Bernoulli mechanism

Atactic polymers are also regular polymers. They contain, by definition, the possible configurational monomeric units in equal proportions, but with an ideally random distribution from molecule to molecule. Such distributions are caused by symmetric Bernoulli mechanisms during polymerization (see Section 15). They are distinguished by having equal numbers of iso- and syndiotactic diads (A i = ATJ, iso-, hetero-, and syndiotactic triads (Na = A is =... 

With Bernoulli mechanisms, the ultimate unit of the growing chain has no influence on the linkage formed by a newly polymerized unit. With first-order Markov mechanisms, the ultimate unit does exert an influence, and in second-order Markov mechanisms, the penultimate, or second last, unit exerts an influence. In third-order Markov mechanisms it is the third last unit that exerts the influence on the linkage of newly joined units. Thus, Bernoulli mechanisms are a special case of Markov mechanisms, and could also be called zero-order Markov mechanisms. Second- and higher-order Markov mechanisms cannot be stated with confidence to occur in polyreactions, and, so, will not be discussed further. In addition, the discussion will be confined to binary mechanisms, that is, polyreactions where the unit possesses only two reaction possibilities. 

Similarly, for second-order Markov mechanisms, the transition probabilities Paa/a,Paa, B,Pba/a,Pba, b,Pab/a,Pab/b,Pbb/a, andpbb/b have to be considered, whereas, for Bernoulli mechanisms, only the transition probabilities ofp a and Pb need be considered. 

It follows from this that Bernoulli and Markov mechanisms differ in whether the transition probabilities of the crossover, or hetero, steps are the same as or different from the homo steps (see Table 15-6). In addition, both types of mechanism can be subclassified as to whether the transition probabilities for the homo linkages are symmetric or asymmetric. In copolymerization, a symmetric Bernoulli mechanism with constitutionally different monomers is called azeotropic copolymerization with configurationally different monomers, it is called random flight polymerization and in stereocontrolled polymerization with nonchiral monomers, it is also called ideal atactic polymerization. ... 

The asymmetric Bernoulli mechanism is another special case (see Table 15-6). From the definitions, it follows that... 

Thus, the asymmetric Bernoulli mechanism can be described by a single parameter, e.g., jca (see also Figure 15-2). 

The A and B units are related by equal probabilities in the symmetric Bernoulli mechanism, that is,... 

A symmetric Bernoulli mechanism leads to an ideal atactic polymer with ideal statistical (i.e., totally random) distribution of units, the term atactic is, however, not always used in this strict sense. 

The adding on of a new monomer is independent of the preceding unit in an (asymmetric) Bernoulli mechanism, that is, Pd/l = Pl/l. Inserting this condition in Equations (15-45)-( 15-48) gives, together with Equations (15-27) and (15-28) ... 

The relationships given in Equations (15-46)-( 15-48) reduce to those already given in Equations (15-39)-( 15-41) when the Markov and Bernoulli mechanisms are symmetric ... 

The asymmetric Bernoulli mechanism described above is characteristic for the polymerization of configurationally different monomers with enantio-morphous catalysts, but is not limited to these cases. With enantiomorphous catalysts, one kind of active position preferentially polymerizes D-monomers, ana the other kind of active position should preferentially polymerize l monomers. The probability of adding on a d monomer to a d unit at a d... 

Thus, according to mechanism, the ratios of mole fractions of different kinds of diads, triads, etc., lead to very different rate constant combinations (Table 15-7). For example, the mole fraction ratio of iso- and syndiotactic diads gives the ratio of rate constants for iso- and syndiotactic linking in the case of Bernoulli mechanisms, but gives the ratios of the rate constants for the cross-steps and not a mean of the rate constants in the case of first-order Markov mechanisms. 

