Intensive variables partial molar quantities

Extensive and Intensive Variables, Partial Molar Quantities... 

Although the function / is a homogeneous function of the mole numbers of degree 1, the partial molar quantities, and are homogeneous functions of degree 0 that is, the partial molar quantities are intensive variables. This statement can be proved by the following procedure. Let us differentiate both sides of Equation (2.32) with respect to x ... 

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. 

Partial molar quantities are ratios of two extensive variables and are, therefore, intensive variables. As intensive variables they are independent of the size of the system and dependent only on other intensive variables. We can take our partial derivatives to be dependent on P, T, and c — 1 concentrations and write Eq. (7) as... 

An extensive variable may be converted into an intensive variable by expressing it per one mole of a substance, namely, by partially differentiating it with respect to the number of moles of a substance in the system. This partial differential is called in chemical thermodynamics the partial molar quantity. For instance, the volume vi for one mole of a substance i in a homogeneous mixture is given by the derivative (partial differential) of the total volume V with respect to the number of moles of substance i as shown in Eq. 1.3 ... 

Among intensive variables important in thermodynamics are partial molar quantities, defined by the equation... 

On the basis of the above analysis it has been shown the partial molar quantities are easily obtained from intensive quantities like the molar volume when this quantity is plotted as a function of an intensive composition variable like the mole fraction. The plots in fig. 1.2 show that the molar volume is almost a linear function of the mole fraction of solute. If the curves in fig. 1.2 were actually perfect straight lines, the partial molar volumes would be constant independent of solution composition. Such a situation would arise if the solution were perfectly ideal. In reality, very few solutions are ideal, as will be seen from the discussion in the following section. In order to see more clearly the departure from ideality, one defines and calculates a quantity called the excess molar volume. This quantity is equal to the actual molar volume less the molar volume for the solution if it were ideal. The latter can be considered as the volume of the solution that would be found if the molecules of the two components form a solution without expansion or contraction. Thus, the ideal molar volume can be defined as... 

These intensive variables are called the fatiial molar quantities associated with the extensive variable Y. As shown by (1.2.5), the chemical potential m is the partial molar quantity associated with the free energy G. Similarly to V corresponds the partial molar volume defined by... 

For c components in a phase, this equation has c + 2 terms, one more than given by the phase rule, because its integration requires the size of the phase as well as its intensive variables. The quantity (9X/dni)PTn is called the partial molar X with respect to i and given the symbol Xt ... 

The above equations again correlate partial molal and molar energies and enthalpies only when all intensive variables are held fixed is the partial molal and molar enthalpy the same. In most cases one may drop the term involving Vs - One may also use Eqs. (5.2.2) to access other thermodynamic functions of interest in terms of differential quantities. 

It is possible to subdivide the properties used to describe a thermodynamic system (e.g., T, P, V,U,...) into two main classes termed intensive and extensive variables. This distinction is quite important since the two classes of variables are often treated in significantly different fashion. For present purposes, extensive properties are defined as those that depend on the mass of the system considered, such as volume and total energy content, indeed all the total system properties (Z) mentioned above. On the other hand, intensive properties do not depend on the mass of the system, an obvious example being density. For example, the density of two grams of water is the same as that of one gram at the same P, T, though the volume is double. Other common intensive variables include temperature, pressure, concentration, viscosity and all molar (Z) and partial molar (Z, defined below) quantities. ... 

In this section we will consider phases with N components and we will choose the following variables the N quantities of matter n, and the p intensive variables for the other conjugate couples. We call the partial molar variable of an extensive funetion its derivative with respect to the amount of matter of a compound whose intensive variables and all the amounts of... 

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