Cross differentials

From cross-differentiation identities one can derive some additional Maxwell relations for partial molar quantities ... 

Cross-differentiation of the Gibbs equation gives the relation... 

McKay (2) showed the possibility of using the Euler cross differentiation equation for the thermodynamic treatment of a three-component system. 

Cross-differentiating the equations and eliminating the p yields the classic wave equation ... 

After cross differentiation, the difference of the two momentum equations produces the ordinary differential equation... 

Substitution into the momentum equations, cross differentiating, and subtracting yields the following third-order differential equation,... 

The pressure-gradient terms can be eliminated by cross differentiating the momentum equations and subtracting. The following third-order equation emerges ... 

The Euler criterion is therefore equivalent to the familiar mixed partials of a function are equal rule of calculus. This cross-differentiation rule is also the condition for the function z(x, y) to have well-defined (single-valued) first derivatives at each point, and thus to be graphable. 

Esin-Markov coefficient — Various cross-differential relationships can be obtained from the - Gibbs-Lippmann equation because it is a complete differential. For instance,... 

For nonpolarizable electrodes (dy/dE) gives QA the value of which depends on the choice of reference component. Various cross-differential relationships can also be obtained (see -> Esin-Markov coefficient). 

Maxwell first noted the cross relations based on a property of the total differentials of the state functions. The cross differentiations of a total differential of the state function are equal to each other. Table 1.14 summarizes the total differentials and the corresponding Maxwell relations. The Maxwell relations may be used to construct important thermodynamic equations of states. 

The volume V is that of the entire system (fig. 1.37) that of the pore is Ah. We repeat that all Information on the work is In 5 and that the division between an area-dependent and a distance-dependent contribution is not always rigorous. However, in the present model the relation between /7 and can immediately be established by cross-differentiation... 

This represents a three-dimensional surface of which the p, T-line Is only one of the three cross-sections, and now dependent on h. The three cross-sections lp,T]i, p,h)T and (h,T) are obtained from the appropriate cross-differentiations. 

Iv) Cross-differentiation also yields Esin-Markov coefficients p. Introduced in sec. I.5.6d. These coefficients contain information on the relative contributions of the cations and anions to the countercharge, l.e. they help to obtain the composition of the double layer. Experimentally, is measured as the horizontal spacing between ff°(pAg) or salt concentrations and defined as... 

Temperature studies provided further insights regarding the state of adjacent water. By way of illustration, in fig. 3.44 the excess entropy per unit area is given as a function of the surface charge. This excess was obtained by retaining the S°dT term in [3.4.21 and cross-differentiation with the o°dp. term. ... 

The starting point is, as before, the Gibbs equation [3.4.11, which can be elaborated for the system under consideration. Typically, attention Is now paid to the F d term, expressed as (3.4.2) for the Agl-case. Cross-differentiation between the temperature, the surface charge, or the salt term, leads to useful... 

Returning to 13.4.11a or lib), cross-differentiation between the first and third terms on the r.h.s. yields (3.4.13 or 13a(, which allows the computation of as a function of pAg, except for a constant. Sometimes there are ways to find this constant, for instance if there are Indications that under certain conditions r,... 

A similar set of four basic equations may be written for the adsorption of a cationic surfactant. For each equation we can write three cross-differential relations, so that in total 24 of these may be formulated. To remain specific, let us consider the variation of the organic adsorption with the surface charge. To that end, the first and third terms on the r.h.s. s of 3.12.5a and 5b] are cross-differentiated. 

The next step consists in carrying out a cross differentiation in either order for example, by applying the relation d E/dSdV) = d E/dVdS) to Eq. (1.13.If) one obtains Eq. (1.13.9) below. The other expressions are similarly derived. 

Once again we simplify by assuming V to be independent of location r within the sample. With this simplification a second relation of interest is derived from Eq. (5.7.17) by the cross differentiation... 

Equations (5.8.5) may be used to obtain twenty four Maxwell relations by cross differentiation these are listed in Table 5.8.1. Several of these arise as trivial modifications of those specified in Section 1.13. The new expressions involve partial derivatives of either fy d r V or of fy d r M with respect to independent variables. A number of the interrelations are useful for starting further derivations they show, for example, how P varies with q or Ho under a variety of fixed... 

The thermodynamic inoperability of 7 does not mean that it is not allowable to use Gibbs adsorption law, because changes of 7 with T and fi s can be carried out reversibly provided the solid is inert. So, the inaccessibility of 7 for dispersed colloid particles does not invalidate the thermodynamics of double layer formation and the various equations obtained by cross-differentiation (see e.g. secs. 1.5.7, II.3.4 and II.3.12a). 

Pethica argued that 3E / dr could be related to 3y / da°, using the appropriate cross-differentiation in the Gibbs equation, but this is begging the question because one Ccumot be sure which form of this equation has to be used. 

The next step is to derive a relation between, J and K. It is a matter of taste, or convenience, which of these variables to take as the independent variable. For adsorption from dilute solutions it is customary to take the concentration (i.e. g ) as independent. On the other hand, for many theoretical analyses it is easier to assume a certain spatial geometry and then And out the g of the solvent with which the curved interface is at equilibrium. Let us foUow the second route, i.e. we want to establish dg / 3J rmd dg / 8K. These differential quotients can be obtained from [4.7.11 by changing variables and cross-differentiation. For instance. 

P. A. Haynes and G. A. Cross, Differential glycosylation of epitope-tagged glycoprotein Gp72 during... 

By cross-differentiating equations like (10.2.23) and (10.2.33), one obtains useful relations between partial derivatives for the thermodynamic properties of the polarizable interface. From equation (10.2.23), cross-differentiation leads to the... 

This relationship states that the change in charge density on the electrode with electrolyte concentration measured at constant cell potential difference is equal to the change in ionic surface excess with cell potential difference measured at constant electrolyte concentration. Cross-differentiation of equation (10.2.33) gives... 

Hence, eliminating /q by cross-differentiating and combining (4-84a) and (4-84b), we obtain a single equation for fi ... 

It we cross-differentiate (12 122a) and (12 122b) and subtract the resulting two equations, we can eliminate the pressure and obtain a single higher-order equation ... 

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