Invariance conditions

Steady state A time-invariant condition for a differential equation in which all time derivatives are zero. 

By using effect invariant conditions in the postconditions, we can describe effects that apply selectively to any action that meets the condition. For example, here s how to keep a count of all actions that create or remove an event on the schedule ... 

With increasing P and T, the equilibrium curve of equation 5.129 reaches an invariant condition, determined by the appearance of silicate melt T = 835 °C, P = 0.45 kbar Wones and Dodge, 1977). 

The general relationships involved for a single chemical reaction in a closed system are shown schematically in Figure 1, where the degree of advancement at point e corresponds to chemical equilibrium. Point t represents a state of the system corresponding to spontaneous chemical reaction. While the invariant condition of the closed system considered is the equilibrium state, e, this generally is not the case for a thermodynamic system open to its environment. For such a system, the time-... 

Thus, when the residence time ta , is sufficiently large relative to the appropriate time scale of the reaction, t1/2, the time-invariant condition of the well-mixed volume considered approaches chemical equilibrium. 

The actual arithmetic involved in the calculations may be done easily by starting with the general expression for electroneutrality and substituting equilibrium expressions (9). Using the equilibrium constants of Table I and assuming that activities equal concentrations, we obtain for 5°C. and an invariant condition (model No. 1) the following data ... 

A feature of HPLC and CE is that, for a defined system with invariant conditions, an analyte will have a constant retention volume and thus may be identified on the basis of retention time. Given the numerous compounds in existence, however, it is not possible to assign, unequivocally, an identity to any peak on the basis of retention times alone, unless the individual components of a sample mixture are known prior to analysis. 

The scale invariance condition hd = 1, leads to the following static HSR ... 

Furthermore, with at least some systems, it has been found that E and N do not vary appreciably with temperature and therefore the temperature-invariance condition is satisfied (Stoeckli et al., 1994b). On the other hand, so far it has not been possible to define the exact meaning of these terms. 

One readily verifies that both the full TDOPM potential and the TDKLI approximation of it satisfy the the generalized translational invariance condition (242) (and hence the harmonic potential theorem) provided that... 

A difficulty with the above scheme is that measurements carried out with various actual gases that approach ideal behavior will lead to slightly dilferent results. A better absolute standard is provided by the so-called triple point of water. As we shall see later, the coexistence conditions of water in the solid, liquid, and vapor state can occur only under a set of precisely controlled, invariant conditions determined by the physical characteristics of H2O. These conditions are completely reproducible all over the world. For consistency with the above absolute temperature scheme the triple point of water is assigned a temperature T (triple point of H2O) = 273.16 K = 7). Then any other absolute temperature is determined through the proportionality T = (P/Pt) 273.16, where P is the pressure at T and Pt is the pressure measured for He in equilibrium with water at its triple point. 

F re 2.2. General representation of a natural water system treated as a continuous, open system. The system receives fluxes of matter from the surroundings and undergoes chemical changes, symbolized by the reaction A = B. The time-invariant condition is represented by dCJdt = 0. 

Comparison of equations 3 and 10 shows the essential difference between the stationary states of closed and continuous, open systems. For the closed system, equilibrium is the time-invariant condition. The total of each independently variable constituent and the equilibrium constant (a function of temperature, pressure, and composition) for each independent reaction (ATab in the example) are required to define the equilibrium composition Ca- For the continuous, open system, the steady state is the time-invariant condition. The mass transfer rate constant, the inflow mole number of each independently variable constituent, and the rate constants (functions of temperature, pressure, and composition) for each independent reaction are requir to define the steady-state composition Ca- It is clear that open-system models of natural waters require more information than closed-system models to define time-invariant compositions. An equilibrium model can be expected to describe a natural water system well when fluxes are small, that is, when flow time scales are long and chemical reaction time scales are short. 

For many systems it is known that there exist regions or environments in which the time-invariant condition closely approaches equilibrium. The concept of local equilibrium is important in examining complex systems. Local equilibrium conditions are expected to develop, for example, for kinetically rapid species and phases at sediment-water interfaces in fresh, estuarine, and marine environments. In contrast, other local environments, such as the photosyn-thetically active surface regions of nearly all lakes and ocean waters and the biologically active regions of soil-water systems, are clearly far removed from total system equilibrium. 

Obviously the sea is an open, dynamic system with variable inputs and outputs of mass and energy for which the state of equilibrium is a construct. As we have seen, the concept of free energy, however, is no less important in dynamic systems. In considering equilibria and kinetics in ocean systems, it is useful to recall that different time scales need to be identified for the various processes. When a particular reaction of a phase or species has—within the time scale of consideration—a negligible rate, it is permissive to define a metastable equilibrium state. Similarly, in a flow system the time-invariant condition of a well-mixed volume approaches chemical equilibrium when the residence time is sufficiently large relative to the appropriate time scale of the reaction. 

The invariance condition can be achieved by other geometries, providing that C° a so that dCJda is a constant, and that of double torsion, as illustrated in Fig. 7, has proved useful for polymers In this case ... 

With the definition (12), the invariance condition (14) takes the form... 

In principle, the crystallization of a protein, nucleic acid, or virus (as exemplified in Figure 2.2) is little different than the crystallization of conventional small molecules. Crystallization requires the gradual creation of a supersaturated solution of the macromolecule followed by spontaneous formation of crystal growth centers or nuclei. Once growth has commenced, emphasis shifts to maintenance of virtually invariant conditions so as to sustain continued ordered addition of single molecules, or perhaps ordered aggregates, to surfaces of the developing crystal. 

Selection Rules for all Laue Classes. The selection rules for the harmonic coefficients are derived from the invariance of the pole distribution to the operations of the crystal and sample Laue groups. The invariance conditions are applied to every function t/ (h, y) from Equation (36), as the terms of different / in this equation are independent. If we compare Equations (38) and (39) with (37) we observe that they have an identical structure. On the other side the sample and the crystal coordinate systems were similarly defined. As a consequence the selection rules for the coefficients of", flf", and respectively y) ", resulting from the sample symmetry must be identical with the selection rules for the coefficients A P and resulting from the crystal symmetry, if the sample and the crystal Laue groups are the same. The exception is the case of cylindrical sample symmetry that has no correspondence with the crystal symmetry. In this case, only the coefficients af and y l are different from zero, if they are not forbidden by the crystal symmetry. 

An r-fold axis along X3 transforms j3 into f The invariance condition is ... 

Selection Rules for all Laue Classes. To find the selection rules for all Laue classes the invariance conditions to rotations are applied to the peak shift weighted by texture (ah(y))Fh(y). As the terms of different I in Equation (118) are independent, the invariance conditions must be applied to every //. 

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