Stationary-state condition

Polymer propagation steps do not change the total radical concentration, so we recognize that the two opposing processes, initiation and termination, will eventually reach a point of balance. This condition is called the stationary state and is characterized by a constant concentration of free radicals. Under stationary-state conditions (subscript s) the rate of initiation equals the rate of termination. Using Eq. (6.2) for the rate of initiation (that is, two radicals produced per initiator molecule) and Eq. (6.14) for termination, we write... 

The propagation of polymer chains is easy to consider under stationary-state conditions. As the preceding example illustrates, the stationary state is reached very rapidly, so we lose only a brief period at the start of the reaction by restricting ourselves to the stationary state. Of course, the stationary-state approximation breaks down at the end of the reaction also, when the radical concentration drops toward zero. We shall restrict our attention to relatively low conversion to polymer, however, to avoid the complications of the Tromms-dorff effect. Therefore deviations from the stationary state at long times need not concern us. 

In the preceding section we observed that both the rate of polymerization and the degree of polymerization under stationary-state conditions can be interpreted to yield some cluster of the constants kp, kj, and k j. The situation is summarized diagramatically in Fig. 6.4. The circles at the two bottom corners... 

When results are compared for polymerization experiments carried out at different frequencies of blinking, it is found that the rate depends on that frequency. To see how this comes about, we must examine the variation of radical concentration under non-stationary-state conditions. This consideration dictates the choice of photoinitiated polymerization, since in the latter it is almost possible to turn on or off—with the blink of a light—the source of free radicals. The qualifying almost in the previous sentence is actually the focus of our attention, since a short but finite amount of time is required for the radical concentration to reach [M-] and a short but finite amount of time is required for it to drop back to zero after the light goes out. 

The change in [Mn ] under stationary-state conditions equals zero for all values of n hence we write... 

The total radical concentration under stationary-state conditions can be similarly obtained ... 

Under stationary-state conditions of flow, that is, when no further acceleration occurs, this force is balanced by gravitational and pressure forces. For... 

Equation (9.28) describes the velocity with which a cylindrical shell of liquid moves through a capillary under stationary-state conditions. This velocity times the cross-sectional area of the shell gives the incremental volume of liquid dV which is delivered from the capillary in an interval of time At. The total volume delivered in this interval AV is obtained by integrating this product over all values of r ... 

The ultracentrifuge has been used extensively, especially for the study of biopolymers, and can be used in several different experimental modes to yield information about polymeric solutes. Of the possible procedures, we shall consider only sedimentation velocity and sedimentation equilibrium. We shall discuss these in turn, beginning with an examination of the forces which operate on a particle setting under stationary-state conditions. 

The general theoretical treatment of ion-selective membranes assumes a homogeneous membrane phase and thermodynamic equilibrium at the phase boundaries. Obvious deviations from a Nemstian behavior are explained by an additional diffusion potential inside the membrane. However, allowing stationary state conditions in which the thermodynamic equilibrium is not established some hitherto difficult to explain facts (e.g., super-Nemstian slope, dependence of the selectivity of ion-transport upon the availability of co-ions, etc.) can be understood more easily. 

The circles in Fig. 4 represent stationary state conditions (mixed potentials). The reaction 1 - I,4, is viewed here as an anodic current flowing across the phase... 

As has been shown 82 85 88), the behavior of amalgam electrodes under conditions of cementation is very similar to that of liquid and glass membrane electrodes under stationary state conditions. Here, Eq. (2) should be written in the following way ... 

Assuming stationary-state conditions for the intermediate, in which L —L is acting as a unidentate ligand, we find... 

As the net velocity of the particle is increased, the viscous force Fv opposing its motion also increases. Soon this force, shown in Figure 2.2b, equals the net driving force responsible for the motion. Once the forces acting on the particle balance, the particle experiences no further acceleration and a stationary state velocity is reached. It may be shown that, under stationary state conditions and for small velocities, the force of resistance is proportional to the stationary state velocity v ... 

Since the net force of gravity and the viscous force are equal under stationary state conditions, Equations (1) and (2) may be equated to give... 

The sedimentation coefficient depends on concentration consequently, it is usually measured at several different concentrations, and the results are extrapolated to zero concentration. It is customary to designate this limiting value by a superscript zero. Experimental values are also generally labeled with respect to temperature, so a value listed as 520 corresponds to a sedimentation coefficient measured at (or corrected to) 20°C and extrapolated to zero concentration. Under stationary state conditions, the force due to the centrifugal field and the viscous force of resistance will be equal. Therefore, co2r replaces g in Equation (4) to give... 

When the apparatus begins to rotate, the fluid experiences an initial acceleration, but a stationary state is rapidly attained in which forces balance and acceleration is zero. Equation (5) gives a generalized expression for torque under stationary-state conditions it must be independent of r. If this were not the case, forces would be different in different parts of the fluid, and acceleration would occur. Accordingly, we set the torque on this volume element equal to a constant ... 

Under stationary-state conditions, the velocity is independent of time, and Equation (28a) becomes... 

As the system is subjected to ongoing, low-level mechanical agitation, the network structure is rearranged to a dispersion of more compact floes that display both a lower yield value and a lower apparent viscosity than the initial dispersion (curve 2). A certain amount of time is required for the dispersed units to acquire a size and structure compatible with the prevailing low level of agitation. This is why intermediate cases (not shown in Fig. 4.14a) are observed before the actual stationary-state condition is obtained. 

Viscous and surface pressure forces balance under stationary-state conditions along the edge of the film a distance r from the center line. In terms of Figure 7.11b, this force balance is given by... 

Under stationary-state conditions an equal and opposite force is exerted on the volume element by the electric field acting on the ions contained in the volume element. The force on the ions is given by the product of the field strength times the total charge. The latter equals the charge density p times the volume of the element therefore... 

We will regard a and 6 as functions of the reactant concentration as expressed in fi and assume that they change on a fast timescale compared with reactant consumption (small e), i.e. we apply the pseudo-stationary-state hypothesis. The pseudo-stationary-state condition da/dt = dO/dz = 0 yields the following simultaneous equations ... 

Diagrams such as Fig. 6.3, which show the dependence of the stationary-state composition on a particular parameter, are known as bifurcation diagrams. It is customary, when trying to judge the efficiency of various processes for instance, to discuss the extent of reaction rather than the concentration of the reactant. The former is the fractional conversion of A into products and is given by the difference between the inflow concentration of A and its stationary-state value, i.e. how much A has reacted, divided by the original (inflow) concentration. For the extent of reaction we use the symbol y, and under stationary-state conditions... 

If the reactor is fed by an inflow of pure A, so fi0 = 0, matters are especially simple. The stationary-state condition is given by... 

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