Homogeneous equation system

A non-dimensional 77 parameter results when the sum of the dimensional exponents is zero for the base dimensions included in the equation system the exponents ei,62,63, , therefore, must satisly the following linear and homogeneous equation system... 

Previously, reference was made to the number d of dimensionally independent parameters in the physical relation considered. When the homogeneous equation system (m) is solved, the ranking of the equation system g wUl correspond to the number of dimensionally independent P parameters. 

Solution. To determine the non-dimensional 71 parameters, the homogeneous equation system according to eqn. (m) for the seven parameters is drawn up. The dimensional exponents (a, / , ) of this equation system can be directly seen from the dimensions of the P parameters, which are... 

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,... 

Temperature, pressure, and composition are thermodynamic coordinates representing conditions imposed upon or exhibited by the system, andtne functional dependence of the thermodynamic properties on these conditions is determined by experiment. This is quite direct for molar or specific volume which can be measured, and leads immediately to the conclusion that there exists an equation of. state relating molar volume to temperature, pressure, and composition for any particular homogeneous PVT system. The equation of state is a primaiy tool in apphcations of thermodyuamics. 

The same applies to the other eigenvectors U2 and Vj, etc., with additional constraints of orthonormality of u, U2, etc. and of Vj, Vj, etc. By analogy with eq. (31.5b) it follows that the r eigenvalues in A must satisfy the system of linear homogeneous equations ... 

Hence, as is often stated, the determination of the normal coordinates is equivalent to the successful search for a matrix L that diagonalizes the product GF via a similarity transformation. This system of linear, simultaneous homogeneous equations can be written in the form... 

This result is a system of simultaneous linear, homogeneous equations for the coefficients, cu. Cramer s rule states that a nontrivial solution exists only if... 

In order to evaluate the catalytic characteristics of colloidal platinum, a comparison of the efficiency of Pt nanoparticles in the quasi-homogeneous reaction shown in Equation 3.7, with that of supported colloids of the same charge and of a conventional heterogeneous platinum catalyst was performed. The quasi-homogeneous colloidal system surpassed the conventional catalyst in turnover frequency by a factor of 3 [157], Enantioselectivity of the reaction (Equation 3.7) in the presence of polyvinyl-pyrrolidone as stabilizer has been studied by Bradley et al. [158,159], who observed that the presence of HC1 in as-prepared cinchona alkaloids modified Pt sols had a marked effect on the rate and reproducibility [158], Removal of HC1 by dialysis improved the performance of the catalysts in both rate and reproducibility. These purified colloidal catalysts can serve as reliable... 

This represents a homogeneous system of equations. There is, as always with homogeneous equations, a trivial solution t fO. 

The Wacker-Hoechst process has been studied in great detail and in all textbooks it occurs as the example of a homogeneous catalyst system illustrating nucleophilic addition to alkenes. Divalent palladium is the oxidising agent and water is the oxygen donor according to the equation ... 

The solution procedure to this equation is the same as described for the temporal isothermal species equations described above. In addition, the associated temperature sensitivity equation can be simply obtained by taking the derivative of Eq. (2.87) with respect to each of the input parameters to the model. The governing equations for similar types of homogeneous reaction systems can be developed for constant volume systems, and stirred and plug flow reactors as described in Chapters 3 and 4 and elsewhere [31-37], The solution to homogeneous systems described by Eq. (2.81) and Eq. (2.87) are often used to study reaction mechanisms in the absence of mass diffusion. These equations (or very similar ones) can approximate the chemical kinetics in flow reactor and shock tube experiments, which are frequently used for developing hydrocarbon combustion reaction mechanisms. 

In this section, we wish to derive the Gibbs-Duhem equation, the fundamental relationship between the allowed variations dRt of the intensive properties of a homogeneous (singlephase) system. Paradoxically, this relationship (which underlies the entire theory of phase equilibria to be developed in Chapter 7) is discovered by considering the fundamental nature of extensive properties Xu as well as the intrinsic scaling property of the fundamental equation U = U(S, V, n, n2,. .., nc) that derives from the extensive nature of U and its Gibbs-space arguments. 

