Gibbs matrix

Since Cy = T dS/dT)v, the (1,1) element of this Gibbs matrix is inversely proportional to the specific heat Cy at constant volume. The (2,2) element is inversely... 

The same relations (11) and (12) hold for the Gibbs free energy in the (N, p,T) ensemble. Equation (11) is also valid for a quanmm mechanical system. Note that for a linear coupling scheme such as Eq. (10), the first term on the right of Eq. (12) is zero the matrix of second derivatives can then be shown to be definite negative, so that the free energy is a concave function of the Xi. 

Fig. 25. Relationship between the measured interfacial strength and the (negative) Gibbs free energy of mixing, (-AG )o5, for glass beads treated with various silane coupling agents embedded in a PVB matrix. Error bars correspond to 95% mean confidence intervals. Redrawn from ref. [165].

In some cases, an alternative explanation is possible. It may be assumed that any very complex organic counterion can also interact with the CP matrix with the formation of weak non-ionic bonds, e.g., dipole-dipole bonds or other types of weak interactions. If the energy of these weak additional interactions is on the level of the energy of the thermal motion, a set of microstates appears for counterions and the surrounding CP matrix, which leads to an increase in the entropy of the system. The changes in Gibbs free energy of this interaction may be evaluated in a semiquantitative way [15]. 

During ATP synthesis, protons move down these gradients from outside [Hour] into the mitochondrial matrix each proton doing both electrical and osmotic work (due to the concentration difference) so that the Gibbs energy change is... 

Using ln40=3.689 and MT = 18.292kJmoP1, and thermochemical Gibbs free energy at 2200 K from Barin and Knacke (1973), we make Table 6.4. We recognize in the four columns below the components C, O, N and H, the 6x4 component matrix B. 

Table 6.4. The component matrix and Gibbs energy of formation of various gaseous species in...

Let us consider a rock at temperature T whose chemical composition q (recipe) is expressed as the vector of all the molar fractions x0 of s elements or oxides. It is assumed that it can be made by an arbitrarily large number p s of mineral phases exclusive of solid solution. B is the component matrix of these minerals for the selected set of elements or oxides. Let nj be the number of moles of mineral j and gj its Gibbs free energy of formation AGf T estimated when formed from either the elements or the oxides. The function to be minimized is the Gibbs free energy G given by... 

Table 6.6. The component matrix and Gibbs free energy of formation for various minerals in the...

The signal generated by the complex is governed by several physical phenomena associated with the matrix thickness. As soon as the probe is placed in contact with the analyte, external mass transfer controls the movement of the analyte toward the surface of the optical probe.(S4) The osmotic pressure and Gibbs free energy dictate the permeation of the analyte into the matrix. Once the analyte has penetrated the matrix, internal mass transfer resistance controls the movement of the analyte in the matrix. Eventually, the probe reaches a steady state of equilibrium with molecules continuously moving in and out of the matrix. 

The use of ISEs in non-aqueous media(for a survey see [125,128]) is limited to electrodes with solid or glassy membranes. Even here there are further limitations connected with membrane material dissolution as a result of complexation by the solvent and damage to the membrane matrix or to the cement between the membrane and the electrode body. Silver halide electrodes have been used in methanol, ethanol, n-propanol, /so-propanol and other aliphatic alcohols, dimethylformamide, acetic acid and mixtures with water [40, 81, 121, 128]. The slope of the ISE potential dependence on the logarithm of the activity decreases with decreasing dielectric constant of the medium. With the fluoride ISE, the theoretical slope was found in ethanol-water mixtures [95] and in dimethylsulphoxide [23], and with PbS ISE in alcohols, their mixtures with water, dioxan and dimethylsulphoxide [134]. The standard Gibbs energies for the transfer of ions from water into these media were also determined [27, 30] using ISEs in non-aqueous media. 

We have already seen that the degree of polymerization of the melt is controlled by the amount of silica in the system (see, for instance, figure 6.4). If we mix two fused salts with the same amount of silica and with cations of similar properties, the anion matrix is not modified by the mixing process and the Gibbs free energy of mixing arises entirely from mixing in the cation matrix—i.e.. 

Equation (5.30) holds for the simple case of a phase with the formula A, B)i(C, D)i. But for more complex phases the function for the Gibbs reference energy surface may be generalised by arranging the site fractions in a (f + c) matrix if there are I sublattices and c components. 

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