Distance from equilibrium fractional

The relaxation time is thus the time at which the distance from equilibrium has been reduced to the fraction 1/e of its initial value. From equation 5.1.50, it is evident that... 

An approximate relationship between the degree of undersaturation of seawater with respect to calcite and the extent of dissolution can be established by comparing the saturation state at the various sediment marker levels with estimates of the amount of dissolution required to produce these levels. In Figure 9 the "distance from equilibrium (1 - 2) has been plotted against the estimated percent dissolution of the calcitic sediment fraction. Within the large uncertainties that exist in the amount of dissolution required to produce the FL and Rq levels, a linear relation between the degree of undersaturation and extent of dissolution can be established. The intercept of the linear plot with the FL and Rq levels indicates that approximately 15 percent more material has been lost than Berger s (12) minimum loss estimate of 50% and 10%, respectively. 

This is the fractional distance from equilibrium, varying from x— 1 at start to x=0 at equilibrium and having the same value for A and P. In terms of x, the rate is... 

Equation 5.5 and the first-order plot also remain valid regardless of volume variations if the fractional distance from equilibrium is defined in terms of amounts N-t instead of concentrations Cj ... 

The relationship between the characteristic rate coefficient k and the time tx required to reduce the distance from equilibrium to the fraction x is analogous to that for irreversible first-order reactions (see eqn 3.23) ... 

The product concentration remained constant within experimental error from 47 to 62 hours. The value at 47 hours, 0.1328 M, can therefore be taken as the equilibrium concentration of lactone, CP . The corresponding values of x (fractional distance from equilibrium) are listed in the third column. A plot of Injt versus t is shown in Figure 5.4 (left diagram) and is seen to be nicely linear, indicating that the reaction is first order-first order (the reverse step actually is pseudo-first order as it involves H20 in large excess as a... 

In first order-first order reversible reactions, the rate of approach to equilibrium is proportional to the fractional distance from equilibrium, measured in terms of any quantity that is a linear function of the concentrations. The same rule holds true for any participant in reactions with first-order parallel steps. 

The rates of product formation in parallel first-order steps are proportional to the fractional distance from equilibrium, which is the same for all participants. In reactions with parallel steps of different reaction orders, the selectivity to the product formed by the parallel step of higher order is higher in batch or plug-flow than in continuous stirred-tank reactors, and decreases with progressing conversion in any type of reactor. 

Typical examples of coupled parallel steps are isomerizations in concert with conversion of the isomers to different products. If isomerization is fast compared with conversion, the isomers are at quasi-equilibrium, producing a kinetic behavior like that of a single reactant. If isomerization is slow compared with conversion, the reactions of the isomers are essentially uncoupled the apparent reaction order of decay of total reactant (sum of the isomers) is higher than that of the individual isomers. If the rates of isomerization and conversion are comparable, the isomer distribution approaches a steady state that is not necessarily close to isomerization equilibrium, and from then on the rates become proportional to the fractional distance from equilibrium. 

Design of Packed Tower. Ideally, the interaction between vapor and liquid in packed towers is true countercurrent rather than the stepwise-countercurrent propess of a bubble-plate tower with theoretical plates. Instead of finite steps, the true countercurrent action should be treated differentially. Consider the schematic drawing of the packed tower in Fig. 7-29. Let 0 be the mols of overflow, V the mols of vapor, x and y the average mol fraction in the liquid and vapor, respectively, n distance above and m distance below the feed. Focusing on the differential section dn, V(dy/dn) must equal 0 dx/dn) for each component, and this transfer must be due to an exchange of components back and forth between the liquid and vapor. This transfer is due to the fact that the vapor and liquid at a given cross section are not in equilibrium with each other, and the rate of transfer will be a function of the distance from equilibrium thus,... 

The pairs will all have the same new internal coordinates in the deformed state. Suppose that the fractions of configurations for free strands which have the specified end-to-end distances of the deformed state are gx and g2, and that the fractions in the various classes for pairs formed from equilibrium with these coordinates are fh The number of configurations available to a pair in class i in the deformed state is therefore ft>i >2 0i02/i- However, the number of strands in each class is always (/j)o B, so the total number of configurations in the deformed state is ... 

Constant-Stress Layer in Flowing Fluids. In the boundary layer of a fluid flowing over a solid wall. Ihe shear stress varies with distance from Ihe wall bul ii may be considered nearly constant within a small fraction of the layer thickness. The concept is of particular importance in turbulent flow where it leads lo a theoretical derivation of the law of ihe wall," the logarithmic distribution of mean velocity. The constant stress layer is ihe best-known example of the equilibrium flow s near a wall. 

When alkylating isobutane, chain tennination forms primarily, but not entirely, 2,2,4-trimethylpentane the alkylate from chain termination very closely resembles isobutene alkylate. The similarity of alkylate compositions, particularly their C0 fractions, originating from various olefins and the distance from thermodynamic equilibrium composition indicates that alkylate molecules, once formed, are relatively stable under alkylation conditions and undergo little isomerization. Undesirable side products, e.g., dimethylhexanes and residue, are probably formed by buter e isomerization and polymerization (rather than by isomerization of alkylate or by isomerization of the C3 carbonium Ion which subsequently becomes alkylate). 

Figure 5.13 Predicted phase diagrams for physical gels made from low-molecular-weight molecules with junctions of unrestricted functionality 4> is the total volume fraction of polymer, and Tr is here the reduced distance from the theta temperature, Tr = — Q/T. The parameter Aq controls the equilibrium constant among aggregates of various sizes. The outer solid lines are binodals, the inner solid lines are spinodals, and the dashed lines are gelation transitions. CP is a critical solution point, CEP is a critical end point, and TCP is a tricriti-cal point. (Reprinted with permission from Tanaka and Stockmayer, Macromolecules 27 3943. Copyright 1994 American Chemical Society.)...

Unless the contrary is explicitly stated, the following discussion of experimental and theoretical results is restricted to single, rigid, spherical particles freely suspended in a Poiseuille flow within a circular tube of effectively infinite length. Notation is as follows a = sphere radius 7 = tube radius (/ was used previously for this quantity) b = radial distance from tube axis to sphere center p = b/R = fractional distance from axis b = stable equilibrium distance of sphere from tube axis = b jR p = fluid density Pp = particle density p = viscosity v = pfp = kinematic viscosity. All velocities defined below are measured relative to the fixed cylinder walls V = mean velocity of flow vector (equal in magnitude to the superficial velocity and pointing parallel to tube axis in the direction of net flow) U = particle velocity vector—that is, the velocity of the sphere center (o = angular velocity of the sphere. The local velocity in the unperturbed Poiseuille flow is... 

---