The number of independent variables

Prom this point on in this book, unless stated otherwise, the discussions of multiphase systems will implicitly assume the existence of thermal, mechanical, and transfer equilibrium. Equations will not explicitly show these equilibria as a condition of validity. 

In the rest of this chapter, we shall assume the state of each phase can be described by the usual variables temperature, pressure, and amount. That is, variables such as elevation in a gravitational field, interface surface area, and extent of stretching of a solid, are not relevant. 

How many of the usual variables of an open multiphase one-substance equilibrium system are independent To find out, we go through the following argument. In the absence of any kind of equilibrium, we could treat phase a as having the three independent variables r , p, and n , and likewise for every other phase. A system of P phases without thermal, mechanical, or transfer equilibrium would then have 3P independent variables. 

We must decide how to count the number of phases. It is usually of no thermodynamic significance whether a phase, with particular values of its intensive properties, is contiguous. For instance, splitting a crystal into several pieces is not usually considered to change the number of phases or the state of the system, provided the increased surface area makes no significant contribution to properties such as internal energy. Thus, the number of phases P refers to the number of different kinds of phases. 

Each independent relation resulting from equilibrium imposes a restriction on the system and reduces the number of independent variables by one. A two-phase system with thermal equilibrium has the single relation = r . For a three-phase system, there are two such relations that are independent, for instance and (The additional relation is not independent since we may deduce it from the other two.) In 

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The number of independent variables is reduced from the original nine to four. This is a great saving in terms of the number of experiments required to determine the desired function. For example, suppose that a decision is made to test only four values for each variable. Then it would require 4 = 262144 experiments to test aU. combinations of these values in the original equation. As a result of equation 59, only 4 = 256 tests are now required for four values each of the four B-numbers. 

Similarity Variables The physical meaning of the term similarity relates to internal similitude, or self-similitude. Thus, similar solutions in boundaiy-layer flow over a horizontal flat plate are those for which the horizontal component of velocity u has the property that two velocity profiles located at different coordinates x differ only by a scale factor. The mathematical interpretation of the term similarity is a transformation of variables carried out so that a reduction in the number of independent variables is achieved. There are essentially two methods for finding similarity variables, separation of variables (not the classical concept) and the use of continuous transformation groups. The basic theoiy is available in Ames (see the references). 

The flow field in front of an expanding piston is characterized by a leading gas-dynamic discontinuity, namely, a shock followed by a monotonic increase in gas-dynamic variables toward the piston. If both shock and piston are regarded as boundary conditions, the intermediate flow field may be treated as isentropic. Therefore, the gas dynamics can be described by only two dependent variables. Moreover, the assumption of similarity reduces the number of independent variables to one, which makes it possible to recast the conservation equations for mass and momentum into a set of two simultaneous ordinary differential equations ... 

For a system such as discussed here, the Gibb s Phase Rule [59] applies and establishes the degrees of freedom for control and operation of the system at equilibrium. The number of independent variables that can be defined for a system are ... 

By the variance, or number of degrees of freedom of the system, we mean the number of independent variables which must be arbitrarily fixed before the state of equilibrium is completely determined. According to the number of these, we have avariant, univariant, bivariant, trivariant,. . . systems. Thus, a completely heterogeneous system is univariant, because its equilibrium is completely specified by fixing a single variable— the temperature. But a salt solution requires two variables— temperature and composition—to be fixed before the equilibrium is determined, since the vapour-pressure depends on both. 

Having phases together in equilibrium restricts the number of thermodynamic variables that can be varied independently and still maintain equilibrium. An expression known as the Gibbs phase rule relates the number of independent components C and number of phases P to the number of variables that can be changed independently. This number, known as the degrees of freedom f is equal to the number of independent variables present in the system minus the number of equations of constraint between the variables. 

The number of independent variables for each phase are the two that we have considered earlier among p, T, V, U, H, A, and G, plus (C—1) composition variables.y Thus the total number of independent variables for P phases is (C -1+2) per phase (P phases) = (C+ 1)P. 

Evidence is therefore found in the data for the Haber equilibrium that the function Kp is not strictly constant with respect to any variation whatsoever of the internal variables, the number of independent variables being given by the phase rule. 

The coverage thus far has provided an account of the usefulness of phase rule to classify equilibria and to establish the number of independent variables or degrees or of freedom available in a specific situation. In the following paragraphs the equilibria used in mass transfer are analyzed in terms of phase rule in the case of leaching, drying and crystallisation. 

The number of independent variables that have to be specified to define a problem will depend on the type of separation process being considered. Some examples of the application of the description rule to more complex columns are given by Hanson et al. (1962). 

The electroneutrality condition decreases the number of independent variables in the system by one these variables correspond to components whose concentration can be varied independently. In general, however, a number of further conditions must be maintained (e.g. stoichiometry and the dissociation equilibrium condition). In addition, because of the electroneutrality condition, the contributions of the anion and cation to a number of solution properties of the electrolyte cannot be separated (e.g. electrical conductivity, diffusion coefficient and decrease in vapour pressure) without assumptions about individual particles. Consequently, mean values have been defined for a number of cases. 

