Molar mass number average, defined

The methods by which polymers are prepared result in a mixture of molecular sizes whose properties depend on the average size of the molecules present. In principle there are a number of ways in which such an average can be calculated. The most straightforward is the simple arithmetic mean, usually called the number average molar mass, M. This is defined by the expression... 

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. 

The number-average molar mass of polymer chains is defined as Mn = (mass of reacted monomers)/[(l/2) of chain ends] (3.161)... 

The number-average molar mass is defined as the sum of the products of the molar mass of each fraction multiplied by its mole fraction... 

The final product of polymerisation, be it of an addition or condensation type, will contain chains covering a range of lengths, and it is only possible to specify an average molar mass.1 The number-average molar mass (Mn)av is defined by... 

In order to characterize the molar mass distribution, several average molar masses are defined that emphasize different parts of this distribution. The number-average molar mass is defined as the ratio of the first (A = 1) to zeroth k = 0) moments of the number fraction distribution ... 

The polydispersity index is defined as the ratio of the weight-average and number-average molar masses Monodisperse samples with... 

In order to quantify the chain length, the molar mass (or molecular weight) must be defined statistically. The number average molar mass, M, is given by ... 

A colUgative method, such as osmotic pressure, effectively counts the number of molecules present and provides a number-average molar mass defined by... 

As DGEBA and EDA are used in stoichiometric proportions, a pseudocomponent with a number average molar mass equal to (Meda + 2 Mdgeba)/3 and a molar volume Vp = 226.7 cm mol may be defined. From Eq. (29), the critical composition of the initial system before reaction is interaction parameter depending on composition or three interaction parameters, e.g. one for each couple of components. 

Isomerizations are defined as reactions that occur with change in the chemical structure but without change in the number-average molar mass or composition. Exchange equilibria and constitutional and configurational transformations belong in this group. 

Using the distribution function (i.e., Eq. (10.3)), several average values can be defined. The number-average segment number, which is proportional to the number-average molar mass, which can be measured using the vapor pressure of solutions or osmometry, is given by the ratio of the 0 and -1 moment of the distribution function ... 

The first quantity is a colHgative property called the number-average relative molar mass or simply the number-average molar mass denoted M and defined as the weighed average of the number fraction of each macromolecule by the following equation. 

Fractionation procedures must be amenable to evaluation and criteria for width and location of a molar mass distribution must be defined. Such criteria exist, e.g., in the ratio of weight- and number-average molar masses and the ratio of centrifuge- and... 

From w M) any molar mass average can be evaluated. For example, the number-average molar mass can be defined in integral form (cf. Equation (1.6)) as... 

The knowledge of the distribution of molar mass of a polydisperse polymer is of extreme importance. Many of the physical properties of polymers depend upon this and a correct distribution is essential in any polymer which is used in a practical situation. One of the most important points in the distribution is the number average molar mass M , and there are several techniques which can be used to measure it. All of these methods involve determining the number of molecules in a given mass of polymer. The polymer is usually dissolved in a solution of known concentration and hence the mass of polymer per unit volume is defined. The most important methods involve the use of the colligative properties of dilute solutions although for a limited number of polymers it is possible to measure M by end-group analysis. 

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 ... 

Note 1 An infinite number of molar-mass averages can in principle be defined, but only a few types of averages are directly accessible experimentally. The most important averages are defined by simple moments of the distribution functions and are obtained by methods applied to systems in thermodynamic equilibrium, such as osmometry, light scattering and sedimentation equilibrium. Hydrodynamic methods, as a rule, yield more complex molar-mass averages. 

The z-average molar mass M. is defined as the ratio of the third to the second moments of the number fraction distribution ... 

From light-scattering measurements, a method depending on the size rather than the number of molecules, a weight-average molar mass M is obtained. This is defined as... 

Assuming that a polymer has a molar-mass distribution defined by (m) = ae ", where n m)Am represents the fraction of molecules with molar masses between m and m + Am, show that a = b and calculate (i) the number- and weight-average molar masses, and hence the polydispersity index and (ii) the molar mass at the mode (maximum) of the distribution function. 

This is the simplest average it is the mass of the specimen divided by the total number of moles present. Properties that depend on the total number of molecules in the specimen, independent of their size, correlate with The weight average molar mass M is defined as... 

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