Independent equations

To obtain the different values of p, it is only hecessary to produce as many independent equations as there are components in the mixture and, if the mixture has n components, to solve a system of n equations having n unknowns. Individual analysis is now possible for mixtures having a few components but even gasoline has more than 200 It soon becomes unrealistic to have ail the sensitivity coefficients necessary for analysis in this case, 200. ... 

The expression for the force on the nuclei, Eq. (89), has the same form as the BO force Eq. (16), but the wave function here is the time-dependent one. As can be shown by perturbation theory, in the limit that the nuclei move very slowly compared to the electrons, and if only one electronic state is involved, the two expressions for the wave function become equivalent. This can be shown by comparing the time-independent equation for the eigenfunction of H i at time t... 

There are n-1 independent equations of type (5.21), since both sides vanish on summing over r. Equation (5,20) then provides one further relation between the fluxes to complete the set. 

The degree of the least polynomial of a square matr ix A, and henee its rank, is the number of linearly independent rows in A. A linearly independent row of A is a row that eannot be obtained from any other row in A by multiplieation by a number. If matrix A has, as its elements, the eoeffieients of a set of simultaneous nonhomo-geneous equations, the rank k is the number of independent equations. If A = , there are the same number of independent equations as unknowns A has an inverse and a unique solution set exists. If k < n, the number of independent equations is less than the number of unknowns A does not have an inverse and no unique solution set exists. The matrix A is square, henee k > n is not possible. 

In cases where the elassieal energy, and henee the quantum Hamiltonian, do not eontain terms that are explieitly time dependent (e.g., interaetions with time varying external eleetrie or magnetie fields would add to the above elassieal energy expression time dependent terms diseussed later in this text), the separations of variables teehniques ean be used to reduee the Sehrodinger equation to a time-independent equation. 

When the Sehrodinger equation ean be separated to generate a time-independent equation deseribing the spatial eoordinate dependenee of the wavefunetion, the eigenvalue E must be returned to the equation determining F(t) to find the time dependent part of the wavefunetion. By solving... 

It is extremely difficult to know values for all of these parameters precisely. Therefore, absolute quantitation is almost never attempted. The determination of relative atomic ratios is an inherently more tractable approach, however. This method is best illustrated by consideration of a binary material composed exclusively of atoms A and B that is perfectiy homogeneous up to the surface. In this case, independent equations can be developed relating the number of atoms sampled to the xps intensity for each atom as follows ... 

Equation 14 contains two independent equations and three unknown variables T). An additional relation needed to solve this equation may... 

For a PVnr system of uniform T and P containing N species and 7T phases at thermodynamic equiUbrium, the intensive state of the system is fully deterrnined by the values of T, P, and the (N — 1) independent mole fractions for each of the equiUbrium phases. The total number of these variables is then 2 + 7t N — 1). The independent equations defining or constraining the equiUbrium state are of three types equations 218 or 219 of phase-equiUbrium, N 7t — 1) in number equation 245 of chemical reaction equiUbrium, r in number and equations of special constraint, s in number. The total number of these equations is A(7t — 1) + r -H 5. The number of equations of reaction equiUbrium r is the number of independent chemical reactions, and may be deterrnined by a systematic procedure (6). Special constraints arise when conditions are imposed, such as forming the system from particular species, which allow one or more additional equations to be written connecting the phase-rule variables (6). 

The degrees of freedom, F, is by definition the difference between the number of variables and the number of independent equations ... 

As soon as there are as many independent equations as there are unknown coefficients, these linear equations are solved for b and Q. A proper choice of the points i7(i) guarantees getting independent equations to solve here. 

The total number of independent equations is therefore (tt — )N + r In their fundamental forms these equations relate chemical potentials, which are functions of temperature, pressure, and composition, the phase-rule variables. Since the degrees of freedom of the system F is the difference between the number of variables and the number of equations. 

For separation processes, a design solution is possible if the number of independent equations equals the number or unknowns. 

Note that application of a systematic approach enables us to resolve a material-balance system into a number of independent equations equal to the number of unknowns that it needs to solve for. The following steps should be followed with any material-balance system, regardless of complexity ... 

Ifweexpandthe total materialbalance equation, weobtainthesumotthesetwobackand soweareunabletosolvetheproblem.Thisisbecausewehavetheunknownandonlytwo independent equations. The unknowns are the concentrations ot each component and the change in volume. Each is a tunction ot time. We need another independent equation. To obtainthisweneedtothinkaboutwhatishappening. 

