Enthalpy molar properties

In open systems consisting of several components the thermodynamic properties of each component depend on the overall composition in addition to T and p. Chemical thermodynamics in such systems relies on the partial molar properties of the components. The partial molar Gibbs energy at constantp, Tand rij (eq. 1.77) has been given a special name due to its great importance the chemical potential. The corresponding partial molar enthalpy, entropy and volume under the same conditions are defined as... 

Figure 6J Excess partial molar properties of jadeite component in NaAlSi206-KAlSi206 molten mixture. (A) Excess partial molar entropy. (B) Excess partial molar enthalpy. Reprinted from Eraser et al. (1983), Bulletin Mineralogique, 106, 111-117, with permission from Masson S.A., Paris, Erance.

Note that thermodynamic tabulations do not normally report the standard partial molar properties of solutes and G p, but rather the enthalpy of formation... 

We will follow the IUPAC recommendation that surface properties per unit surface area be represented by the lower case (g = Gibbs free energy, u = energy, h = enthalpy, etc.) with a superscript.s designating that the property is for the surface. The quantities gs,us,hs... for the surface are in many ways comparable to molar properties (or partial molar properties for mixtures) in the bulk phase. 

Figure 17.6 Excess molar properties at p = 0.1 MPa for (X111-C7H16 +X2I-C4H9CI) (a) gives the excess molar enthalpies. The solid line represents values at T= 298.15 K, while the dashed line gives values changed to T = 323.15 K, using the excess molar heat capacities at T = 298.15 K shown in (b). The excess molar volumes at T= 298.15 K are shown in (c).

Tliis equation defines the partial molar property of species i in solution, where the generic symbol Mt may standfor the partial molar internal energy t/, the partial molar enthalpy //, the partial molar entropy 5,, the partial molar Gibbs energy G,, etc. It is a response function, representing the change of total prope ity n M due to additionat constant T and f of a differential amount of species i to a finite amount of solution. 

Equation 41 shows that the chemical potential is a partial molar property. We will need other partial molar quantities (e.g., those for volume, enthalpy, and entropy) in dealing with pressure and temperature effects on energetics of reactions. 

Gibbs-Duhem Relationship The partial molar properties of a multicomponent phase cannot be varied independently (the mole fractions, jc, = ,/E of the components total unity). For example, for the chemical potentials, /i, the Gibbs-Duhem relationship is En, dni = 0 (for details, see e.g., Atkirs, 1990 Blandamer, 1992 Denbigh, 1971). Similar constraints apply to the partial molar volumes, enthalpies, entropies, and heat capacities. For pure substances, the partial molar property is equal to the molar property. For example, the chemical potential of a pure solid or liquid is its energy per mole. For gaseous, liquid, or solid solutions, X, = X,(ny), that is, the chemical potentials and partial molar volumes of the species depend on the mole fractions. 

They are used as industrial solvents for small- and large-scale separation processes, and they have unusual thermodynamic properties, which depend in a complicated manner on composition, pressure, and temperature for example, the excess molar enthalpy (fp-) of ethanol + water mixture against concentration exhibits three extrema in its dependence on composition at 333.15 K and 0.4 MPa. The thermodynamic behavior of these systems is particularly intricate in the water-rich region, as illustrated by the dependencies of the molar heat capacity and partial molar volume on composition. This sensitivity of the partial molar properties indicates that structural changes occur in the water-rich region of these mixtures. Of course, the unique structural properties of water are responsible for this behavior. ... 

Define an ideal solid solution. Discuss how you would determine the Gibbs free energy, enthalpy, molar volume, and entropy of formation of a binary ideal solid solution given the appropriate properties of its pure end members. 

Examples of partial molar properties include the partial molar enthalpy Ha, which is defined as... 

The value of any extensive property of a system is equal to the sum of the partial molar properties of each component multiplied by the amount of each component in the system. Therefore, we can divide the property of a mixture, such as the volume or enthalpy, between its individual components according to their partial molar properties. 

Figure 5.1 Variation of the molar property X (e.g., molar enthalpy or molar volume) of a binary mixture as a function of the mole fraction of component 2 (solid line). The dashed line is the tangent line to X at the composition X2 = x2-...

Equations (5.16) and (5.17) correspond to a straight line connecting the molar properties of the pure components (e g., the thick dashed line in Fig. 5.1). In this case, we see that the partial molar volumes and partial molar enthalpies of each species are equal to the respective molar quantities in the pure state. We expect ideal behavior when the fluids that are mixed consist of similar molecules. 

