Molecular weight arithmetic mean

This is certainly true of the simple properties, such as the molecular weight of a mixture. For the van der Waals equation of state, the parameter b stands for the excluded volume due to the molecule, which suggests that the linear additive relation, or the arithmetic mean, may be appropriate for an ideal mixture ... 

The velocity of the atoms can be set as the arithmetic mean velocity v from the Maxwell-Boltzmann distribution. It only dependents on the temperature and their molecular weight ... 

Various molecular weight averages are current in polymer science. We show here that these are simply arithmetic means of molecular weight distributions. It Tiiay be mentioned in passing that the concepts of small particle statistics that are discussed here apply also to other systems, such as soils, emulsions, and carbon black, in which any sample contains a distribution of elements with different sizes. 

To define any arithmetic mean A, let us assume unit volume of a sample of N polymer molecules comprising molecules with molecular weight Mj, rii molecules with molecular weight M2, molecules with molecular weight... 

The arithmetic mean molecular weight A is given as usual by the total measured quantity (Af) divided by the total number of elements. That is, the ratio n N is the proportion of the sample with molecular weight M,. If we call this proportion fh the arithmetic mean molecular weight is given by... 

Equation (2-4) defines the arithmetic mean of the distribution of molecular weights. Almost all molecular weight averages can be defined from this equation. 

To compile the numberdistribution we have expressed the proportion of species with molecular weight A/, as the corresponding mole fraction a,. Substitution of A, for fi in Eq. (2-4) shows that the arithmetic mean of the number distribution is... 

We have seen that average molecular weights arc arithmetic means of distributions of molecular weights. An alternative and generally more useful definition is in terms of moments of the distribution. This facilitates generalizations beyond the two averages we have considered to this point and clarilies the estimation of parameters related to the breadth and symmetry of the distribution. 

As a general case the ratio of the first moment to the zeroth moment of any distribution defines the arithmetic mean. For an unnormalized number distribution, , is the number of moles per unit volume with molecular weight M, and the zeroth and first moments of the distribution about zero are given respectively by... 

We have seen that M , the arithmetic mean of the number distribution, is equal to the ratio of the first to the zeroth moment of this distribution (Eq. 2-20). If we take ratios of successively higher moments of the number distribution, other average molecular weights are described ... 

Hence by extrapolating the line in fig. 20.3 back to its intercept with the tt/c axis we can evaluate the molecular weight of the polymer. This calculation is easily extended to the case of a mixture of polymer molecules of different weights we see that the molecular weight calculated on the basis of (20.97) is the arithmetic mean molecular weight of the polymer. 

The distribution of molecular weights in a polymer sample is commonly expressed as the proportions of the sample with particular molecular weights. The mass of data contained in the distribution can be analyzed more easily by condensing the information into parameters that present a concise picture of the distribution and describe its various aspects. One such summarizing parameter is the arithmetic mean that is often used with synthetic polymers. The various molecular weight averages used for polymers can be shown to be simply arithmetic means of molecular weight distributions. 

The ratio ai/A is the proportion of molecules with molecular weight If we call this proportion fi, the arithmetic mean molecular weight is given by ... 

The distribution of molecular weights in a polymer sample is commonly expressed as the proportions of the sample with particular molecular weights. The various molecular weight averages used for polymers can be shown to be simply arithmetic means of molecular weight distributions. 

In the previous section we have seen that average molecular weights are arithmetic means of distributions of molecular weights. An alternative and generally more... 

Number average molecular weight is defined as the arithmetic average (mean) of the molecular weight of all molecules in a polymer. 40 C.F.R. 723.250(b). 

According to equation (2-20), the arithmetic mean does not result from the individually calculated equivalent molecular weights. Expressions... 

Arithmetic mean (arithmetic average, mean x) n. (1) In statistics, the average of a set of measurements found by summing the measurements and dividing the sum by the number of measurements (number-average molecular weight is an arithmetic average). (2) The conceptual mean, fi, of the population from which a set of measurements was drawn, rarely known exactly. The sample mean, x, is the most efficient estimator of the population mean, fi. 

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