Approximate closed equations

The conservation of density of a mechanical system in phase space (Liouville s theorem) implies a rigorous functional relation between g(ri2) and 53(> i2,> i3,> 23)- Starting from this rigorous functional relation (an integro-differential equation ) an approximate closed equation can be obtained " by using the superposition approximation which asserts that... 

In all other cases approximate closed equations for (x), and a(t) can be used as long as it is justified to assume that the probability distribution P(x t) has only one highly peaked maximum. Under such circumstances K(x) can be expanded in a Taylor series around (x), as ... 

For a uni-modal distribution centered around (x), it follows from the exact equation (3.76) that the approximate closed equation is ... 

The approximate closed equations for grossvariables derived in this section still have a rather complicated structure, since only relatively general assumptions have been used. In special models, however, it may be possible to find alternative and simpler derivations of grossvariable equations. Such an alternative procedure will be developed in Chap. 6 when setting up equations for certain grossvariables of political psychology in interacting societies. 

Kinetics. Details of the kinetics of polymerization of THF have been reviewed (6,148). There are five main conclusions. (/) Macroions are the principal propagating species in all systems. (2) With stable complex anions, such as PF , SbF , and AsF , the polymerization is living under normal polymerization conditions. When initia tion is fast, kinetics of polymerizations in bulk can be closely approximated by equation 2, where/ is the specific rate constant of propagation /is time [I q is the initiator concentration at t = 0 and [M q, [M and [M are the monomer concentrations at t = 0, at equiHbrium, and at time /, respectively. 

After the electrode reaction starts at a potential close to E°, the concentrations of both O and R in a thin layer of solution next to the electrode become different from those in the bulk, cQ and cR. This layer is known as the diffusion layer. Beyond the diffusion layer, the solution is maintained uniform by natural or forced convection. When the reaction continues, the diffusion layer s thickness, /, increases with time until it reaches a steady-state value. This behaviour is also known as the relaxation process and accounts for many features of a voltammogram. Besides the electrode potential, equations (A.3) and (A.4) show that the electrode current output is proportional to the concentration gradient dcourfa /dx or dcRrface/dx. If the concentration distribution in the diffusion layer is almost linear, which is true under a steady state, these gradients can be qualitatively approximated by equation (A.5). 

Figure 9.4 shows curves for the drag coefficient (based on the velocity for a freely settling sphere and the mean approach velocity for a fixed or suspended sphere) and for the fractional increase in drag caused by wall effects, Kp — 1). Up to Re of order 50, the results are approximated closely by an equation proposed by Fay on and Happel (F2) ... 

It has been shown that the two-phase pressure variation for palladium-hydrogen yields values of ASa— and AHa—/ , and that these values are closely temperature independent. The temperature independence results because the value of the integral in Equation 21 can be approximated closely by the corresponding relative partial value at the critical composition. Variations of ASg o and AHh—o with temperature (see Figure 6) are apparently too small to be detected in the plot of In /22(two-phase) against T ly but can be detected by the more sensitive plot of RT In P /2 against T or possibly could be detected by calorimetric determinations of AHa—over a wide temperature range. 

This is no longer a closed equation for , but higher moments enter as well. The evolution of < Y> in the course of time is therefore not determined by itself, but is influenced by the fluctuations around this average. The macroscopic approximation consists in ignoring these fluctuations, and keeping only the first term in the expansion (8.5). With this approximation therefore (8.4) is valid even when a y) is nonlinear. Thus one obtains as macroscopic equation... 

In this Section we consider namely the latter. Under the shortened Kirkwood s approximation equation (2.3.64) is expressed through Y(r,t) only, therefore the substitution of equation (2.3.64) into (2.3.16) results in a closed equation uniquely defining Y(r, t). This equation is linear in Y (see the title of this Section) and equations (4.1.14), (4.1.15) are not used any longer. Substituting equation (2.3.64) into (4.1.16), using the relative coordinate r = r i - r [ and after some manipulations... 

From these studies it appears that the kinetics of polymerization of THF are closely approximated by equation 42. The equation does not always apply from the beginning of the polymerization and frequently cannot be applied before a steady-state of active centers is achieved. The initiator term, / , in this equation is often a function of several components. Only in the case of preformed trialkyloxo nium ions of the form R30+X is the initiation simple. These results suggest that in order to theoretically study the kinetics of polymerization of THF or to compare the kinetics of THF polymerization in the presence of different gegenions, it is desirable to use preformed trialkyl oxonium salts. Ideally... 