Most solutions do not exhibit ideal behavior, and the actual curve corresponding to the variation of the molar volume or enthalpy of the mixture deviates from a straight line (e.g., the solid line in Fig. 5.1). When the curve for the molar volume lies above the ideal mixture line, the system expands upon mixing when the curve lies below the line, the system contracts. In the case of the molar enthalpy, a curve that lies above the ideal mixture line corresponds to the system that absorbs heat (e.g., mixing lead bromide and water) a curve that lies below the line corresponds to the system releasing heat (e.g., mixing sulfuric acid and water). This non-ideal mixing in the case of the molar enthalpy is the principle used in cold packs and heat packs. We will develop mathematical models to describe non-ideal mixtures. We use partial molar properties in more detail later. 

The approach to the thermodynamics of solubilization in micellar solutions is based on the determination of a given partial molar property of the solute (volume, enthalpy, heat capacity, compressibility) as a function of the surfactant content. The simplest approach is to use the pseudophase model. The partial molar quantity, L will thus be an average value of Y in the micellar and aqueous phases, as described by... 

This simply shows that there is a physical relationship between different quantities that one can measure in a gas system, so that gas pressure can be expressed as a function of gas volume, temperature and number of moles, n. In general, some relationships come from the specific properties of a material and some follow from physical laws that are independent of the material (such as the laws of thermodynamics). There are two different kinds of thermodynamic variables intensive variables (those that do not depend on the size and amount of the system, like temperature, pressure, density, electrostatic potential, electric field, magnetic field and molar properties) and extensive variables (those that scale linearly with the size and amount of the system, like mass, volume, number of molecules, internal energy, enthalpy and entropy). Extensive variables are additive whereas intensive variables are not. 

Remembering the definition of enthalpy H = U + PV, we can introduce it as the partial molar properties of the reactive substance at the entrance and exit of the reactor. 

Lowercase roman letters usually denote molar properties of a phase. Thus, g, A. s, and v are the molar Gibbs energy, molar enthalpy, molar entropy, and molar volume. Whan it is essentia] to distinguish between a molar property of a mixture nod that of a pure component, we identify the pure-component property by a subscript. For example, ft, is the molar enthalpy of pure i. Total properties are usually designated by capilal letters, Thus H is the total enthalpy of a mixture it is related to the molar mixture enihelpy A by H nh. where n is the total number of moles in the mixture. 

Note that in this case the. pure component and partial molar enthalpies differ considerably. Con-sequendy. we say that this solution is quite nonideal, where, as we shall see in Chapter 9. an ideal solution is one in wh ch some partial molar properties (in particular the enthalpy, internal energy, and volume) are equal to the pure component values. Further, here the solution is so nonideal that at the temperature chosen the purecomponenf and partial molar. enthalpies are even of different signs for both water and sulfuric acid. For later reference we note that, at a h,so4 = 0.5, we have ... 

A partial molar property of a component in a mixture may be either greater than or less than the corresponding pure-component molar property. Furthermore, the partial molar property may vary with composidon in a complicated way. Show this to be the case by computing (a) the partial molar volumes and (b) the partial molar enthalpies of ethanol and water in an ethanol-water mixture. (The data that follow are from Volumes 3 and 5 of the International Critical Tables, McGraw-Hill, New York, 1929.)... 

In practice, pure-component molar enthalpies are employed to approximate A/7rx. This approximation is exact for ideal solutions only, when partial molar properties reduce to pure-component molar properties. In general, one accounts for more than the making and breaking of chemical bonds in (3-35). Nonidealities such as heats of solution and ionic interactions are also accounted for when partial molar enthalpies are employed. Now, the first law of thermodynamics for open systems, which contains the total differential of specific enthalpy, is written in a form that allows one to calculate temperature profiles in a tubular reactor ... 

The quantities /l, - - T8, correspond to the partial molar enthalpy of each component, since = H — T S, and differentiation with respect to the moles of species A at constant T, p and Nq produces the following relation between partial molar properties ... 

Intensive properties (for example, molar volume, molar enthalpy, etc.) are functions of intensive variables temperature, pressure, mole fractions. Two mixtures with the same mole fractions have identical intensive properties at fixed pressure and temperature. The extensive property of the mixture is obtained by multiplying the molar property by the total moles ... 

Partial molar properties are defined for any property that has an extensive form for example, volume, enthalpy, etc. They are intensive properties and as such, they are functions of pressure, temperature, and mol fractions. To see how partial molar properties can be useful, consider the following thought experiment a vessels that contains a mixture (for example, a solution of several components) is poured into another vessel B. We will calculate the enthalpy in vessel B as it builds up during this process. The differential of is given bveq. fo.Sl the process obviously takes place under constant pressure and constant temperature, therefore, dT=o and dP = o. This simplifies the differential to the form. 

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