pressure and temperature are commanded. 

B(A) is the probability of observing the system in state A, and B(B) is the probability of observing state B. In this model, the space is divided exactly into A and B. The dividing hyper-surface between the two is employed in Transition State Theory for rate calculations [19]. The identification of the dividing surface, which is usually assumed to depend on coordinates only, is a non-trivial task. Moreover, in principle, the dividing surface is a function of the whole phase space - coordinates and velocities, and therefore the exact calculation of it can be even more complex. Nevertheless, it is a crucial ingredient of the IVansition State Theory and variants of it. 

Another, simple form of elemental carbon would be chains formed from carbon atoms. As a prototype model a single>stranded chain is most suitable. If branching were to be considered, all intermediate forms up to and including the diamond and graphite like clusters would be included. For non branched chains, the two variants to choose from are a system of alternating singly and triply bonded carbon atoms (poly-ynes), and a system with all double bonds (cumulenes). Cumulene structures are assumed to be the preferred ones for odd membered chains, whereas the even ones may have some poly-yne character. Recent studies on linear Cg show that a cumulene-like structure is preferred, both at the SCF level and when correlation is accounted for(50). 

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