Kirchhoffs equation

If the system consists of the three phases with two components solid salt + solution + vapour we obtain KirchhofFs equation (cf. 169). ... 

The effect of T on AHp j is taken into account by the Kirchhoff equation (Denbigh, 198 1,... 

We assumed in Justification Box 5.1 that AH aelt/ is independent of temperature and pressure, which is not quite true, although the dependence is usually sufficiently slight that we can legitimately ignore it. For accurate work, we need to recall the Kirchhoff equation (Equation (3.19)) to correct for changes in AH. 

The simplesf mefhod of solution of fhe Kirchhoff equations that correspond to the random network of conducfance elemenfs in three dimensions is in the single-bond effective medium approximation (SB-EMA), wherein a single effective bond between two pores is considered in an effective medium of surrounding bonds. The conductivify (7b, of fhe effective bond is obtained from the self-consistent solution of fhe equation ... 

As mentioned in Section 2.4, in the ionic model the chemical bond is an electrical capacitor. It is therefore possible to replace the bond network by an equivalent electric circuit consisting of links which contain capacitors as shown in Fig. 2.6. The appropriate Kirchhoff equations for this electrical network are eqns (2.7) and (2.11). It is thus possible in principle to determine the bond fluxes for a bond network in exactly the same way as one solves for the charges on the capacitors of an electrical network. While solving these equations is simple in principle providing the capacitances are known, the calculation itself can be... 

The network equations (3.3) and (3.4) invite comparison with the Kirchhoff equations (2.7) and (2.11). By choosing the ionic charge, in eqn (2.7) to be the same as the atomic valence, F,-, in eqn (3.3), and recognizing that in unstrained structures is equal to Sy, it follows from eqn (2.11) that loop ijl Cy = 0 which, when compared with eqn (3.4), means that the capacitances in eqn (2.11) must cancel, i.e. they must all be equal. This greatly simplifies the model, since it means that... 

In Chapter 2 it was shown that the Madelung field of a crystal is equivalent to a capacitive electric circuit which can be solved using a set of Kirchhoff equations. In Sections 3.1 and 3.2 it was shown that for unstrained structures the capacitances are all equal and that there is a simple relationship between the bond flux (or experimental bond valence) and the bond length. These ideas are brought together here in a summary of the three basic rules of the bond valence model, Rules 3.3, 3.4, and 3.5. 

There is an alternative way of calculating the bond flux using the Kirchhoff equations ((2.7) and (2.11)) in place of the network equations ((3.3) and (3.4)), the problem in this case being to determine the appropriate bond capacitances which are not now all equal. Where the multipole produces a shorter bond, a larger capacitance is needed, and conversely where the multipole produces a longer bond, a smaller capacitance is needed. Transferable bond capacitances have been successfully used to model the asymmetries in d° transition metal environments as discussed in Section 8.3.2 below. 

The network equations constitute a set of A a 1 valence sum rule equations (eqn (3.3)) and A b Xa+1 loop equations (eqn (3.4)) where the network contains atoms and A b bonds. Alternatively one can use the equivalent Kirchhoff equations (2.7) and (2.11). One can readily write down equations of type 3.3 but one of these is redundant since the sum of all atomic valences in the crystal must be zero. There are many more than Ab — Aa + 1 possible loops in most bond graphs, but only Ab —Aa+ 1 are independent. Equations (3.3) and (3.4) thus constitute a set of Ab equations which is exactly the number needed to solve for the Ab unknown bond valences,. s. 

The simulator used was a DISMOL, described previously by Batistella and Maciel (2). All explanations of the equations used, the solution methods, and the routine of solution are described in Batistella and Maciel (5). DISMOL is a simulator that permits changes in feed composition, feed temperaturethe evaporation rate, as well as feed flow rate. The effective rate of surface evaporation is obtained from the kinetic theory of gases. The liquid film thickness is obtained by mass balance and geometry of the evaporator. The temperature in the liquid obeys the Fourier-Kirchhoff equation. The solution of the velocity profile requires knowledge of the viscosity and the liquid film thickness over the evaporator. The solution for the temperature and the concentration profiles requires knowledge of the velocity profiles, which determine the convective heat and mass fluxes. 

