The Required Equations

The procedure for calculating a phase boundary is, as just mentioned, to calculate the locus of P, T conditions such that A G = 0 for the reaction involving the phases on either side of the boundary. To do this we must be able to calculate the apparent or standard free energies of formation of the phases as a function of P and T, as outlined in Chapter 7. This is done for compound i with the equations 

For a reaction involving a number of phases or compounds or species as reactants and products, A G is simply the sum of the individual A, G terms (AaG or AjG) for each compound. (In the following example and those in the next few chapters, each i will be a distinct phase, but in general i may be any constituent of the system). Thus equation (7.27) becomes 

Since i iAG° and iG° can both be written as ArG°, equation (8.5) can also be written 

This (or equation 8.5) is the general form for the calculation of the free energy of any reaction at T and P, where no changes in phase composition are involved. 

Phase boundaries are calculated from equation (8.6), with the additional knowledge that Aj-G = 0 on the boundary. For the reaction 

---

Strategy For the required equation, the coefficients are half as great as in the given equation the equation is also reversed. Apply rules 1 and 2 in succession. 

Urine data are required to determine the individual values of ke and km. The required equations are derived next. 

Although the equations become considerably more complex than for the IV case, Cmax and Cm n can be calculated when the drug is administered by an extra-vascular route. The required equations may be developed as follows The equation describing the plasma concentration versus time curve following one extra-vascular administration was discussed previously. Equation (35) may be written as follows ... 

Interval halving can also be used when more than one unknown must be found. For example, suppose there are two unknowns. Two interval-halving loops could be used, one inside the other. With a fixed value of the outside variable, the inside loop is converged first to find the inside variable. Then the outside variable is changed, and the inside loop is reeonverged. This procedure is repeated until both unknown variables are found that satisfy all the required equations. 

In this way we obtained the approximate source concentration ratio 7, and then the degree of partial melting can be calculated. From Eqs. (7.5) and (7.6), the required equation is obtained as follows ... 

In the foregoing we have discussed the determination of the chemical potentials as functions of the temperature, pressure, and composition by means of experimental studies of phase equilibria. The converse problem of determining the phase equilibria from a knowledge of the chemical potentials is of some importance. For any given phase equilibrium the required equations are the same as those developed for the integral method. The solution of the equation or equations requires that a sufficient number of... 

The growth rate of the ApBq layer is described by equation (1.24), while its dissolution rate by equation (5.17). Term-by-term subtracting of the latter equation from the former yields the required equation describing the rate of formation of this layer at the interface between solid A and liquid B... 

When x, = y, P/Pf, the Raoult s law analogy, is substituted the required equation is reproduced ... 

The first part of this question is discussed in Section 10.5.2 and the required equation is presented as equation 10.108. 

When several parameters need to be optimized simultaneouslyf the use of simple model equations (ifpossible) seems to have advantages over linear interpolation methods. If the required equations become more complicated, however, this advantage is rapidly lost. 

The second step is the evaluation of the change in fugacity of the liquid with increase in pressure above P The required equation follows directly fro Eq. (11.17),... 

The required equations for p(r Rx) and v(r Rx) are derived by conditionally averaging the Stokes equations, thereby obtaining... 

Calculate AH298 for the reaction, and set out an equation for A H, as a function of temperature T. This step directly employs the procedures developed in the preceding examples. Thus Ais found to be 1.0728 x 104 cal/(g mol), and the required equation is... 

Because we need 1 mol of C2H2 on the left of the required equation, and we have 2 mol there in equation 1, we divide equation 1 by 2, including its A// value ... 

To obtain AH for the required reaction, we must somehow combine equations (a), (b), (c), and (d) to produce that reaction, and add the corresponding AH values. This procedure can best be done by focusing on the reactants and products of the required reaction. The reactants are B(s) and H2(g), and the product is B2H6(g). How can we obtain the correct equation Reaction (a) has B(s) as a reactant, as needed in the required equation. Thus reaction (a) will be used as it is. Reaction (b) has B2H6(g) as a reactant, but this substance is needed as a product. Thus reaction (b) must be reversed, and the sign of AH changed accordingly. Up to this point we have... 

Electrosorption data can also be interpreted In terms of surface pressures or. for that matter, two-dimensional equations of state. For the above models, the required equations read (see app. 1)... 

For practical purposes it would be expedient to have equations available featuring K° as a parameter. To this end, the required equations of sec. 4.6d could be modified. A numerical solution for large xa has been given by Kljlstra et al. l, based on Flxman s theory. Results are presented in fig. 4.31. The reduction of the mobility by conduction behind the slip plane is obvious. For very high ratio the situation arises of systems with low mobility but... 

To write the required equations for the dissolution and dissociation of salts, you must recognize the names and formulas of common, stable ions that are often present. Table 2-3 contains the names and formulas of several ions since then you have learned others. Review these names and formulas carefully (formulas of ions must include their charges). Many of the ions you need to know are present in the salts listed in Appendix H. 

The subscript av serves to remind us that [yl] and are evaluated at the arithmetic average composition. To solve Eq. 8.5.11 we need an expression for the variation of over the film. The required equation is provided by the last of the set of n Maxwell-Stefan relations (Eq. 8.3.2), which simplifies to give... 

At boundary nodes where the variable values are given by Dirichlet conditions, no model equations are solved. When the boundary condition involve derivatives as defined by Neumann conditions, the boundary condition must be discretized to provide the required equation. The governing equation is thus solved on internal points only, not on the boundaries. Mixed or Robin conditions can also be used. These conditions consist of linear combinations of the variable value and its gradient at the boundary. A common problem does arise when higher order approximations of the derivatives are used at... 

It is usually possible to calculate rate coefficients, but mechanism postulation is necessary. Equations are used in coefficient calculation, and development of the required equations is based on knowledge of the reactions occurring in the system. If the system studied is relatively simple, it may be possible to deduce mechanisms from curve shape. The postulate can some-I lines be confirmed by ancillary measurements in the experimental system (e.g., increase of the same-charge ion in solution would indicate an exchange reaction). In other cases, confirmation of mechanisms may require additional lests. 

We could manipulate the equations and cancel the species that appear on both sides to reassure ourselves that the required equation is actually obtained.)... 

Substituting from equation (3.65) into equation (3.62) and dividing throughout by Sx yields the required equation for the conservation of eneigy ... 

Multiplicity of active centers may also be due to die influence of penultimate (antepenultimate, etc.) units. This influence has been studied for free radical polymerization and the required equations are available 7). In ring-opening copolymerizations, due to the onium structure of the growing species, this influence should be more... 

Let a and / be the corrections required to make these values satisfy the conditions of the problem in hand. The required equation is, therefore,... 

In order to obtain the overall kinetic equation for a complex reaction scheme it is necessary to write down the differential rate equations for the reactants, products, and intermediates, and to integrate them in order to obtain an expression for the concentrations of reactants and products at various times. However, in many cases there are no mathematical methods available for the solution of the differential equations. When this happens, numerical solutions may be obtained using a computer. It is sometimes possible to avoid this procedure by using an approximate method, known as the steady-state method, to obtain the required equation. 

The growth of crystals in terms of diffrision theory can be described by a similar equation to Equation 7.1 used to quantify the mass transfer of particles. The required equation will be of the form ... 

---