Property partial molar

A partial molar property Mi of an arbitrary extensive thermodynamic property M=M T, p, 111, 2, i) is defined by the equation  

Partial molar properties give information about the change of the total property due to addition of an infinitely small amount of substance of species i to the mixture. From eq 2.16 it becomes apparent that, by definition, the chemical 

From eqs 2.98 and 2.99 the generalized Gibbs-Duhem equation is readily obtained  

Partial molar properties are defined by partial derivatives (equations 2.22,9.7), which does not provide a very easy route to understanding them. There is however a highly intuitive way of thinking about partial molar properties. We will use volume as an example because it is readily visualized, but all relations derived can be used equally well for any other state variable. 

In Chapter 4, we examined the properties of ideal solutions. Many properties of an ideal solution do not change on mixing. For example, the volume of a mixture is equal to the sum of the volume of the original unmixed solutions. In this situation, it is straightforward to assign how much volume is occupied by each component in the system — it is simply the volume occupied by components in their unmixed state. 

For a general system, however, the volume, as well as other properties, is not additive. That is, the volume of a mixture is not equal to the sum of the volumes of the individual pure components. In this situation, it is not clear how to assign how much volume is occupied of each species. One logical manner to do this is through the use of partial molar properties. 

In this Chapter, we define partial molar properties and describe their application. We then discuss their relationship with the change of properties of a system on mixing. Finally, we examine the graphical representation of partial molar properties for binary mixtures. 

We have defined the chemical potential of a component as the partial derivative of the Gibbs free energy of the system (or, for a homogeneous system, of the phase) with respect to the number of moles of the component at constant P and T—i.e., 

As we have already seen, chemical potential varies with phase composition. If we derive this chemical potential with respect to T and P at constant composition, we obtain 

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

The basic definition of the partial molar property (in phase 1) is 

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 ). 

In Chapters 7 and 9 we will use the partial molar derivative of the compressibility factor z, which is an intensive, dimensionless quantity. This usage seems to contradict the previous paragraph. However, if we define an extensive property Z=nz and insert its value in Eq. 4.18 we will find that Zi, is perfectly well behaved and has the right dimensions. This procedure is also sometimes used for other intensive 

Physical and Chemical Equilibrium for Chemical Engineers, Second Edition. Noel de Nevers. 2012 John Wiley Sons, Inc. Published 2012 by John Wiley Sons, Inc. 

One significant aspect of partial molar properties is that represented by equation (26.6). If Qi is the partial molar value of any property in a system containing n - moles of the constituent i, then the total value O for the system is given by the sum of all the terms. For a system consisting of a single, pure substance the partial molar property is identical with the ordinary molar value. This result has often been used in the earlier treatment. 

Another asp ect of partial molar volumes and heat contents, in particular, arises from the thermod3mamic requirement that for an ideal gas mixture or for an ideal liquid solution, as defined for example in 30a and 34a, respectively, there is no change of volume or of heat content upon mixing the components. This means that the partial molar volume and heat content of each substance in the mixture are equal to the respective molar values for the pure constituents. Any deviation of the partial molar quantity from the molar value then gives an indication of departure from ideal behavior this information is useful in connection with the study of solutions. 

Apparent Molar Properties.—Although not of direct thermo-d3mamic significance, the apparent molar property is related to the corresponding partial molar property, as will be seen below. The importance of 

If G is the value of a partieidar ))roperty for a mixture consisting of Ui moles of constituent 1 and ri2 moles of constituent 2, and Gi is the value of the property per mole of pure constituent 1, then the apparent molar value, represented by / ( , of the given property for the component 2 is given by 

In order to indicate the fact that the value of t G as given by equation (42.1) applies to the constituent 2, i.e., the solute, a subscript 2 is sometimes included. However, this is usually omitted, for in the great majority of cases it is understood that the apparent molar property refers to the solute. It i.s seen from equation (42.1) that l o is the apparent contribution of 1 mole of the component 2 to the property G of the mixture. If the particular property were strictly additive for the two components, e.g., volume and heat content for ideal gas and liquid solutions, the value of 4 q would be equal to the actual molar contribution, and hence also to the partial molar value. For nonideal systems, however, the quantities are all different. 

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 

Note that the partial molar derivatives may also be taken under conditions other than constant p and T. 

When a species becomes part of a mixture, it loses its identity yet it still contributes to the properties of the mixture, since the total solution properties of the mixture depend on the amount present of each species and its resultant interactions. We wish to develop a way to account for how much of a solution property (V, H, U, S, Q.. .) can be assigned to each species. We do this through a new formalism the partial molar property. 

The state postulate tells us that if we specify two intensive properties for any pure species, we constrain the state of a single-phase system. For extensive properties, we must additionally specify the total number of moles. In Chapter 5, we learned how to mathematically describe any intensive thermodynamic property in terms of partial derivatives of two independent, intensive properties. Since we are now concerned with thermal and mechanical equilibrium, it makes sense to choose T and P as the independent, intensive properties. We wish to extend the formulation to mixtures with changing composition. In addition to specifying two independent properties, we must also consider the number of moles of each species in the mixture. 

We now wish to specify the extensive thermodynamic property of the entire mixture, K, where we use the symbol K to represent any possible extensive thermodynamic property, that is, K = V,H, U, S, G, and so on. In essence, by using K, we avoid repetitive derivations by treating the problem in general. If we were to divide K by the total 

Mathematically, we can write the extensive total solution property K in terms of T, P, and the number of moles of m different species  

Note that Equation (6.13) is an extension and generalization of the equation on page 18 to systems of m species. We need to knowm + 2 independent quantities to completely specify K. In Equation (6.13) we specifically choose the system temperature, pressure, and number of moles of each of the m species. Once these measured properties are specified, the state of the system is constrained and all the extensive properties, K, take specific v alues. 