Note that the coordinatewise optimization method has already found numerous practical applications to optimization of heat, oil, water, and gas supply systems (Merenkov and Khasilev, 1985 Merenkov et al., 1992 Sumarokov, 1976). As a matter of fact, in the algorithms used for applied problems the flow distribution was calculated not on the base of entropy maximization, but with the help of the closed system of equations of the first and second Kirchhoff laws. However, because of equivalence of approaches that are based on the principle of conservation and equilibrium (extremality) the Kirchhoff equations can be strictly replaced by thermodynamic relations. And the extreme thermodynamic approach in many cases should be preferable owing to the known low sensitivity of the extremal methods to variation of the space of variables. 

According to the Kirchhoff equation and Equation (1.73b), Equation (1.105) gives ... 

For monomolecular processes of any complexity, this electrotechnic analogy makes it possible to determine the stationary rate of chemical reac tions with respect to the reactive species. To do this, one has to consider the Kirchhoff equation for the balance of current inflow and outflow at aU points of the electric circuit contacts. 

While comparing the stationary kinetic equations (in their thermodynamic form) for the intermediate concentrations of system (1.34) to the Kirchhoff equation for the electric current inflow and outflow at all junction points of an equivalent electric circuit, one can easily ascertain that the combination of reactions (1.34) will be described by the equivalent electric diagram... 

Figure 1.3 Illustration of the application of the Kirchhoff equations for the balance of electric charge inflowing to and outflowing from point i Iji + l d + In = 0.

If the transformation pathway cannot be reduced to monomolecular reactions, nonunit stoichiometric coefficients may appear at some junction points of the kinetic resistors. In terms of electric circuits, this means that the absence of the balance of the current inflow and outflow at this June tion point may cause norJinearity and deviations from the canonical form of the KirchhofF equation. 

The commonly known consequence of the KirchhofF equation is that the electric potential in each intermediate junction point i is described by the equation... 

The bond flux can also be calculated by recognizing that in the ionic limit each bond is an electric capacitor (represented by flux linking two opposite charges). The network of atoms and bonds is thus a capacitive electrical circuit, and since in most equilibrium structures aU the bond capacitances are empirically found to be equal, the fluxes can be calculated using the Kirchhoff equations (2) and (3) in which the bond capacitances cancel. ... 

The Kirchhoff equation as derived above riiould be applicable to both chemical and physical processes, but one highly important limitation must be borne in mind. For a chemical reaction there is no difficulty concerning (dAH/dT)p, i.e., the variation of AH with temperature, at constant pressure, since the reaction can be carried out at two or more temperatures and AH determined at the same pressure, e.g., 1 atm., in each case. For a phase change, such as fusion or vaporization, however, the ordinary latent heat of furion or vaporization (AH) is the value under equilibrium conditions, when a change of temperature is accompanied by a change of pressure. If equation (12.7) is to be applied to a phase change the AH s must refer to the same pressure at different temperatures these are consequently not the ordinary latent heats. If the variation of the equilibrium heat of fusion, vaporization or transition with temperature is required, equation (12.7) must be modified, as will be seen in 271. 

Variation of Equilibrium Latent Heat with Temperature.—If a given substance can occur in two phases, A and B, one of which changes into the other as the temperature is raised, then the value of the accompanying latent heat and its variation with temperature depend on whether the pressure is maintained constant, e.g., 1 atm., or whether it is the equilibrium value. In the former case the Kirchhoff equation (12.7) will apply, as stated in 12j, but if equilibrium conditions are postulated allowance must be made for the change of pressure with temperature. 

Further, from KirchhofFs equation (Chap I) we have—... 

Porter s equation has been applied by 0 Wood to the calculation of the osmotic pressure from the results of vapour pressure measurements obtained in connection with concentrated solutions of sucrose in water at a series of different temperatures (cf Wood, Trans Faraday Soc, 1915) Porter [Trans Faraday Soc, 19x5) has further considered von Babo b law, and Kirchhoffs equation for the latent heat of dilution of a solution, laying special emphasis on the assumptions introduced into the deduction of the expressions as obtained in their usual form The reader is referred to the paper for details... 

KirchhofFs equation holds for the vapour pressure of steam 7 for this Brunelli8 gave ... 

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