In the geometric method1,11 experimental results are used to minimize the region in which the optimum exists. The response is obtained for a number of points that are located very near one another. The number of points should be one greater than the number of independent variables. From the results a surface (this is aline when there are two independent variables) representing a constant value of the response is constructed. This method hypothesizes that on one side of this surface will be all the pointsthatyieldabetterresponse,andthereforetheoptimummustlieonthatsideofthe surface. 

One way to reduce the number of independent variables in the FRET-adjusted spectral equation is to use samples with a fixed donor-to-acceptor ratio. Under these conditions, the values of d and a are no longer independent, but rather the concentration of d is now a function of a and vice-versa. This approach is typical for the situation of FRET-based biosensor constructs. These sensors normally are designed to have a donor fluorophore attached to an acceptor by a domain whose structure is altered either as a result of a biological activity (such as proteolysis or phosphorylation), or by its interaction with a specific ligand with which it has high affinity. In general, FRET based biosensors have a stoichiometry of one... 

The symmetry of the structure is imposed on the field (j)(r) by building the field inside a unit cell of smaller polyhedron, replicating it by reflections, translations, and rotations [21-23]. This procedure reduces the number of independent variables one order of magnitude for G structure and two orders of magnitude for D structure. 

A basic exposition of Gibbs phase rule is essential for understanding phase solubility analysis, and detailed presentations of theory are available [41,42]. In a system where none of the chemical species interact with each other, the number of independently variable factors (i.e., the number of degrees of freedom, F) in the system is given by... 

Fortunately, few of these variables are truly independent. Geochemists have developed a variety of numerical schemes to solve for equilibrium in multicomponent systems, each of which features a reduction in the number of independent variables carried through the calculation. The schemes are alike in that each solves sets of mass action and mass balance equations. They vary, however, in their choices of thermodynamic components and independent variables, and how effectively the number of independent variables has been reduced. 

A goal in deriving the governing equations is to reduce the number of independent variables by eliminating the molalities nij of the secondary species. To this end, we can rearrange the equation above to give the value of ntj,... 

Providing an additional piece of information about the size of each phase predicts that a total of Ni + N, or Nc, values is needed to constrain the system s state and extent. This total matches the number of variables we must supply in order to solve the governing equations. Hence, although we can make no claim that we have cast the governing equations in simplest form, we can say that we have reduced the number of independent variables to the minimum allowed by thermodynamics. 

Examine the following optimization problem. State the total number of variables, and list them. State the number of independent variables, and list a set. 

Determine the number of independent variables, the number of independent equations, and the number of degrees of freedom for the reboiler shown in the figure. What variables should be specified to make the solution of the material and energy balances determinate (Q = heat transferred)... 

RR, PCR, and PLS are appropriate methodologies when the number of descriptors exceeds the number of observations, and they are designed to utilize all available descriptors in order to produce an unbiased model whose predictive ability is accurately reflected by q2, regardless of the number of independent variables in the model. 

The literature of the past three decades has witnessed a tremendous explosion in the use of computed descriptors in QSAR. But it is noteworthy that this has exacerbated another problem rank deficiency. This occurs when the number of independent variables is larger than the number of observations. Stepwise regression and other similar approaches, which are popularly used when there is a rank deficiency, often result in overly optimistic and statistically incorrect predictive models. Such models would fail in predicting the properties of future, untested cases similar to those used to develop the model. It is essential that subset selection, if performed, be done within the model validation step as opposed to outside of the model validation step, thus providing an honest measure of the predictive ability of the model, i.e., the true q2 [39,40,68,69]. Unfortunately, many published QSAR studies involve subset selection followed by model validation, thus yielding a naive q2, which inflates the predictive ability of the model. The following steps outline the proper sequence of events for descriptor thinning and LOO cross-validation, e.g.,... 

Computational cost increases linearly with the number of independent variables. Thus a large number of inert scalars can be treated with little additional computational cost. For reacting scalars, the total computational cost will often be dominated by the chemical source term (see Section 6.9). 

The number of independent variables required to specify the state of a mechanical or thermodynamic system. Degrees of freedom arise from the possible motions of molecules or particles in a system. (The term generalized coordinates is also used in physics to designate the minimal number of coordinates needed to specify the state of a mechanical system.) 2. The number of independent or unrestricted random variables constituting a statistic. See Statistics (A Primer)... 

The choice of electrical effect parameterization depends on the number of data points in the data set to be modeled. When using linear regression analysis the number of degrees of freedom, Ndf, is equal to the number of data points, Ndp, minus the number of independent variables, Ai,v, minus one. When modeling physicochemical data Ndf/Nj, should be at least 2 and preferably 3 or more. As the experimental error in the data increases, tVof/iViv should also increase. 

AlOOR A statistic that corrects lOOR for the number of independant variables. 

The dimension of a coordinate-based chemistry space is simply the number of independent variables used to define the space. As seen in earlier discussions, the dimension of such spaces can be quite large, and there are a significant number of examples where the dimension can exceed one million (27,38). Even for spaces of much lower dimension, say around 10 or greater, the effects of the curse of dimensionality (74,75) can be felt. Bishop (76) provides an excellent example, which shows that the ratio of the volume of a hypersphere inscribed in a unit hypercube of the same dimension goes to zero as the dimen-... 

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