One of the great difficulties in molecular quantum mechanics is that of actually finding solutions to the Schrodinger time-independent equation. So whilst we might want to solve... 

Suppose that is the lowest energy solution to the Schrodinger time-independent equation for the problem in hand. That is to say,... 

In Chapter 4,1 discussed the concept of an idealized dihydrogen molecule where the electrons did not repel each other. After making the Bom-Oppenheimer approximation, we found that the electronic Schrddinger equation separated into two independent equations, one for either electron. These equations are the ones appropriate to the hydrogen molecule ion. 

Perform an overall material balance and the necessary component material balances so as to provide the maximum number of independent equations. In the event the balance is written in differential form, appropriate integration must be carried out over time, and the set of equations solved for the unknowns. 

The site balance specihes that the number of empty plus occupied sites is a constant, Sq. Equality of the reaction rates plus the site balance gives four independent equations. Combining them allows a solution for while eliminating the surface concentrations [S], [AS], and [PS]. Substitute the various reaction rates into the site balance to obtain... 

Material-balance problems are particular examples of the general design problem discussed in Chapter 1. The unknowns are compositions or flows, and the relating equations arise from the conservation law and the stoichiometry of the reactions. For any problem to have a unique solution it must be possible to write the same number of independent equations as there are unknowns. 

To have a better appreciation of the utility of these representations let us first consider the laws that govern flow rates and pressure drops in a pipeline network. These are the counterparts to KirchofTs laws for electrical circuits, namely, (i) the algebraic sum of flows at each vertex must be zero (ii) the algebraic sum of pressure drops around any cyclic path must be zero. For a connected network with N vertices and P edges there will be (N — 1) independent equations corresponding to the first law (KirchofTs current... 

This equation is extremely useful for calculating the equilibrium composition of the reaction mixture. The mole numbers of the various species at equilibrium may be related to their values at time zero using the extent of reaction. When these values are substituted into equation 2.6.9, one has a single equation in a single unknown, the equilibrium extent of reaction. This technique is utilized in Illustration 2.1. If more than one independent reaction is occurring in a given system, one needs as many equations of the form of equation 2.6.9 as there are independent reactions. These equations are then written in terms of the various extents of reaction to obtain a set of independent equations equal to the number of unknowns. Such a system is considered in Illustration 2.2. 

In order to arrive at a consistent set of relationships from which complex reaction equilibria may be determined, one must develop the same number of independent equations as there are unknowns. The following treatment indicates one method of arriving at a set of chemical reactions that are independent. It has been adopted from the text by Aris (1). ... 

This allows separation of the corresponding Schrodinger equation in n independent equations, one for each electron, so that the solution of Equation 1.2 will be a product of functions of the type ... 

A proper set of chemical equations for a system is made up of R linearly independent equations. 

On the other hand, if more process variables whose values are unknown exist in category 2 than there are independent equations, the process model is called underdetermined that is, the model has an infinite number of feasible solutions so that the objective function in category 1 is the additional criterion used to reduce the number of solutions to just one (or a few) by specifying what is the best solution. Finally, if the equations in category 2 contain more independent equations... 

The number of independent equations required to describe the system (sometimes called the order of the model). 

There are p independent variables Xj,j = 1,.. ., p. Independent here means controllable or adjustable, not functionally independent. Equation (2.3) is linear with respect to the fy, but jc- can be nonlinear. Keep in mind, however, that the values of Xj (based on the input data) are just numbers that are substituted prior to solving for the estimates jS, hence nonlinear functions of xj in the model are of no concern. For example, if the model is a quadratic function,... 

In Section 1.5 we briefly discussed the relationships of equality and inequality constraints in the context of independent and dependent variables. Normally in design and control calculations, it is important to eliminate redundant information and equations before any calculations are performed. Modem multivariable optimization software, however, does not require that the user clearly identify independent, dependent, or superfluous variables, or active or redundant constraints. If the number of independent equations is larger than the number of decision variables, the software informs you that no solution exists because the problem is overspecified. Current codes have incorporated diagnostic tools that permit the user to include all possible variables and constraints in the original problem formulation so that you do not necessarily have to eliminate constraints and variables prior to using the software. Keep in mind, however, that the smaller the dimensionality of the problem introduced into the software, the less time it takes to solve the problem. 

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