Number average measurement

Since viscous oils are commonly broad-boiling mixtures, measurements of molecular weight commonly provide mass-average or number-average measurements. A variety of methods are available. 

Mean The mean, X, is the numerical average obtained by dividing the sum of the individual measurements by the number of measurements... 

Median The median, Xmed) is the middle value when data are ordered from the smallest to the largest value. When the data include an odd number of measurements, the median is the middle value. For an even number of measurements, the median is the average of the n/2 and the nil) + 1 measurements, where n is the number of measurements. 

The solute molecular weight enters the van t Hoff equation as the factor of proportionality between the number of solute particles that the osmotic pressure counts and the mass of solute which is known from the preparation of the solution. The molecular weight that is obtained from measurements on poly disperse systems is a number average quantity. 

Table 9.3 lists the intrinsic viscosity for a number of poly(caprolactam) samples of different molecular weight. The M values listed are number average figures based on both end group analysis and osmotic pressure experiments. Tlie values of [r ] were measured in w-cresol at 25°C. In the following example we consider the evaluation of the Mark-Houwink coefficients from these data. 

The width of molecular weight distribution (MWD) is usually represented by the ratio of the weight—average and the number—average molecular weights, MJM. In iadustry, MWD is often represented by the value of the melt flow ratio (MER), which is calculated as a ratio of two melt indexes measured at two melt pressures that differ by a factor of 10. Most commodity-grade LLDPE resias have a narrow MWD, with the MJM ratios of 2.5—4.5 and MER values in the 20—35 range. However, LLDPE resias produced with chromium oxide-based catalysts have a broad MWD, with M.Jof 10—35 and MER of 80-200. 

Ferefs diameter (Fig. 20-5) is the perpendicular projection, in a fixed direction, of the tangents to the extremities of the particle profile. Martin s diameter is a hne, parallel to a fixed direction, which divides the particle profile into two equal areas. Since the magnitude of these statistical diameters varies with particle orientation, these diameters have meaning only when a sufficient number of measurements are averaged. 

The mean of several readings (x) will make a more reliable estimate of the true mean (yu) than is given by one observation. The greater the number of measurements (n), the closer will the sample average approach the true mean. The standard error of the mean sx is given by ... 

There are various methods for the determination of the surface area of solids based on the adsorption of a mono-, or polymolecular layer on the surface of the solid. These methods do not measure the particle diameter or projected area as such, but measure the available surface per gram or milliliter of powder. The surface measured is usually greater than that determined by permeability methods as the latter are effectively concerned with the fluid taking the path of least resistance thru the bed, whereas the adsorbate will penetrate thru the whole of the bed as well as pores in the powder particles. These methods appear to be more accurate than surface areas calculated from weight averages or number averages of particle size because cracks, pores, and capillaries of the particles are included and are independent of particle shape and size... 

The vast majority of the kinetic detail is presented in tabular form. Amassing of data in this way has revealed a number of errors, to which attention is drawn, and also demonstrated the need for the expression of the rate data in common units. Accordingly, all units of rate coefficients in this section have been converted to mole.l-1.sec-1 for zeroth-order coefficients (k0), sec-1 for first-order coefficients (kt), l.mole-1.sec-1 for second-order coefficients (k2), l2.mole-2.sec-1 for third-order coefficients (fc3), etc., and consequently no further reference to units is made. Likewise, energies and enthalpies of activation are all in kcal. mole-1, and entropies of activation are in cal.deg-1mole-1. Where these latter parameters have been obtained over a temperature range which precludes the accuracy favoured by the authors, attention has been drawn to this and also to a few papers, mainly early ones, in which the units of the rate coefficients (and even the reaction orders) cannot be ascertained. In cases where a number of measurements have been made under the same conditions by the same workers, the average values of the observed rate coefficients are quoted. In many reactions much of the kinetic data has been obtained under competitive conditions such that rate coefficients are not available in these cases the relative reactivities (usually relative to benzene) are quoted. 

Molecular weights of polysaccharides in solution can also be measured by osmotic pressure and light scattering. Osmotic pressure yields the number average molecular weight, which can be usefully used with Mw from sedimentation equilibrium as a measure of polydispersity Preston and Wik [28] have done this for example with hyaluronic acid. The ratio Mw/Mn the polydispersity index is often given as a measure of polydispersity, and can be related to the width of a molecular weight distribution via the well-known Herdan [96] relation ... 

If it is possible to analyse end groups of a particular specimen of polymer, it may be possible to use the data to determine number average relative molar mass. If the molecules are branched the degree of branching can be measured from a combination of end group analysis and relative molar mass determination (determined by an alternative method). 

Number of measured points (N) = 95 ASD Average standard deviation of regression... 

---