Closure condition

This must be an identity, independent of F(x) if any vector in 3T is to be expressible in the form (8-9). Therefore, the closure condition is... 

We shall meet examples later where the basis is not denumerable, but has a continuous index, say g>, where the orthogonality and closure conditions, analogs of Eqs. (8-7) and (8-13), are respectively... 

Consider the same composite system as before, but with the impermeable diathermal wall no longer fixed. Both internal energy as well as the volume and V may now change, subject to the extra closure condition, RP) + VP) = constant. 

By way of illustration consider a binary composite system characterized by extensive parameters Xk and Xf in the two subsystems and the closure condition Xk + X k — Xk. The equilibrium values of Xk and X k are determined by the vanishing of quantities defined in the sense of equation (3) as... 

Figure 4.12 Principal component analysis of the major elements in Coumiac limestones. 91 percent of the variance is explained by the first two components. The data can be explained by the combination of three chemical end-members calcitic (CaO and C02), detrital (Si02 and A1203), and organic (organic C and Fe203). Because of the closure condition these three end-members translate into only two significant components.

In most cases of interest, however, the system represented by equation (5.3.3) is overdetermined and we must enforce the closure condition with a different method. Let us return to a standard mass-balance least-square problem, such as, for instance, calculating the mineral abundances from the whole-rock and mineral chemical compositions. If xu x2,.. -,x are the mineral fractions, which may be lumped together in a vector x, the closure condition... 

Solving the forward problem of the isotopic and chemical evolution of n reservoir exchanging a radioactive and its daughter isotope requires the solution of 3n— 1 differential equations (the minus one stems from the closure condition). The parameters are n (n — 1) independent flux factors k for the stable isotope N and n (n — 1) independent M/N fractionation factors D. In addition, the n values of R y the n values of Rh and the n—1 allotments x of the stable isotope among the reservoirs must be assumed at some time, preferably at the beginning of the evolution (e.g., 4.5 Ga ago), or in the modern times, in which case integration is carried out backwards in time. 

Due to the hermitian character of the dynamical matrix, the eigenvalues are real and the eigenvector satisfies the orthonormality and closure conditions. The coupling coefficients are given by... 

The term within the square brackets in equation 11.79 is the normalized (equilibrium) distribution coefficient between the minerals a and )3 (cf section 10.8) at the closure condition of the mineral isochron. Ganguly and Ruitz (1986) have shown it to be essentially equal to the observed (disequilibrium) distribution coefficient between the two minerals as measured at the present time. can be assumed to be 1, within reasonable approximation. Equation 11.79 can be calibrated by opportunely expanding AG° over P and T ... 

Zhang Y. and Chen N.S. (2007) Analytical solution for a spherical diffusion couple, with applications to closure conditions and geospeedometry. Geochim. Cosmachim. Acta submitted. 

This equation requires a closure condition which consists of fixing the value of either AG or the pressure P at the new state. 

The simulations reported here consisted of pressurizing an initially evacuated adsorber with four mixtures of different compositions. These simulations are very much like the traditional flash calculations of chemical engineering thermodynamics applied to an adsorption system. The first set of runs, which we refer to as set G-NVT, is equivalent to solving Eq. (4) with = 0, qf = 0, = y "AF > 0, closure condition... 

Because of the closure condition = 1), only two of these are indepen-... 

Several closure conditions have been proposed (see [75] and references therein). In the present study the MSA-like closure [51,52,72],... 

Solution of the MSA-like closure condition, given by Eq. (14), leads to a closed form analytical expression for the thermodynamic properties of the system [46, 48-50],... 

Here, we propose a more realistic model of protein-electrolyte mixture. In the present case all the ionic species (macroions, co-ions and counterions) are modelled as charged hard spheres interacting by Coulomb potential as for the primitive model (Sec. 2), but the macroions are allowed to form dimers as a result of the short-range attractive interaction. Numerical evaluation of this multicomponent version of the dimerizing-macroion model has been carried out using PROZA formalism, supplemented by the MSA closure conditions (Sec. 3). 

Equations (2.6.8) and (2.6.9) give what is known in statistical mechanics as the closure conditions. To solve a particular problem given u(r), one must first determine c(r) for r < CT. After Fourier transformation, one can calculate the total correlation function in Fourier space, namely, h(k), using equation (2.6.2). Inverse Fourier transformation then gives h r) and g(r). 

A treatment for polar solvents on the basis of the mean spherical approximation was first given by Wertheim [24, 25]. The closure conditions are based simply on the dipole-dipole interaction energy between the polar molecules in the system. Neglecting molecular polarizability, these conditions are... 

It should be clear that the pair correlation function has, in general, two contributions. One is due to interaction, which in this case is unity. The second arises from the closure condition with respect to N. Placing a particle at a fixed position changes the conditional density of particles everywhere in the system from N/V into (N— )/V. Hence, the pair correlation due to this effect is... 

To see this, we first note that though it is true that for ionic species, equations (4.123)-(4.126) can result from the electro-neutrality conditions, the conditions themselves are not necessarily a result of the electric charge neutrality. They arise from the closure condition with respect to the fragments A and B. Thus, for a solute S dissociating into two neutral fragments A and B, as in (4.118) not necessarily ionic species as in (4.121), we still have the following conservation relations ... 

Thus, we see that the requirement that the solute fully dissociates as stated in (4.118) or (4.121) imposes closure conditions on G, GAB, GBB, GAW, and Gbw. Therefore, care must be exercised to label these KBIs properly, e.g., we... 

This cannot be interpreted in terms of PS. Because of the long-range correlations imposed by the closure condition, these Gap are not the Kirkwood-Buff integrals. In the same sense, the relations (8.49), (8.51), and (8.53) hold true because of the closure condition with respect to the individual particles A and C. Clearly, one cannot conclude from (8.49) that the PS of W is zero. The sign of PS of W is determined by the difference GWa and Gwc, provided that GWa and GWc are evaluated in a system open with respect to the three components. 

Here, the integrations extend over the entire volume of the system. The interpretation of (G.7) and (G.8) is straightforward. The quantity pG is the change in the number of particles in the entire system caused by placing one particle at some fixed point, say R0. When N is constant, this change is exactly — 1 the particle we have placed at. No such closure condition is imposed in an open system. Equation (G.8) is just the compressibility equation. 

We now recognize that correlation between densities at different locations can arise from two sources. One is due to direct intermolecular interactions, and the other is due to the closure condition. We assume that the first is operative only at short distances say 0 < R < Rcor> where Rqor is the correlation distance beyond which a particle placed at fixed position does not have any influence on the density at any other position. This is the distance RCor> beyond which g(R) is nearly equal to 1. The second, is normally referred to as long-range correlation. We shall refer to it as the closure correlation (CC). Since this part has the 1ST 1 dependence on N, it has a negligible effect on g(R) at R < Rqok- It becomes important when integration extends to R —> oo. 

Note that in an open system, all the correlation is due to the direct interactions, no closure condition is in effect beyond R > Rcor, y0(LC) = 1. Next, we assume that within the correlation distance, both gc and g0 are the same (or nearly the same, but the difference is negligible) function of R. Subtracting (G.16) from (G.15) we obtain... 

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