Partial mass properties

Consider a physical property (such as the total Gibbs free energy G) of a continuous mixture, the value of which depends on the composition of the mixture. Because the latter is a function of, say, the mole distribution n(x), one has a mapping from a function to (in this case) a scalar quantity G, which is expressed by saying that G is given by afunctional of n(x). [One could equally well consider the mass distribution function m(x), and consequently one would have partial mass properties rather than partial molar ones.] We use z for the label x when in-... 

The important partial mass properties in (2.365) are easily transformed into molar basis as the partial mass and partial molar variables of species s are related by the molecular weight. The activity for species s is redefined by ... 

For equilibrium calculations we almost always use the partial molar Gibbs energy or chemical potential. But the partial volumes and enthalpies appear most often in the form of partial mass properties, and are used for calculations other than equilibrium. Wherever a dnt appears in this chapter (or elsewhere in this book) it could be replaced by... 

If we do that (Problem 6.9) we find analogs to most of the equations and procedures shown in this chapter, with partial molar properties replaced by those same properties divided by the molecular weight. Such properties have no common name perhaps they are best called partial mass properties. The most commonly seen application is with plots of enthalpy per unit mass vs. mass fraction, as shown for water and sulfuric acid in Figure 6.8. As shown in Problem 6.9, if we apply the method of tangent intercepts to this figure, the intercept values are the partial mass enthalpies. 

Partial Molar Properties Consider a homogeneous fluid solution comprised of any number of chemical species. For such a PVT system let the symbol M represent the molar (or unit-mass) value of any extensive thermodynamic property of the solution, where M may stand in turn for U, H, S, and so on. A total-system property is then nM, where n = Xi/i, and i is the index identifying chemical species. One might expect the solution propei fy M to be related solely to the properties M, of the pure chemical species which comprise the solution. However, no such generally vahd relation is known, and the connection must be establi ed experimentally for eveiy specific system. 

Equation (4-49) is merely a special case of Eq. (4-48) however, Eq. (4-50) is a vital new relation. Known as the summahility equation, it provides for the calculation of solution properties from partial properties. Thus, a solution property apportioned according to the recipe of Eq. (4-47) may be recovered simply by adding the properties attributed to the individual species, each weighted oy its mole fraction in solution. The equations for partial molar properties are also valid for partial specific properties, in which case m replaces n and the x, are mass fractions. Equation (4-47) applied to the definitions of Eqs. (4-11) through (4-13) yields the partial-property relations ... 

This type of defect equilibrium treatment has been used extensively to model the defect chemistry and non-stoichiometry of inorganic substances and has the great advantage that it easily takes several simultaneous defect equilibria into account [22], On the other hand, the way the mass action laws are normally used they are focused on partial thermodynamic properties and not on the integral Gibbs energy. The latter is often preferred in other types of thermodynamic analyses. In such cases the following solid solution approach is an alternative. 

Tamoxifen, an antiestrogen with partial agonistic properties, has been shown to maintain bone mass and lower serum cholesterol levels in postmenopausal women (Love et al., 1991b, 1992). As indicated above, the uterotrophic activity of tamoxifen, however, limits its acceptability for the prevention and treatment of osteoporosis. In a previous... 

The properties of solutions as represented by the symbol M may be on unit-mass basis as well as on a mole basis. The equations relating solutk properties are unchanged in form one merely replaces the various n s, represent ing moles, by m s, representing mass, and speaks of partial specific properti rather than of partial molar properties. In order to accommodate either, generally speak simply of partial properties. 

Since we are concerned here primarily with the properties of solutions, represent molar (or unit-mass) properties of the solution by the plain symbol Partial properties are denoted by an overbar, and a subscript identifies the specie giving the symbol Mf. In addition, we need a symbol for the properties of th individual species as they exist in the pure state at the T and P of the solution. These molar (or unit-mass) properties are identified by only a subscript, and the symbol is Mj. In summary, three kinds of properties used in solution thermody namics are distinguished by the following symbolism ... 

Molar or unit-mass value of any extensive property of pure species i Partial molar property of species i in solution Property change of mixing Standard property change of reaction j Mass... 

Frequently the integral form of the conservation la v of the property is particularized as total and partial mass balance and also as energy or thermal balance [3.7]. For each particularization, a control volume must be selected in order to have a form capable of permitting the computation of each integral from the relation (3.5). As an initial condition, we have to declare the property, the transport vector and the property generation rate. Figure 3.2 presents the way to obtain the equations of the differential balance of total mass, mass species and energy (heat). The... 

In 2.4 we presented differential forms of the thermodynamic stuff equations for overall mass, energy, and entropy flows through open systems. Usually, such systems, together with their inlet and outlet streams, will be mixtures of any number of components. Individual components can contribute in different ways to mass, energy, and entropy flows, so here we generalize the stuff equations to show explicitly the contributions from individual components these generalized forms contain partial molar properties introduced in 3.4. 

Such salt effects are of practical importance in stagemise separation processes and in pollution abatement. Certain salts increase the solubility by more than an order of. magnitude (salting-in), and also change the solvent selectivity for various solutes others decrease the solubility (salting-out) (4, 6, 7, 10) Partial molal properties of the dissolved gas are also profoundly affected by the addition of salt. Thermodynamic properties of gas-electrolyte solution are also an important consideration in the design and operation of fuel cells, where mass transfer of... 

An expanded flow bin should be designed if partial mass flow is acceptable in terms of the bulk solid s properties and, as a general mle of thumb, the cylinder diameter is greater than about 6 m. 

Here we show PROPERTY in capitals and its partial molar derivative, properin lowercase letters to emphasize that the derivative is normally taken of an extensive property, such as the enthalpy of a system, but the resulting (properis intensive, for example, enthalpy per mol Because a partial molar property is the derivative of an extensive property with respect to number of mols it is an intensive property itself. Partial molar values normally exist only for extensive properties (V, U, H, S, A, G). They do not exist for intensive properties (T, P, viscosity, density, refractive index, all specific or per unit mass properties). There is no meaning to the terms partial molar temperature (degrees per mol at constant T ) or partial molar specific volume (cubic feet per mol per mol ). 

The spectroscopic properties of the /V-nitrosamines, especially the nmr and mass spectra, vary widely depending on the substituents on the amine nitrogen (44—47). The nmr spectra are affected by the E—Z isomerism around the N—N partial double bond and by the axial—equatorial geometry resulting from conformational isomerism in the heterocycles (44,45). Some general spectral characteristics for typical dialkylnitrosamines and simple heterocycHc nitrosamines are given in Table 1. 

At room temperature phenol is a white, crystalline mass. Phenol gradually turns pink if it contains impurities or is exposed to heat or light. It has a distinctive sweet, tarry odor, and burning taste. Phenol has limited solubiUty in water between 0 and 65°C. Above 65.3°C phenol and water are miscible in all proportions. It is very soluble in alcohol, ben2ene, chloroform, ether, and partially disassociated organics in general. It is less soluble in paraffinic hydrocarbons. The important physical properties of phenol are Hsted in Table 1. 

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