Homogeneous Variables

Finally, it should be mentioned that the equilibrium statistical mechanics is thermodynamically self-consistent if the statistical variables (x1,. .., xn), the potentials (/, g,. ..), and the variables (u1,. ..,un) are homogeneous variables of the first- or zero-order satisfying Eqs. (21)-(25). 

A variable mode in a two-way or three-way array can consist of homogeneous variables, heterogeneous variables or a mixture of the two. Homogeneous variables have the same physical units and are often measured on the same instmment. The variables of a two-way... 

Examples for homogeneous variables are typically expressed originally in densities, thus normalized to the imit of volume, e.g.,... 

Another way to get homogeneous variables is to form molar quantities from extensive variables. Note that the equation of continuity and other derived equations consider typically a small unit of volume dV = dxdydz which is the starting point for the derivation of the respective law. 

The first clear step away from analyticity was made in 1965 by Widom [17] who suggested that the assumption of analytic fimctions be replaced by the less severe assumption that the singular part of the appropriate themiodynamic fimction was a homogeneous fimction of two variables, (p - 1) and (1 - T ). A homogeneous fimction f(u, v) of two variables is one that satisfies the condition... 

In both cases the late stages of kinetics show power law domain growth, the nature of which does not depend on the mitial state it depends on the nature of the fluctuating variable(s) which is (are) driving the phase separation process. Such a fluctuating variable is called the order parameter for a binary mixture, tlie order parameter o(r,0 is tlie relative concentration of one of the two species and its fluctuation around the mean value is 5e(/,t) = c(r,t) - c. In the disordered phase, the system s concentration is homogeneous and the order... 

An important point for all these studies is the possible variability of the single molecule or single particle studies. It is not possible, a priori, to exclude bad particles from the averaging procedure. It is clear, however, that high structural resolution can only be obtained from a very homogeneous ensemble. Various classification and analysis schemes are used to extract such homogeneous data, even from sets of mixed states [69]. In general, a typical resolution of the order of 1-3 mn is obtained today. 

The importance of between-laboratory variability on the results of an analytical method can be determined by having several laboratories analyze the same sample. In one such study seven laboratories analyzed a sample of homogenized milk for a selected alfatoxin. The results, in parts per billion, are summarized in the following table. 

Speed Devices. Many displacement pumps are connected by variable speed drives. When these pumps are used as a time device on a homogenizer, the setting is fixed, ie, the maximum speed is limited in order to meet the requirements of pasteurization. 

Generally, Httle is known in advance concerning the degree of homogeneity of most sampled systems. Uniformity, rarely constant throughout bulk systems, is often nonrandom. During the production of thousands of tons of material, size and shape distribution, surface and bulk composition, density, moisture, etc, can vary. Thus, in any bulk container, the product may be stratified into zones of variable properties. In gas and Hquid systems, particulates segregate and concentrate in specific locations in the container as the result of sedimentation (qv) or flotation (qv) processes. 

Equations 54 and 58 through 60 are equivalent forms of the fundamental property relation apphcable to changes between equihbtium states in any homogeneous fluid system, either open or closed. Equation 58 shows that ff is a function of 5" and P. Similarly, Pi is a function of T and C, and G is a function of T and P The choice of which equation to use in a particular apphcation is dictated by convenience. Elowever, the Gibbs energy, G, is of particular importance because of its unique functional relation to T, P, and the the variables of primary interest in chemical technology. Thus, by equation 60,... 

As indicated earlier, the vaUdity of the method of dimensional analysis is based on the premise that any equation that correcdy describes a physical phenomenon must be dimensionally homogeneous. An equation is said to be dimensionally homogeneous if each term has the same exponents of dimensions. Such an equation is of course independent of the systems of units employed provided the units are compatible with the dimensional system of the equation. It is convenient to represent the exponents of dimensions of a variable by a column vector called dimensional vector represented by the column corresponding to the variable in the dimensional matrix. In equation 3, the dimensional vector of force F is [1,1, —2] where the prime denotes the matrix transpose. 

Variable Coejftcients The method of variation of parameters apphes equally well to the linear difference equation with variable coefficients. Techniques are therefore needed to solve the homogeneous system with variable coefficients. 

Reduction of Order If one homogeneous solution, say, can be found by inspection or otherwise, an equation of lower order can be obtained by the substitution = yju,. The resultant equation must be satisfied by = constant or Ai = 0. Thus the equation will be of reduced order if the new variable L/ = A y,/ul) is introduced. 

The simplest element is a single homogeneous. stream. The variables necessary to define it are ... 

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