Consider a homogeneous mixture made up of N, N2,. .., moles of components 1, 2,. .., k, respectively. To this mixture we add a small amount of component 1, dNf, while maintaining the temperature T, pressure P, and number of moles of the other components constant. We measure the resulting change in the total property, d(NM), and define the ratio d(NM) over dN.  

Notice that the change must be determined at constant temperature, pressure, and number of moles of the other components. If, for example, the addition of the moles takes place at constant pressure - say the open atmosphere - we must make sure that any change in temperature, resulting from the mixing, is accommodated for by the addition or removal of the necessary amount of heat so that it remains also constant. 

It follows from Eq. 11.3.1 that, for a mixture of specified components, the partial molar properties are functions of its composition, temperature, and pressure  

We demonstrate the physical meaning of partial molar properties with the simple case of the volume of a binary mixture in the next Example. 

Consider a mixture containing 3 moles of methanol(m) and 7 moles of water(w) at 2S°C and 1 atm. To this mixture we add a small amount of water, say 0.1 mole (see Problem 11.2.a), while maintaining the temperature, pressure, and number of moles of methanol constant. The volume of the mixture increases by 1.78 cm.  

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A useful feature of the partial molar properties is that the property of a mixture (subscript mix) can be written as the sum of the mole-weighted contributions of the partial molar properties of the components ... 

The derivatives in the summation are partial molar properties, denoted by Tf/ thus... 

This result, known as the Gibbs-Duhem equation, imposes a constraint on how the partial molar properties of any phase may vary with temperature, pressure, and composition. In particular, at constant T and P it represents a simple relation among the Af/ to which measured values of partial properties must conform. 

Equation 163, written as = G- /-RT, clearly shows that In ( ) " is a partial molar property with respect to G /KT. MultipHcation of equation 175 by n and differentiation with respect to at constant T, P, and in accord with equation 116 yields, after reduction, equation 179 (constant T,x), where is the partial molar compressibiUty factor. This equation is the partial-property analogue of equation 178. 

Foi an ideal solution, G, = 0, and tlieiefoie 7 = 1- Compatison shows that equation 203 relates to exactiy as equation 163 relates ( ) to GG Moreover, just as ( ) is a partial property with respect to G /E.T, so In y is a partial property with respect to G /RT. Equation 116, the defining equation for a partial molar property, in this case becomes equation 204 ... 

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. 

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 ... 

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... 

Equation (4-47), which defines a partial molar property, provides a general means by which partial property values may be determined. However, for a.hinary solution an alternative method is useful. Equation (4-50) for a binaiy solution is... 

The partial molar property, other than the volume, of a constituent species in an ideal gas mixture is equal to the corresponding molar property of the species as a pure ideal gas at the mixture temperature hut at a pressure equal to its partial pressure in the mixture. 

All three quantities are for the same T, P, and physical state. Eq. (4-126) defines a partial molar property change of mixing, and Eq. (4-125) is the summability relation for these properties. Each of Eqs. (4-93) through (4-96) is an expression for an ideal solution property, and each may be combined with the defining equation for an excess property (Eq. [4-99]), yielding ... 

Hence, for a pure substance, the chemical potential is a measure of its molar Gibbs free energy. We next want to describe the chemical potential for a component in a mixture, but to do so, we first need to define and describe a quantity known as a partial molar property. 

Equation (5.16) can be integrated. We expect the partial molar properties to be functions of composition, and of temperature and pressure. For a system at constant temperature and constant pressure, the partial molar properties would be functions only of composition. We will start with an infinitesimal quantity of material, with the composition fixed by the initial amounts of each component present, and then increase the amounts of each component but always in that same fixed ratio so that the composition stays constant. When we do this. Z, stays constant, and the integration of equation... 

By either a direct integration in which Z is held constant, or by using Euler s theorem, we have accomplished the integration of equation (5.16), and are now prepared to understand the physical significance of the partial molar property. For a one-component system, Z = nZ, , where Zm is the molar property. Thus, Zm is the contribution to Z for a mole of substance, and the total Z is the molar Zm multiplied by the number of moles. For a two-component system, equation (5.17) gives... 

In summary, we have defined a partial molar property Z,- as... 

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... 

Before leaving our discussion of partial molar properties, we want to emphasize that only the partial molar Gibbs free energy is equal to n,-. The chemical potential can be written as (cM/<9 ,)rv or (dH/dnj)s p H partial molar quantities for fi, into equations such as those given above. 

Equation (5.23) is known as the Gibbs-Duhem equation. It relates the partial molar properties of the components in a mixture. Equation (5.23) can be used to calculate one partial molar property from the other. For example, solving for dZ gives... 

Obtaining partial molar properties involves the determination of the derivative... 

A method for determining partial molar properties, most often applied to electrolyte solutions, involves using the apparent molar property [Pg.222]

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... 

Chapter 4 presents the Third Law, demonstrates its usefulness in generating absolute entropies, and describes its implications and limitations in real systems. Chapter 5 develops the concept of the chemical potential and its importance as a criterion for equilibrium. Partial molar properties are defined and described, and their relationship through the Gibbs-Duhem equation is presented. 

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