In view of all of the preceding observations concerning the formal differences between closed and open systems, what general conclusions can be drawn about the applicability of equilibrium concepts in understanding and describing the chemical behavior of the elements in natural water systems Since equilibrium is the time-invariant state of a closed system, the question is under what conditions do open systems approximate closed systems. A simple example will illustrate the relationships, which are already implicit in Equation 35. If one considers the case of a simple reaction... 

Appendix 2 to this chapter gives a derivation that shows that the isotope effect is more closely approximated by Equation 2.72. The II symbols signify a... 

The structure factor governing the distribution of intensity in a layer line for a commensurable helix may be approximated closely 1 the cylindrically symmetrical transform of a helical molecule with atoms at cylindrical coordinates, (rj, (j>j, Zj) for the asymmetric unit repeating along the helix axis in accordance with equation [1], The cylindrically averaged intensity function has been given in convenient from by Davies and Rich (5) ... 

These relations offer considerable simplification of the equation for vertical motion, as we shall see. However, note that Eq (2-14) approximates closely the expressions given by Eq (2-15) in the specified ranges of Reynolds numbers. 

The solution obtained using this system of equations is plotted as dashed lines in Figure 3.4. The solution based on this quasi-steady state approximation closely matches the solution obtained by solving the full kinetic system of Equations (3.27). The major difference between the two solutions is that the quasi-steady approximation does not account explicitly for enzyme binding. Therefore a + b remains constant in this case, while in the full kinetic system a + b + c remains constant. Since the fraction of reactant A that is bound to the enzyme is small (c/a << 1), the quasi-steady approximation is relatively accurate. 

The parameter F mimics tire short time, microscopic dynamics, and depends on structural and hydrodynamic correlations. The memory function describes stress fiuctuations which become more sluggish together with density fiuctuations, because slow structural rearrangements dominate all quantities. A self consistent approximation closing the equations of motion is made mimicking (14a). In the... 

The redox potential of solution, E, is solely determined by the Fe /Fe activity ratio in this case. If the pH is kept very low so that dissolved Fe " does not hydrolyze to form Fe-hydroxy species, then this activity ratio in solution is approximated closely by the ratio of the total dissolved Fe to Fe. Once a complexing ligand is put in solution, the activity of one of the metal ions is lowered relative to the other, and the redox potential changes. For example, fluoride complexes much more strongly with Fe " than with Fe, lowering the activity of Fe relative to Fe. According to equation 7.55, the consequence... 

In solving the many-particle Schrodinger equation, it is desirable to choose the approximate potential V0(x) to be as close as possible to the actual potential V(x), since this leads to rapid convergence of the expansion (18). Goscinski showed [20, 21] that for atoms, the approximate Schrodinger equation (10) can be solved exactly provided that V0(x) is chosen to be the attractive Coulomb potential of the bare atomic nucleus ... 

The value of geometrical percolation threshold pc. The volume fraction at random close packing, d>m, is identified with . The pc of a dispersion of randomly placed monodisperse ellipsoidal filler particles as a function of Af is approximated by Equation 13.36. Equation 13.37 can then be used for fibers with Af>10, and Equation 13.38 for platelets of aspect ratio 1/Af, with the results summarized in Figure 13.14. 

One of the main methods for obtaining approximate closed-form solutions of the corresponding hydrodynamic problems is to linearize the Navier-Stokes equations for low Reynolds numbers. This method is often used in this chapter when we study the motion of small particles, drops, and bubbles in a fluid. 

For jet flows and mixing layers, there are various estimates for Prt ranging from 0.5 to 0.75 [397]. These estimates are helpful for approximate calculations of turbulent heat and mass transfer and can be used for closing equations (3.1.37). 

If E is equal to Wo then < is identical1 with (as can be seen from Eq. 26- 6), so that it is natural to assume that if E is nearly equal to Wo the function will approximate closely to the true wave function p0- The variation method is therefore very frequently used to obtain approximate wave functions as well as approximate energy values. From Equation 26-6 we see that the application of the variation method provides us with that function among those considered which approximates most closely to according to the following criterion On expanding — in terms of the correct wave functions [Pg.182]

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