Number partial molar properties

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. 

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... 

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... 

The partial molar properties are not measured directly per se, but are readily derivable from experimental measurements. For example, the volumes or heat capacities of definite quantities of solution of known composition are measured. These data are then expressed in terms of an intensive quantity—such as the specific volume or heat capacity, or the molar volume or heat capacity—as a function of some composition variable. The problem then arises of determining the partial molar quantity from these functions. The intensive quantity must first be converted to an extensive quantity, then the differentiation must be performed. Two general methods are possible (1) the composition variables may be expressed in terms of the mole numbers before the differentiation and reintroduced after the differentiation or (2) expressions for the partial molar quantities may be obtained in terms of the derivatives of the intensive quantity with respect to the composition variables. In the remainder of this section several examples are given with emphasis on the second method. Multicomponent systems are used throughout the section in order to obtain general relations. 

The definition of a partial molar property, Eq. (11.7), provides the means for calculation of partial properties from solution-property data. Implicit in this definition is another, equally important, equation that allows the reverse, i.e., calculation of solution properties from knowledge of the partial properties. The derivation of this equation starts with the observation tliat the themiodynamie properties of a homogeneous phase are functions of temperature, pressure, and the numbers of moles of the individual species which comprise the phase. Thus for thermodynamic property M ... 

In view of the difficulty of determining the exact slope of the curve at all points, it is preferable to use an analytical procedure instead of the graphical one just described. The property G is then expressed as a function of the number of moles of one component, c.g., the molality, associated with a constant amount of the other component. Upon differentiation with respect to n, i.e., the molality, an expression for the partial molar property is obtained. 

It is frequently more convenient to determine the partial specific properties, defined in terms of grams instead of moles, of the constituents of a solution, and then to multiply the results by the respective molecular weights to yield the partial molar properties. Any of the methods described above may be adapted for this purpose. The value of the property G or of AG per mole is replaced by the value per gram, and n or n, the mole fraction or number of moles, is replaced by the corresponding gram fraction or number of grams, respectively. 

By dividing both sides of Eq. (5.6) by N, the total number of moles in the system, we find that molar properties are similarly related to partial molar properties... 

The constancy subscript Uj i indicates that the number of moles of all components, except component i, are kept constant when evaluating the partial derivative.) The derivative dY/dni)T,P,rij i is called the partial molar property for the constituent i, and it is represented by writing a bar over the symbol for the particular property, so that... 

Since it is unlikely that enough information will be available for any mixing property to obtain A6>mix as an explicit function of mole fractions or species mole numbers for ternary, quaternary, etc. mixtures, it is not surprising that there is little information on partial molar properties in such systems. 

Equations like Eqs. 8.1-16aand b are also satisfied by the partial molar properties. To see this, we multiply Eq. 8.1-16a by the total number of moles N and take the derivative with respect to N at constant 7, P, and /Vj, to obtain... 

The chemical potential /z, is a partial molar property of the Gibbs free energy because, as illustrated by the mole number coefficient of G in equation (29-24r/), temperature, pressure, and all other mole numbers are held constant during differentiation with respect to Ni. 

The derivative operator appearing in (3.4.5) is called the partial molar derivative, and the quantity F,- defined by (3.4.5) is called the partial molar F for component i. It is the partial molar property that can always be mole-fraction averaged to obtain the mixture property F. Note, however, that F is itself a property of the mixture, not a property of pure i partial molar properties depend on temperature, pressure, and composition. We emphasize that the definition (3.4.5) demands that F be extensive and that the properties held fixed can only be temperature, pressure, and all other mole numbers except N,. Partial molar properties are intensive state functions they may be either measurable or conceptual depending on the identity of F. 

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. 

A note on partial molar properties In case you are beginning to wonder why there are so many questions and problems about concentrations I will answer by telling you that you need concentrations in about four out of every five problems in physical chemistry. The matter of fact is that a lot of chemistry and all of biochemistry takes place in solutions. Then there are problems inherent to solutions. Solutions are considered simple physical mixtures of two or more different kinds of molecules, with no chemical bonds made or broken. For a really well-behaved solution physical chemists have a name, by analogy with the gas laws an ideal solution. Yet solutions are actually complicated systems whose molecular nature we are only now beginning to understand [1, 2, 3, 4]. Two solvents, when mixed, often release heat (or absorb heat) and undergo change in volume. Think of a water sulfuric acid (caution]) mixture or a water DMSO (dimethyl sulfoxide) mixture. After the solvent mixture equilibrates you will find that its volume is not equal to the sum of the volumes of the pure solvents (it is usually smaller). In physical chemistry we treat these problems by using the concept of molar volume, V. Molar volumes are empirical numbers - they are determined by experimental measurements for different solvent compositions. Read the next problem. 

The thermodynamic properties of binary or multi-component systems are better described in terms of the partial molar quantities of their components (Pitzer, 1995). The partial molar property of a given component of the mixture physically represent the change in the property by addition to a large multi-component system of a small amount of one of the components at constant p, T and mole number of the other components. 

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 ). 

For a pure substance, any partial molar property is the same as the molar property for example, the pure species partial molar volume is the same as the pure species molar volume. If we add one mol of pure water to a large number mols of pure water, the volume of the original sample of water will increase by the exact amount of the volume of the water added, because there is no volume change on mixing pure anything with itself. However, in mixtures of more than one species, the two are generally not the same, as shown by Examples 6.1, 6.2, and 6.3. 

It is shown in Chapter 6 that for any partial molar property (for example, q where Q is any extensive property) of any mixture with any number of species in one phase the partial molar equation requires that... 

Apply thermodynamics to mixtures. Write the differential for any extensive property, dK, in terms of m + 2 independent variables, where m is the number of species in the mixture. Define and find values for pure species properties, total solution properties, partial molar properties, and property changes of mixing. 

A partial molar property is always defined at constant temperature and pressure, two of the criteria for phase equilibrium. Partial molar properties are also defined with respect to number of moles. The number of moles of all otherj species in the mixture are held constant it is only the number of moles of species i that is changed. A common mistake in working with partial molar properties is to erroneously replace number of moles with mole fractions. We must realize, however, that Equation (6.15) does not simply convert to mole fraction, that is,... 

We can logically extend this thought to interpret a partial molar property as though it represents the intensive property value of an individual species as it exists in solution. In contrast, the pure species property, ku indicates how an individual species acts when it is by itself The difference Ki — ki compares how the species behaves in the mixture to how it behaves by itself. If this number is zero, the species behaves identically in the mixture to how it behaves as a pure species. In contrast, if this number is large, the species interactions in the mixture are quite different from when it is by itself... 

Often an analytical expression for the total solution property, k, is known as a function of composition. In that case, the partial molar property, Kj, can be found by differentiation of the extensive expression for K with respect to Uj, holding T, P, and the number of moles of the otherj species constant, as prescribed by Equation (6.15). 

A variety of procedures can be used to determine Z, as a function of composition.2 Care must be taken if reliable values are to be obtained, since the determination of a derivative or a slope is often difficult to do with high accuracy. A number of different techniques are employed, depending upon the accuracy of the data that is used to calculate Z, and the nature of the system. We will now consider several examples involving the determination of V,- and Cpj, since these are the properties for which absolute values for the partial molar quantity can be obtained. Only relative values of //, and can be obtained, since absolute values of H and G are not available. For H, and we determine H, — H° or — n°, where H° and are values for H, and in a reference or standard state. We will delay a discussion of these quantities until we have described standard states. 

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