Molar averages, definitions

Partial molar (atomic) volume. Another useful and general way of discussing the actual trend in an alloy system of the average molar dimensions, as represented for instance by the molar (average) volumes, is through the definition and use of the partial molar (atomic) volume. 

It should be emphasized that the flux vectors for which expressions have been given in Eqs. (28) through (36) are all defined here as fluxes with respect to the mass average velocity. Not all authors use this convention, and considerable confusion has resulted in the definition of the energy flux and the mass flux. Mass fluxes with respect to molar average velocity, stationary coordinates, and the velocity of one component (such as the solvent, for example) are all to be found in the literature on diffusional processes. Research workers in the field of diffusion should be meticulous in specifying the frame of reference for fluxes used in writing up their research work. In the next section this important matter is considered in detail for two-component systems. 

Starting from the definitions of the molar diffusion fluxes in the molar average velocity reference frame and in volume average velocity reference frame (Table 1.3), verify Eqs. 1.2.20 and 1.2.22. These equations may be written in n — 1 dimensional matrix form as... 

Other definitions of the diffusive flux such as the mass flux relative to the molar average velocity conld be employed but they have no parricolar value. Equation (2.3-4) or (2.34a) can be used in conjunction with the definitions of v and u lo show immediately that... 

The analngous rate expressions for the molar diffusive flux relative to the molar average velocity can be ased, The preferred definition relates the flux to the mole fraction gradient... 

Here, the system is isobaric and the total concentration is constant. Also, the total flux is constant in direction z, and we have dN jdz = 0. With the definition of molar average velocity... 

Table 2.4 Diffusion flux closures based on the molar average velocity definition Wilke ...

Solution Using the definition of the molar average velocity and the mole fiaction gives the following ... 

E.29 The king of Zirconia is naturally fond of the element zirconium and has established an independent definition of the mole based on zirconium. The mass of one zirconium-90 atom is 1.4929 X 10 22 g. If zirconium were the standard used for molar mass (instead of carbon-12), 1 mol would be defined as the amount of substance that contains the same number of entities as there are atoms in exactly 90 g of zirconium-90. In that case, what would be (a) the molar mass of carbon-12 (b) the (average) molar mass of gold ... 

Why do polymers not have definite molar masses How does the fact that polymers have average molar masses affect their melting points ... 

Polymers generally do not have definite molecular masses because there is no fixed point at which the chainlengthening process will cease. There is no fixed molar mass, only an average molar mass. Because there is no one unique compound, there is no one unique melting point, rather a range of melting points. 

Note 2 Any molar-mass average can be defined in terms of mass fractions or mole fractions. In this document only a few of the important molar-mass averages are given in terms of the mass fractions, Wi, of the species with molar mass M. These definitions are most closely related to the experimental determination of molar-mass averages. 

In a solution containing both unimer and micelles, Ma, which is by definition more sensitive to low molar mass particles, is always less than the weight-average molecular weight, Mv. Further details on osmometry can be found in the review by Adams (1989), whilst examples of its application to micellar block copolymer solutions are given in Chapter 3. 

For conversions lower than xgei the average molar mass of the polymer exhibits a continuous increase. The first two moments of the molar mass distribution are the number-average molar mass, Mn, and the mass-average molar mass, Mw, respectively. Mn is defined in terms of the number contribution of every species to the whole population. The weight factor used to define this average is the molar fraction. Mw is defined in terms of the mass contribution of every species to the whole mass, so that the mass fraction is the weight factor used in its definition. 

Fig. 2.3 shows a MMD curve. In this figure also the characteristic molar mass averages are indicated their definition is given in Table 2.5. 

Chen-Chen [7] Their correlation was based on diffusion measurements of 50 combinations of conditions with 3 to 4 replicates each and exhibited an average error of 6 percent. In this correlation, Vr = VB/[0.9724 (Vrf, + 0.04765)] and V B = the liquid molar volume at the melting point, as discussed previously. Their association parameter P [which is different from the definition of that symbol in Eq. (5-229)] accounts for hydrogen bonding of the solvent. Values for acetonitrile and methanol are P = 1.58 ana 2.31, respectively. 

By the general definition (99) of statistical averaging, in dealing with dense systems we have to include the perturbed potential energy for the system as a whole, i.e. to perform a summation of the energies (121) over all the molecules in the molar volume... 

The experimental dependence M = /(V), i.e. the classical SEC calibration curve usually obtained by using narrow standards, in such a case can be obtained directly without calibration from the on-line LS detector. By combining the experimental function M = /(V) and the concentration profile (from DRI), one can construct the complete MMD of the HA sample. The differential and cumulative MMD of a high molar mass HA sample (Mw = 652 kDa, D = 2.1) are shown in Fig. (10). Starting from the initial MMD, the molecular weight averages and dispersity index (Mn, Mw, Mz, and D) could be easily calculated using the appropriate definitions. 

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