Calculation of the Driving Force

The rate of the free energy release due to the phase transformation in the region R can be written as 

the first term in the right-hand side is the energy gained per unit volume and the second term corresponds to the formation of a new boundary region of length h between o and f phases. 

Integration of Equation 12.12 is performed only for the part of the boundary that is located between a and ao phases. The second item in Equation 12.12 corresponds to the loss in the change of total free energy and to the formation of the new part of the boundary between a and P phases, which is of length h. Therefore, for a unit volume we get where y is the free energy per unit area of the a-P 

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Miodownik, A. P., Watkin, J. S. and Gittus, J. H. (1979) Calculation of the driving force for the radiation induced precipitation of NiaSi in nickel silicon alloys , UKAEA Report ND> R-283(S), February. 

In the new MINLP model, some equations prescribe mass balances at the concentration borders other equations assume monotonity of the concentrations. Equation 1 sets the mass exchange me to zero if the unit does not exist otherwise its value is constrained by Q (big-M technique). Equations 2 and 3 force the exact calculation of the driving forces of the units (dy) only if the given units do exist in the solution F is a constraint. Continuous 0 ij,kslack variables are applied instead of the binary feij,k variables that are used in equation 4. Equality of z and bz can be enforced by a linear penalty term in the objective function (equation 5) since pz and ps are positive. 

Model calculations of the growth rate R are shown in Fig. 3. These are plotted as a function of the driving force for crystallization, A/x ln(p/pg), where p and Pg are the actual and equilibrium vapor pressures, respectively. At very low temperatures, the growth rate is essentially... 

Fig. 3 Ising model calculations of the normalized growth rate as a function of the driving force. The surface temperatures are 0.40T , closed circles, 0.54 T/j, open circles, and 1.08 T/j, squares.

The reason that the Second Law has no quantitative relevance to nonequilibrium states is that it gives the direction of change, not the rate of change. So although it allows the calculation of the thermodynamic force that drives the system toward equilibrium, it does not provide a basis for calculating the all... 

One of the driving forces for the formation of CALPHAD was the concept of providing a unified database for use by the various calculation programmes... 

Figure 11 Plot of the log of the electron transfer rate as a function of the driving force. Also shown is a Marcus theory curve calculated from Eq. (8), taking V = 2.12 cm and X = 3225 cm . ...

Figure 4.19(a) shows the concentration of water vapor on the feed and permeate sides of the membrane module in the case of a simple counter-flow module. On the high-pressure side of the module, the water vapor concentration in the feed gas drops from 1000 ppm to about 310 ppm halfway through the module and to 100 ppm at the residue end. The graph directly below the module drawing shows the theoretical maximum concentration of water vapor on the permeate side of the membrane. This maximum is determined by the feed-to-permeate pressure ratio of 20 as described in the footnote to page 186. The actual calculated permeate-side concentration is also shown. The difference between these two lines is a measure of the driving force for water vapor transport across the membrane. At the feed end of the module, this difference is about 1000 ppm, but at the permeate end the difference is only about 100 ppm. 

A graph showing the change of the driving force for the electron transfer rate, calculated from Marcus theory, versus the rate constant is given in Figure 16 (bottom). 

Particularly interesting is that Eq. (84) respects the limits in Eqs. (82) and (83). For intermediate values of the driving force, the calculated activation barrier from Eq. (84) is very close to that obtained in the classic harmonic model developed in this section [Eq. (79)]. Thus from an experimental point of view, distinction between the two models is rather difficult or impossible [36a], although totally different mathematical expressions for AG variations with AG° and k are obtained. 

The assessment of the driving force follows from the reaction standard Gibbs free energies A Go, calculated by using the corresponding energies of formation of the reactant CX (AGr) and the products (AGp) in reactions (4.11a,b). 

The range of the stress-intensity factor AK, a measure of the driving force for crack, was calculated using the formula... 

Quite recently, Lee and co-workers have theoretically examined the reactions of vinyl chloride with various nucleophiles in quest of the driving force of the unexpected preference for the SnVct process. Calculations were carried out at three levels, RHF/6-311 -b G (RHF), MP2/6-311 -b G (MP2), and G2(-b)(MP2), with MP2/6-311 -b G geometries for the latter two levels. The preference for tror tt attack is dependent on the level of calculation and the nature of the nucleophile. The highest level results are summarized in Table 2 and typical strucmres of TSs for the SnV(7 and SnV tt pathways are shown in Fig. 2. The data show that the SnVa route is energetically favored over the SNVTrroute for Cl and Br as nucleophile, while the reverse is the case for OH and SH . 

From the positions of the maxima of the ionic density profiles relative to the minima of the ion-metal interaction potentials, they concluded that 1 is contact adsorbed and Li" " is not. Spohr [190] and later Perera and Berkowitz [191] obtained similar results by means of free energy calculations for 1 and simultaneous Li" " and 1 adsorption, respectively, on Pt(lOO), using the same interaction potentials. Eck and Spohr [77, 192] and Toth and Heinzinger [80] studied the adsorption of Li+ and several halide ions near the ab initio model of the mercury interface [40]. The liquid/ gas interface, contrary to metallic interfaces, is depleted in the interfacial region [193-195]. This is a consequence of the driving force towards fully hydrated ions. 

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