A set of n linear homogeneous equations in n unknowns always has the solution x, — x2 = xn = 0, which is the trivial solution. Suppose the coefficient determinant det(aiy) is not equal to zero for a set of linear homogeneous equations we can then use Cramer s rule (1.82). Since the equations are homogeneous, the determinant Rk will have a column of zeros, and will equal zero hence xt = x2= =xn = 0, and we have only the trivial solution. Thus for a nontrivial solution of the homogeneous equations to exist, we must have det(o,y) = 0. This condition can also be shown to be sufficient to insure the existence of a nontrivial solution. A system of n simultaneous, linear, homogeneous equations in n unknowns has a nontrivial solution if and only if the determinant of the coefficients equals zero. 

The homogeneous hydrogenation systems discussed in this paper may be treated as analogues of enzyme systems with the rhodium catalyst as the enzyme (E), hydrogen (Si) and cyclohexene (S2) as the substrates, and excess ligand or other donor site as the inhibitor (I). The well-established mathematical operations of enzyme kinetics (12) can then be used to derive rate equations for various possible mechanisms. 

For steady-state solutions, the set of the four differential equations (7.189) to (7.192) (or equivalently the DEs (7.190) to (7.193)) reduces to a set of four coupled rational equations in the unknown variables E (or e), Cx, Cs, and Cp. To solve the corresponding steady-state equations, we interpret the equations (7.189) to (7.192) as a system of four coupled scalar homogeneous equations for the right-hand sides of the DE system in the form F(E, Cx, Cs, Cp) =0. The resulting coupled system of four scalar equations is best solved via Newton s22 method after finding the Jacobian23 DF by partial differentiation of the right-hand-side functions / of the equations (7.189) to (7.192). I.e.,... 

Thus the dispersion relation is symmetric about q = 0. It follows from eqs. (18) and (21) that the components of c ( q/) satisfy the same set of 3.v linear homogeneous equations as the components of the eigenvector c(q /). Therefore, if degeneracy is absent, e(q /) and c ( q /) can only differ by a phase factor (which preserves normalization). The physical properties of the system are independent of the choice of this phase factor, which we take to be... 

Note that Eq. (A.43) is identical to Eq. (A.l) therefore, the solutions of this homogeneous differential equation system have the following form (see Eq. (A.20)) ... 

For the solutions of the homogeneous differential equation system (A.43) with conditions (A.45)-(A.49), it is obtained... 

The first and second laws of thermodynamics for a homogeneous closed system involving only PV work lead to the fundamental equation for the internal energy... 

However the obtained system of the equations remains still too difficult for analyzing. For definiteness the (6,0) SWNT has been studied (N=6). To facilitate the solution of the equation system (8) we shall research first of all the case when only one oscillation mode (k=0) is induced. Obviously, the case corresponds to the oscillations, which are homogeneous along SWNT perimeter. For further simplification of the equation system we shall suppose that (pG = (p G, y/a = f/. Using Lorenz s invariance property for running... 

Comparison of equations (9. A.4) and (9.74) shows that all results deduced for the ORDE for a general homogeneous photochemical system can be applied to a heterogeneous colloidal system. 

The LCAO assumption and the Fock equations lead to the Roothaan-Hall matrix equation, or system of B homogeneous equations in B unknowns ... 

This system of equations (1) and (2) has only one solution—differing from zero and so physically significant—when the equations are not contradictory (but dependent). The condition for this in a system of linear homogeneous equations ... 

Inhomogeneous redox systems The forms of Nernst equation quoted in Section 1.41(a), (b), and (c) are, strictly speaking, valid only for homogeneous redox systems, where there is no change in the number of molecules (or ions) when the substance is reduced or oxidized. For inhomogeneous systems, where this is not the case, general equations would be too complex to quote, but the... 

---