Thermodynamics partial molar property

THERMODYNAMIC FUNCTIONS OF HYDRATION, AH, AND THERMODYNAMIC PARTIAL MOLAR PROPERTIES,, OF NON-CHARGED ORGANIC MOLECULES... 

P rtl IMol r Properties. The properties of individual components in a mixture or solution play an important role in solution thermodynamics. These properties, which represent molar derivatives of such extensive quantities as Gibbs free energy and entropy, are called partial molar properties. For example, in a Hquid mixture of ethanol and water, the partial molar volume of ethanol and the partial molar volume of water have values that are, in general, quite different from the volumes of pure ethanol and pure water at the same temperature and pressure (21). If the mixture is an ideal solution, the partial molar volume of a component in solution is the same as the molar volume of the pure material at the same temperature and pressure. 

Perhaps the most significant of the partial molar properties, because of its appHcation to equiHbrium thermodynamics, is the chemical potential, ]1. This fundamental property, and related properties such as fugacity and activity, are essential to mathematical solutions of phase equihbrium problems. The natural logarithm of the Hquid-phase activity coefficient, Iny, is also defined as a partial molar quantity. For Hquid mixtures, the activity coefficient, y, describes nonideal Hquid-phase behavior. 

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. 

Pertinent examples on partial molar properties are presented in Smith, Van Ness, and Abbott (Introduction to Chemical Engineering Thermodynamics, 5th ed.. Sec. 10.3, McGraw-Hill, NewYonc, 1996). Gibbs/Duhem Equation Differentiation of Eq. (4-50) yields... 

First, we note that all of the thermodynamic equations that we have derived for the total extensive variables apply to the partial molar properties. Thus, if... 

A similar proof can be used for applying any of our thermodynamic equations to partial molar properties. For example, if... 

In Chapter 5, we defined the partial molar property Z, and described how it could be used to determine the total thermodynamic property through the equation... 

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... 

Because most chemical, biological, and geological processes occur at constant temperature and pressure, it is convenient to provide a special name for the partial derivatives of all thermodynamic properties with respect to mole number at constant pressure and temperature. They are called partial molar properties, and they are defined by the relationship... 

Generally, the thermodynamic relationships that we have developed for extensive thermodynamic properties also apply to partial molar properties. Thus, as [Equation (7.17)]... 

As we will see, partial molar properties are of general application in the thermodynamics of mixtures and solutions. 

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

Equations 116,117,121,122, and 124 are the general property relations between partial molar properties and solution properties. The symbol M may represent the molar value of any extensive thermodynamic property, for example, V, U, H, S, jA, or G. When M = if, the derivatives (dH/dT)p and (dH/dP)Tx are given by equations 75 and 79. Equations 121,122, and 124 then become the following ... 

It has been said chemists have solutions 3 Solutions are involved in so many chemical processes1 that we must have the mathematical tools to comfortably work with them, and thermodynamics provides many of these tools. Thermodynamic properties such as the chemical potential, partial molar properties, fugacities, and activities, provide the keys to unlock the description of mixtures. 

Partial molar properties take a special place in the thermodynamics of mixtures and phase equilibria. They are defined as... 

X, is the molar thermodynamic property of a pure component (adsorbate or adsorbent) and X, is the partial molar property of the component, defined as... 

The definition of a partial molar property, Eq. (11.2), provides the me-for calculation of partial properties from solution-property data. Implicit in definition is a second, equally important, equation that allows the calculation solution properties from knowledge of the partial properties. The derivation this second equation starts with the observation that the thermodynamic propertl of a homogeneous phase are functions of temperature, pressure, and the numb of moles of the individual species which comprise the phase. For thermodyna property M we may therefore write... 

This equation defines the partial molar volume of species i in solution. It is simply the volumetric response of the system to the addition at constant T and P of a differential amount of species i A partial molar property may be defined in like fashion for each extensive thermodynamic property. Letting M represent the molar value of such a property, we write the general defining equation for a partial molar property as... 

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. ... 

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... 

The chemical potentials are the key partial molar quantities. The pi s determine reaction and phase equilibrium. Moreover, all other partial molar properties and all thermodynamic properties of the solution can be found from the pi s if we know the chemical potentials as functions of T, P, and composition. For example, the partial derivatives of p with respect to T... 

Partial molar properties play a central role in phase-equilibrium thermodynamics, and ii is convenient to broaden their definition to include partial molar residua) junctions and partial molar excess functions. Hence, we define, analogous to Eq- (1-2-5),... 

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