Most probable molecular

The Maxwell-Boltzmann velocity distribution function resembles the Gaussian distribution function because molecular and atomic velocities are randomly distributed about their mean. For a hypothetical particle constrained to move on the A -axis, or for the A -component of velocities of a real collection of particles moving freely in 3-space, the peak in the velocity distribution is at the mean, Vj. = 0. This leads to an apparent contradiction. As we know from the kinetic theor y of gases, at T > 0 all molecules are in motion. How can all particles be moving when the most probable velocity is = 0 ... 

In addition to and r nis ai other way of characterizing coil dimensions is to consider which end-to-end distance has the greatest probability of occurring for specified n and 1 values. Derive an expression for this most probable value of r, r, from Eq. (1.44). Compare the ratio r ms/ m the ratio from the kinetic molecular theory of gases (consult, say,... 

Hence the sizes of spherical micelles are distributed around a most probable aggregation number M, which depends only on molecular details of the surfactants in this simplest approximation. Indeed, micelle size distributions at concentrations beyond the CMC have shown a marked peak at a given aggregation number in many simulations [37,111,112,117,119,138,144,154,157]. 

Analysis of data pertaining to the modulus of PEO gels obtained by the polyaddition reaction [90] shows that even in this simplified case the network structure substantially deviates from the ideal one. For all samples studied, the molecular weight between crosslinks (M p) exceeds the molecular weight of the precursor (MJ. With decreasing precursor concentration the M xp/Mn ratio increases. Thus, at Mn = 5650 a decrease in precursor concentration from 50 to 20% increases the ratio from 2.3 to 12 most probably due to intramolecular cycle formation. 

This distribution is known as the Schultz-Flory or most probable distribution.2S The moments of the molecular weight distribution are ... 

Because of the absence of chain limiter, the catalyst itself may initially act as the chain limiter (Fig. 8.22). The catalyst reacts with the olefinic regions of the polymer backbone and causes chain scission to occur, forming two new chains. The reactive carbene which is produced then moves from chain to chain, forming two new chains with each scission until the most probable molecular weight distribution is reached (Mw/Mn = 2), producing linear chains end capped with [Ru] catalyst residues. 

The molecular models upon which the theoretical intensity curves of Fig. 4 are based have. been chosen to distinguish between the three most probable configurations of fluorine nitrate. The... 

Development of several new siloxane-imide copolymers for commercial applications have also been reported by Lee 181) and Berger58). Although no information was given in terms of the chemical compositions of these materials, most of these polymers were reported to be processable by solution or melt processing techniques, most probably due to their high siloxane contents. However, due to the presence of low (—20 to —120 °C) and high (>230 °C) temperature Tg s, it was clear that multiphase copolymers have been synthesized. Molecular weights and thermal stabilities, etc, were not reported. 

From X-ray measurements in the liquid crystalline phase it is impossible to determine the conformation of the molecules in the condensed state. Computer simulations give us information about the molecules internal freedom in vacuum, but the conformations of the molecules in the condensed state can be different because of intermolecular repulsion or attraction. But it may be assumed that the molecular conformations in the solid state are among the most stable conformations of the molecules in the condensed matter and therefore also among the most probable conformations in the liquid crystalline state. Thus, as more crystallo-graphically independent molecules in the unit cell exist, the more we can learn about the internal molecular freedom of the molecules in the condensed state. 

Finally, it is worth while to consider the applicability of these relaxation methods to molecules having flexible conformations. Examples given in the previous Section demonstrated that relaxation rates are able to define either the most probable conformation of a flexible molecular segment, or to specify a range of allowed conformations from the total number of confor-... 

A pattern emerges when this molecular beam experiment is repeated for various gases at a common temperature Molecules with small masses move faster than those with large masses. Figure 5 shows this for H2, CH4, and CO2. Of these molecules, H2 has the smallest mass and CO2 the largest. The vertical line drawn for each gas shows the speed at which the distribution reaches its maximum height. More molecules have this speed than any other, so this is the most probable speed for molecules of that gas. The most probable speed for a molecule of hydrogen at 300 K is 1.57 X 10 m/s, which is 3.41 X 10 mi/hr. 

The viscosity average molecular weight depends on the nature of the intrinsic viscosity-molecular weight relationship in each particular case, as represented by the exponent a of the empirical relationship (52), or (55). However, it is not very sensitive to the value of a over the range of concern. For polymers having the most probable distribution to be discussed in the next chapter, it may be shown, for example, that... 

The other case which we consider is that of a most probable primary distribution. The molecular size distribution after random cross-linking must correspond exactly to that which would be obtained by random condensation of a mixture of bifunctional and tetrafunctional units. This follows as an extension of the correspondence between these two cases considered in the discussion of the critical condition given in the preceding section. The equations developed there are applicable to this case. 

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