Molar mass average calculations

Table 4 Molar mass averages (in kg/mol) calculated from the molar mass distributions obtained by SEC, short and long exposure TDFRS, and DLS...

The number-average molar mass is calculated by the following generalization of Eq. (3.79) ... 

The calculation of copolymer molar mass averages and so on, and copolymer polydispersity D is done as in conventional GPC calculations using the copolymer molar mass calculated from Eq. (3). 

The number- and the weight-average molar masses are calculated from the moments of the molar mass distribution. The first moment of the number fraction distribution is the number-average molar mass Mn. ... 

Another problem with GPC of condensation resins is the calibration of the columns. Because in the ohgomeric and polymeric regions of the resins no compounds with a special and singular molar mass and a clear molecular structure are available, similar or chemically related substances have to be used as calibration standards. However, differences in the hydrodynamic volumes even at the same molar mass cannot be excluded totally. This uncertain cahbration of the columns also induces a great uncertainty in the calculation of molar mass averages on the basis of the chromatograms obtained. 

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. 

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-... 

FIGURE 14 Schematic representation of processing of a SEC chromatogram, the calculation of molar mass averages. For detailed explanation, see the text. 

Molar mass averages and polydispersity can be calculated theoretically for various polymerization reactions. In the case of polycondensation of monomer type a-R-b (e.g., hydroxy acid) or the equimolar mixture of monomers a-Ri-a and b-Ri-b the average polymerization degrees are as follow ... 

This number appears to be constant for flexible polymers, with the average value = 20.6( 8%). Table 25.1 shows data for polyolefin melts listing density p, plateau modulus Ge, melt chain dimensions from SANS o/M, entanglement molar mass Me calculated from Eq. (25.6), Kuhn length b, packing length p, tube diameter a, and the overlap parameter for entanglement P, all at temperature T. 

Polymers with a peroxide bond are interesting since they are very reactive. They are used as fuels and initiators. Kishore and coworkers [61] analyzed PSOX, a poly (alpha-methylstyrene peroxide), by SEC and found Mr, = 2.5 kDa and M = 4.0 kDa. The MALDI spectrum of PSOX displayed an acceptable spectral quality, with more than 50 peaks, due to poly(alpha-methylstyrene peroxide) chains. However, the upper limit of the mass range is very low, about 3.0 kDa, and the strongest peaks fall at 1.5 kDa. It was quite apparent that MALDI strongly underestimates the molar mass averages. In fact, the calculation (Equations 45.1 and 45.2) yielded... 

In many cases distribution functions are determined experimentally the characterization of petroleum fractions by true-boiling-point distillation or gas-chromatographically simulated distillation, and the characterization of polymers by gel-permeation chromatography. In principle, the integrals of continuous thermodynamics may be directly solved based on these experimentally determined distribution functions. However, this approach delicate numerical analyses and the assumption the complete distribution function has been obtained by experiment clearly this is no the case, for example, for some polymers only molar-mass averages are determined. Thus, there are numerous cases where smoothed or analytical distribution function provides more reliable phase equilibrium calculation than those obtained by use of the experimentally determined distribution function. When the integrals of continuous thermodynamics possess analytical solutions considerably numerical simplification is afforded and this is one motive for the desire to have analytical expressions for the distribution function. 

By calibrating the column or column set with a set of standards, a calibration curve of retention versus the logarithm of molar mass is constructed so that molar mass averages may be calculated for unknown polymers. Figure 6.10 is an example chromatogram of a separation of a polyDADMAC coagulant polyelectrolyte. The molar mass distribution, see Figure 6.11, is relative to the calibration standards used, polyethylene oxide in this example. 

However, molar mass averages represent reduced information only and do not describe a polydisperse sample comprehensively. The macroscopic properties of macromolecules can better be derived from their MMD, w(A/). Two samples can have the same molar mass averages but very different MMDs and therefore macroscopic properties. It is possible to derive the molar mass averages from the MMD but not vice versa. The MMD can be calculated from the signal heights, h(y). 

The calculation of copolymer molar mass averages and copolymer polydispersity is made as in conventional SEC... 

The greater the mass of an individual atom, the greater the molar mass of the substance. However, most elements exist in nature as a mixture of isotopes. We saw in Section B, for instance, that neon exists as three isotopes, each with a different mass. In chemistry, we almost always deal with natural samples of elements, which have the natural abundance of isotopes. So, we need the average molar mass, the molar mass calculated by taking into account the masses of the isotopes and their relative abundances in typical samples ... 

STRATEGY First calculate the average atomic mass of the isotopes by adding together the individual masses, each multiplied by the fraction that represents its abundance. Then obtain the molar mass, the mass per mole of atoms, by multiplying the average atomic mass by Avogadro s constant. 

Sf lf-Test 8.15A The osmotic pressure of 3.0 g of polystyrene dissolved in enough benzene to produce 150. mL of solution was 1.21 kPa at 25°C. Calculate the average molar mass of the sample of polystyrene. 

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... 

In all calculations the molar masses given in the top of Table I were used. First of all, the effects of variations in the concentration of trifunctional monomers were determined, as exemplified by the nine formulations of Table I and the resulting prepolymer characteristics after full conversion given in Table II. Formulations FIO to F40 result in branched prepolymers, which are cured in the third stage by difunctional monomers. On the other hand, formulations FOO to F04 result in the same linear prepolymer, which is subsequently cured with various mixtures of di- and trifunctional monomers. The number average functionalities of PI (and P2) and of the mixtures of E and F monomers are varied systematically between 2.0 and 2.4. Therefore, the only difference between formulations FjO and FOj is the stage in which the branching units are added. 

The molar mass of a naturally occurring mixture of isotopes is the weighted average of the isotopic molar masses. Each isotope contributes to the total in proportion to its percentage (fractional) abundance, and the average is calculated using Equation. ... 

Does it bother you to find that neither the chemical formula nor the molar mass is needed for these calculations Remember that not all data are necessarily required for any particular calculation. Because average kinetic energy depends on temperature but not on molar mass, we do not need mass information to do this problem. 

The problem has two parts. First we must calculate the density of atmospheric air. To do this, we need to determine the molar mass of diy air, which is the weighted average of the molar masses of its... 

In this section, we review the properties of a series of PNIPAM-b-PEO copolymers with PEO blocks of varying length, with respect to the PNIPAM block. Key features of their solutions will be compared with those of PNIPAM-g-PEO solutions. PNIPAM-b-PEO copolymers were prepared by free-radical polymerisation of NIPAM initiated by macroazoinitiators having PEO chains linked symmetrically at each end of a 2,2/-azobis(isobutyronitrile) derivative [169,170]. The polydispersities of PEOs were low, enabling calculations of the number-average molar mass for each PNIPAM block from analysis of their H-NMR spectra (Table 2). 

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