Bernoulli and Euler dominated the mechanics of flexible and elastic bodies for many years. They also investigated the flow of fluids. In particular, they wanted to know about the relationship between the speed at which blood flows and its pressure. Bernoulli experimented by puncturing the wall of a pipe with a small, open-ended straw, and noted that as the fluid passed through the tube the height to which the fluid rose up the straw was related to fluid s pressure. Soon physicians all over Europe were measuring patients blood pressure by sticking pointed-ended glass tubes directly into their arteries. (It was not until 1896 that an Italian doctor discovered a less painful method that is still in widespread... 

Despite bearing no direct relation to any physical dynamical system, the onedimensional discrete-time piecewise linear Bernoulli Shift map nonetheless displays many of the key mechanisms leading to deterministic chaos. The map is defined by (see figure 4.2) ... 

Droplets are dispersed from nebulizer nozzles by one of five basic mechanisms based on Bernoulli (Venturi) effect. 

More complex schemes have been proposed, such as second-order Markov chains with four independent parameters (corresponding to a copolymerization with penultimate effect, that is, an effect of the penultimate member of the growing chain), the nonsymmetric Bernoulli or Markov chains, or even non-Maikov models a few of these will be examined in a later section. Verification of these models calls for the knowledge of the distribution of sequences that become longer, the more complex the proposed mechanism. Considering only Bernoulli and Markov processes it may be said that at the dyad level all models fit the experimental data and hence none can be verified at the triad level the Bernoulli process can be verified or rejected, while all Markov processes fit at the tetrad level the validity of a first-order Markov chain can be confirmed, at the pentad level that of a second-order Maikov chain, and so on (10). 

The other method is the velocity head method. The term V2/2g has dimensions of length and is commonly called a velocity head. Application of the Bernoulli equation to the problem of frictionless discharge at velocity V through a nozzle at the bottom of a column of liquid of height H shows that H = V2/2g. Thus II is the liquid head corresponding to the velocity V. Use of the velocity head to scale pressure drops has wide application in fluid mechanics. Examination of the Navier-Stokes equations suggests that when the inertial terms dominate the viscous terms, pressure gradients are expected to be proportional to pV2 where V is a characteristic velocity of the flow. 

Integration of Eq. 2.9-11 leads to the macroscopic mechanical energy balance equation, the steady-state version of which is the famous Bernoulli equation. Next we subtract Eq. 2.9-11 from Eq. 2.9-10 to obtain the differential thermal energy-balance... 

Thermodynamics deals with relations among bulk (macroscopic) properties of matter. Bulk matter, however, is comprised of atoms and molecules and, therefore, its properties must result from the nature and behavior of these microscopic particles. An explanation of a bulk property based on molecular behavior is a theory for the behavior. Today, we know that the behavior of atoms and molecules is described by quantum mechanics. However, theories for gas properties predate the development of quantum mechanics. An early model of gases found to be very successftd in explaining their equation of state at low pressures was the kinetic model of noninteracting particles, attributed to Bernoulli. In this model, the pressure exerted by n moles of gas confined to a container of volume V at temperature T is explained as due to the incessant collisions of the gas molecules with the walls of the container. Only the translational motion of gas particles contributes to the pressure, and for translational motion Newtonian mechanics is an excellent approximation to quantum mechanics. We will see that ideal gas behavior results when interactions between gas molecules are completely neglected. 

As we have seen, the above are variations of the mechanical energy equation. They are variously called the Bernoulli equation, the extended Bernoulli equation, or the engineering Bernoulli equation by writers of elementary fluid mechanics textbooks. Regardless of one s taste in nomenclature, Eq. (62) lies at the heart of nearly all practical engineering design problems. 

A very useful equation to deal with phenomena associated with the flow of fluids is the Bernoulli equation. It can be used to analyse fluid flow along a streamline from a point 1 to a point 2 assuming that the flow is steady, the process is adiabatic and that frictional forces between the fluid and the tube are negligible. Various forms of the equation appear in textbooks on fluid mechanics and physics. A statement in differential form can be obtained ... 

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