Variance of a system

There are three variable factors, viz., temperature, pressure and concentration, on which the equilibrium of a system depends. In some cases, we have to mention only one factor to define the system completely, sometimes two or three. So, the degree of freedom (or variance) of a system is defined as, the least number of variable factors such as temperature, pressure or concentration which must be specified so that the remaining variables are fixed automatically and the system is completely defined. 

The phase rule is a relation among the number of independent components, the number of phases, and the variance of a system in equilibrium. The independent components (or, briefly, the components) of a system are the substances that must be added to realize the system. The word phase has been defined earlier (Chaps. 2 and 16). Thus a system containing ice, water, and water vapor consists of three phases but only one component (water-substance), since any two of the phases can be formed from the third. The variance of the system is the number of independent ways in which the system can be varied these ways may include varying the temperature and the pressure, and also... 

Invariant systems.— When the variance of a system is equal to zero the system is called invariant there exists bui one temperature and one pressure for which an invariant system may be in equilibrium the composition and the density of each of the phases which compose the system in equilibrium are, besides, determined but this is not so for the mass of each phase even if the total mass of each of the independent components which form the system is given, it would be possible to divide in an infinite number of different ways these components into phases having the composition proper for equilibrium. 

We have supposed in the preceding discussion that the experiments have been carried out in the absence of a vapour phase. We come to the same general conclusion however if air is present since the variance of a system is not altered if we add both a new phase (the vapour phase) and a new constituent (air). We note that air can be regarded as a single component since its composition remains constant throughout (aside from the vapours derived from the solution). 

The variance of a chemical system is exactly analogous to the variance of a system of linear equations. For example, for the function... 

Gibb s phase mle states that the variance of a system with p external intensive physical variables, containing c independent components and ... 

Fixed-Bed Behavior The number of transitions occurring in a fixed bed of initially uniform composition before it becomes saturated by a constant composition feed stream is generally equal to the variance of the system. This introductory discussion will be limited to single transition systems. 

The calculated values y of the dependent variable are then found, for jc, corresponding to the experimental observations, from the model equation (2-71). The quantity ct, the variance of the observations y is calculated with Eq. (2-90), where the denominator is the degrees of freedom of a system with n observations and four parameters. 

Asif et al. (1991) studied distributor effects in liquid-fluidized beds of low-density particles by measuring RTDs of the system by pulse injection of methylene blue. If PF leads into and follows the fluidized bed with a total time delay of 10 s, use the following data to calculate the mean-residence time and variance of a fluid element, and find N for the US model. 

The variance of a quantity /, which is variously denoted by aj, var(/), or a1 if), measures the intrinsic range of fluctuations in a system. Given N properly distributed samples of /, the variance is defined as the average squared deviation from the mean ... 

The width of a peak, or, in other words, the length of a solute zone, is affected primarily by diffusion phenomena leading to a broadening of the solute zone. In addition, in capillary zone electrophoresis, zone broadening can be caused by thermal effects, electrodispersion, or adsorptive effects. All these effects can be expressed as coefficients of variance cr2, adding to a total coefficient of variance of the system ofotal ... 

Basic Concepts. The goal of factor and components analysis is to simplify the quantitative description of a system by determining the minimum number of new variables necessary to reproduce various attributes of the data. Principal components analysis attempts to maximally reproduce the variance in the system while factor analysis tries to maximally reproduce the matrix of correlations. These procedures reduce the original data matrix from one having m variables necessary to describe the n samples to a matrix with p components or factors (p[Pg.26]

Thorstenson and Plummer (1977), in an elegant theoretical discussion (see section on The Fundamental Problems), discussed the equilibrium criteria applicable to a system composed of a two-component solid that is a member of a binary solid solution and an aqueous phase, depending on whether the solid reacts with fixed or variable composition. Because of kinetic restrictions, a solid may react with a fixed composition, even though it is a member of a continuous solid solution. Thorstenson and Plummer refer to equilibrium between such a solid and an aqueous phase as stoichiometric saturation. Because the solid reacts with fixed composition (reacts congruently), the chemical potentials of individual components cannot be equated between phases the solid reacts thermodynamically as a one-component phase. The variance of the system is reduced from two to one and, according to Thorstenson and Plummer, the only equilibrium constraint is IAP g. calcite = Keq(x>- where Keq(x) is the equilibrium constant for the solid, a function of... 

Figure 7 The temperature dependence of the dissociation probability of an electron started a distance of rQ - 24 A from a fixed positive charge in a system characterized by a Gaussian variance of a = 0.10 eV.

The number of degrees of freedom (sometimes also referred to as the variance) is the number of variable factors, such as temperature, pressure, and concentration that must be fixed in order to define the condition of a system at equilibrium. Thus, a one-component system in one phase, say a gas, would have two degrees of freedom a one component system in two phases (liquid and gas) would have one degree of freedom. A system of one component and three phases would have no degrees of... 

The nature of alloys. Homogeneous and heterogeneous alloys. Solid solutions, intermetallic compounds. The phase rule, P - - P = C 2 number of phases, variance, number of components of a system in equilibrium triple point. Phase diagrams of binary systems eutectic mixture eutectic point. The systems As-Pb, Pb-Sn, Ag-Au, Ag-Sr. 

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