Molar function

As with other extensive variables, we will usually work with the molar function Um defined as... 

Figure 7.4 Excess molar functions for. YjfGO + V2C2H5OH. The solid lines represent results at T — 303.15 K and the dashed lines are for results at T = 363.15 K.

A survey of the Additive Molar Functions (AMFs), which will be discussed in this book, is given in Scheme 3.3. There the names, symbols and definitions are given of the 21 AMFs from which the majority of the physical and physicochemical properties of polymers can be calculated or at least estimated. Scheme 3.3 is at the same time a condensed list of the Nomenclature used. 

Seven Classes of Additive Molar Functions can be distinguished, each containing three AMFs ... 

SCHEME 3.3 Additive molar functions (per structural unit) ... 

We may conclude that Cp, and Cpare additive molar functions their group contributions, also valid for polymers, are given in Table 5.1. 

Table 7.11 gives a survey of the system of equations to be used. It contains four additive molar functions, a number of auxiliary equations and the final expressions for <5t(otai) and for the components of <5. 

The mechanical properties of polymers are controlled by the elastic parameters the three moduli and the Poisson ratio these four parameters are theoretically interrelated. If two of them are known, the other two can be calculated. The moduli are also related to the different sound velocities. Since the latter are again correlated with additive molar functions (the molar elastic wave velocity functions, to be treated in Chap. 14), the elastic part of the mechanical properties can be estimated or predicted by means of the additive group contribution technique. 

As will be demonstrated in Chap. 14, the different sound speeds are related with additive molar functions of the form ... 

If the additive group contributions of these molar functions are known, the moduli can be calculated ... 

Estimation and prediction of the bulk modulus from additive molar functions... 

We now summarise the relationships between K and some additive molar functions of a very different nature ... 

The speeds of longitudinal and transverse (shear) sonic waves can be estimated, c.q. predicted via two additive molar functions. From these sound velocities the four most important elastic parameters (the three elastic moduli and the Poisson ratio) can be estimated. 

In Chap. 13 we have already discussed the use of sound speed measurements for the derivation of elastic parameters. We shall come back on that, more elaborately, in this chapter. We have also seen that sound speeds can be expressed in terms of additive molar functions these are of course basic for estimations, as well for mechanical properties as for thermal conductivity (Chap. 17). 

This expression makes it possible to calculate the compression (bulk) modulus from the additive molar functions U and V. 

Hartmann (1984) found that the analogous additive property for the shear modulus is V(G/p)1/6 it is also temperature independent and additive as long as the material is in the solid state well below Tg. We shall coin this molar function as UH the Hartmann function ... 

Table 14.5 shows the comparison of the values of the Ur and UH functions derived from experimental sound speed data with the values derived from the group contributions. It also shows a comparison between experimental sound speed values with those calculated purely from additive molar functions. The agreement is on the whole very satisfactory. 

Comparison between experiment and calculation of sound speeds and molar functions... 

SCHEME 14.1 Calculation of the elastic parameters EP and the additive molar functions U from the sound speed measurement u and vice versa. Valid only for elastic isotropic materials. 

Finally the method of calculation, from sound speeds to molar functions (via elastic parameters) and vice versa, is illustrated in Scheme 14.1 (see also Examples 14.1 and 14.2). 

Our conclusion is that by means of four additive molar functions (M, V, UR and Uh) all modes of dynamic sound velocities and the four dynamic elastic parameters (K, G, E and) can be estimated, c.q. predicted from the chemical structure of the polymer, including cross-linked polymers. 

The Permachor, an additive molar function for the estimation of the permeability... 

An additive molar function for the char-forming tendency Cft... 

Molar elastic wave functions, 383,391 Molar free enthalpy of formation, 792 Molar functions, classification, 62 Molar heat... 

It is sometimes convenient to reformulate the generalized diffusional driving forces dg either in terms of mass or molar functions, using the partial mass Gibbs free energy definition, Gg = hg —Tsg, and the chain rule of partial differentiation assuming that the chemical potential (i.e., /Xg = Gg) is a function of temperature, pressure and concentration (Slattery [89], sect. 8.4). dg can then be expressed in several useful forms as listed below. Expressing the thermodynamic functions on a mass basis we may write ... 

It may be noted that all known thermodynamic relations are also valid for the partial molar functions. Thus, the relation G = H — TS in terms of partial molar functions is... 

In the alloy Ag Au, silver is the electronegative component (the less noble one) and gold is the electropositive component (the more noble one). The electrolyte in the given example was a special glass with Ag ion conductivity. The glass was melted on the silver electrode. The electrolyte fihn had a thickness of 0.1 mm and the resistance was of the order of 2000 Q. The cell reaction was described in the previous chapter. The electrode process (Eq. (3.31)) consisted of the transfer of metal atoms from the pure Ag into the alloy environment Agjj AUj, keeping the alloy composition constant. Eqs. (3.33)-(3.37) could then be applied to calculate the partial molar functions of the Ag Au, system. 

Corresponding partial molar Gibbs energies for 500 °C were calculated using Eq. (3.33). From the temperature dependence of the potentials partial molar entropies were calculated using Eq. (3.36). Finally, partial molar enthalpies were obtained using Eq. (3.37). Values of the partial molar functions of Ag as a function of composition are summarized in Table 3.5. 

The partial molar functions of the component B can be calculated using the Gibbs-Duhem equation. The Gibbs-Duhem equation for any partial molar function Z (Z equal to AG, AS, AH, etc.) has the general form... 

The integration of these equations gives the partial molar functions of component B. For the integration the integration constants must be known. Because the potential difference was measured between the alloy and the pure metal, the partial molar functions are relative values referring to the pure metal as a reference state. Therefore, integration is carried out between Xg=1, x = 0 and Xg, and x = 1 Xg. 

From partial molar functions to integral functions... 

With the partial molar functions AZ and AZg one can calculate the integral functions. One obtains for an alloy of composition and Xg the thermodynamic functions of formation... 

Another way to obtain integral thermodynamic functions is provided by the Duhem-Margules equation. The partial molar function AZ is plotted versus y (the composition of the alloy is represented by the formula A B) and then is integrated from... 

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