Apparent molar mass

Special care has to be taken if the polymer is only soluble in a solvent mixture or if a certain property, e.g., a definite value of the second virial coefficient, needs to be adjusted by adding another solvent. In this case the analysis is complicated due to the different refractive indices of the solvent components [32]. In case of a binary solvent mixture we find, that formally Equation (42) is still valid. The refractive index increment needs to be replaced by an increment accounting for a complex formation of the polymer and the solvent mixture, when one of the solvents adsorbs preferentially on the polymer. Instead of measuring the true molar mass Mw the apparent molar mass Mapp is measured. How large the difference is depends on the difference between the refractive index increments ([dn/dc) — (dn/dc)A>0. (dn/dc)fl is the increment determined in the mixed solvents in osmotic equilibrium, while (dn/dc)A0 is determined for infinite dilution of the polymer in solvent A. For clarity we omitted the fixed parameters such as temperature, T, and pressure, p. 

The expression in brackets represents the interparticle interactions. It gives the contribution by which the true molar mass is modified to yield the measurable apparent molar mass M pp(c), which is a function of the concentration. A dimensionless quantity can be obtained by multiplying Eq. (78b) with which will be called the reduced osmotic modulus... 

The mole is associated with a specific chemical entity as defined by its chemical formula [15], Its structural formula, isotopic composition, isomeric form, crystal structure, or chirality may have to be given in order to completely specify the entity of interest. The achievable uncertainty of amount-of-substance measurement is limited by that of its apparent molar mass. This consideration affects not only measurements on entities with variable molar mass, but those on pure substances. It is related to the traditional and important concern about purity. 

The simultaneous negative frequency shift of the EQCM (Fig. 12.1(a)) indicates mass increase in agreement with Takahashi s results [44]. The apparent molar mass of deposited film, M was evaluated from the average initial slope of a linear plot—A/ vs. the anodic charge density, q in Fig. 12.2 and use of the Sauerbrey equation and Faraday s law of electrolysis. 

LALLS Detection. Consideration of the treatment of light scattering for heterogeneous copolymers (6) permits the dependence of the apparent molar mass M at the ith elution volume interval in a SEC-on-line LALLS experiment to be represented in terms of Pi and Qi by... 

Experimental estimates of p and // usually must rely on a model for the crosslinking chemistry, making quantitative tests of the phantom model difficult. Network defects preclude the use of Eqs (7.31) and (7.41), written in terms of the molar mass of a network strand M. Indeed, since Ms is not known for real networks and the affine and phantom models predict the same classical form of the stress-elongation curve [Eqs (7.32) and (7.33)] there is no practical means of determining which (if either) model is correct for small deformations of unentangled networks. For these reasons, we henceforth describe the modulus of all classical models as a network of strands with apparent molar mass M ... 

H NMR speetra were leeorded on a Bruker AC 400 speetrometer. Apparent molar masses were determined by size exelusion ehromatography (SEC) either in THF or H2O. For dibloek copolymers soluble in THF, the deviee was equipped with a Jaseo PU-980 pump, a PSS eolunm (8 300 mm. 

For air, which is a mixture of gases, the apparent molar mass is defined. It is the ratio of the total mass of gases contained in 1 kg and total amount of k mols in the mixture. 

Since the mass of this amount of protein is 0.225 g, the apparent molar mass is... 

Equation (6-52) generally will be used here. An apparent molar mass A/app can be defined as a molar mass calculated from experimental data at finite concentrations from an equation applicable to infinite dilution only ... 

Consequently, virial coefficients can be determined from the concentration dependence of the reciprocal apparent molar mass. But, since the various methods for measuring the molar mass yield various averages of it (see Chapters 8 and 9), the virial coefficients obtained will be average values which vary according to the method used to determine them. Virial coefficients obtained from osmotic-pressure measurements (and all other measurements based on colligative methods) will give the average... 

There is no closed expression for the concentration dependence of the apparent molar masses. Generally, the molar masses of the unimers, the equilibrium constants and the degree of association iVare obtained by iteration. 

The choice of method depends primarily on the information required, and secondarily on the field of study, the amount of substance available, the time required, and, when necessary, on the effort required to purify the samples. Determinations are generally made at various concentrations. Then the apparent molar mass is calculated with the aid of an ideal theoretical relationship—that is, a relationship that only applies strictly at infinite dilution. This apparent molar mass must then be extrapolated to zero concentration to obtain the true molar mass. Apparent and true molar masses may differ considerably. Coil-shaped macromolecules of number-average molar mass of 10 g/ mol can, for example, have an apparent number-average molar mass in good solvents of 555 000 g/mol at a concentration of 0.01 g/ ml and 110 000 g/mol only, on the other hand, when the concentration is 0.1 g/ml. 

Since ks cannot be derived theoretically, Ke is usually determined by calibration with substances of known molar mass. As with all the other molar-mass-determination methods, only apparent molar masses Afapp are obtained for finite concentrations (see Sections 6.4 and 6.5) because of the effects of the virial coefficients and/or association, A/app must therefore be extrapolated to the concentration a 0. In vapor phase osmometry, low amounts of nonvolatile impurities interfere with the result, but volatile impurities do not, since they pass into the vapor phase. 

The summation must proceed both over molecules of the same average composition and different molar masses, and as molecules of the same molar mass but different average composition. On conventional analysis of data according to Equation (9-45) an apparent molar mass (Afw)app would, on account of Equation (9-48), be obtained instead of the true molar mass < Mw >, even when c 0, i.e.. 

Equation (9-108) is, strictly speaking, only applicable at infinite dilution. At finite concentrations, it yields an apparent molar mass (Mw)app, which is extrapolated in the usual manner by plotting 1 /(M v)app = /(c) to c — 0. 

Normally, this technique cannot be applied to copolymers. In fact, the LS detector systematically underestimates the abundance of one of the repeat units. It has been shown that for copolymers t LS detector measures an apparent molar mass, Mapp, which differs from Mw by a term AM ... 

From the slope of this plot the apparent molar mass can be determined by multiplying the slope with Faraday s constant... 

Equation [4.4.53] is again valid for monodisperse polymers only. Polydisperse polymers lead to apparent molar mass averages and to averages of the virial coefficients which have to be transformed into the desired common averages by appropriate calculation meth-... 

A threshold situation is the full occupation of the possible active sites on the colunrn packing surface with adsorbed macromolecules. The resulting drop of pore volume can cause large decrease of retention volume of the nonadsorbed polymer and in effect bring about substantial increase of its apparent molar mass. 

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