Intensive property , partial molar

When the value of an intensive property / can be expressed as an algebraic function of the composition, the partial molar quantities can be determined analytically. 

Partial molar properties, like molar properties, are intensive magnitudes (i.e., they do not depend on the extent of the system), and the same relationships that are valid for molar properties hold for them as well. For instance. 

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 intensive properties of a system include temperature, pressure and the chemical potentials (or partial molar free energies) of the various species present. 

We must also specify carefully the nature of the new phase. Either the intensive properties of this new phase (partial molar volume, composition etc.), differ but infinitesimally from those of the original phase, or they differ from them by a finite, non-zero, amount. 

The method of derivation for H is generally useful for extensive properties to relate the effect of intensive properties on partial molar properties. 

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. ... 

A single homogeneous phase such as an aqueous salt (say NaCl) solution has a large number of properties, such as temperature, density, NaCl molality, refractive index, heat capacity, absorption spectra, vapor pressure, conductivity, partial molar entropy of water, partial molar enthalpy of NaCl, ionization constant, osmotic coefficient, ionic strength, and so on. We know however that these properties are not all independent of one another. Most chemists know instinctively that a solution of NaCl in water will have all its properties fixed if temperature, pressure, and salt concentration are fixed. In other words, there are apparently three independent variables for this two-component system, or three variables which must be fixed before all variables are fixed. Furthermore, there seems to be no fundamental reason for singling out temperature, pressure, and salt concentration from the dozens of properties available, it s just more convenient any three would do. In saying this we have made the usual assumption that properties means intensive variables, or that the size of the system is irrelevant. If extensive variables are included, one extra variable is needed to fix all variables. This could be the system volume, or any other extensive parameter. 

For characterization of the interrelation between the composition and extensive properties of the solution the outstanding American physico-chemist Gilbert Newton Lewis (1875-1946) introduced additional intensive parameters under the common mme partial molar quantity. Among them are partial molar volume, partial molar heat capacity, chemical potential, etc. 

The partial molar values per se are intensive properties as they do not depend on the total amount of solution and may be both positive and negative. If the solution pressure and temperature do not change, any of its extensive property is a function only of its composition ... 

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. 

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. 

In addition to total and molar properties, we have partial molar properties, which are a little trickier to understand. It s relatively easy to see that the volume (extensive variable) of a system depends on how much stuff you have in the system, but that its temperature or density (intensive variables) do not. This is true no matter how many different phases there are in the system, as long as you are considering the whole system, not just parts of it. 

Partial molar quantities are intensive properties of the solution since they depend only on the composition of the solution, not upon the total amount d each component. If we add the several components simultaneously, keeping their ratios constant, the partial molal quantities remain the same. We can thus integrate the above expression keeping nj, n2,. .. in constant proportions and find, while holding temperature and pressure constant, that... 

Since it is the free energy of a solution which is a property of Interest, we must know the contribution of each component of a system to the total free energy. The partial molar energy is the amount of free energy contributed by each component of the system. It is an intensive variable and is often called the chemical potential, sjmibolized by p . 

The absolute values of the various extensive properties (except V) are, of course, never known, and the same applies to the partial molar quantities Ei. Therefore the latter must all be calculated with respect to the same reference state as in the case of E itself. Moreover, the although they are intensive and therefore independent of the size of the system, are still dependent on the relative proportions of the various components (e.g. on the mole fractions), and also on the temperature and pressure. 

The last partial derivative on the right side of Eq. 5.5.5, (dG/dn)T,p, is especially interesting because it is the rate at which the Gibbs energy increases with the amount of substance added to a system whose intensive properties remain constant. Thus, pi is revealed to be equal to Gm, the molar Gibbs energy of the substance. 

This is the rate at which property X changes with the amount of species i added to the mixture as the temperature, the pressure, and the amounts of all other species are kept constant. A partial molar quantity is an intensive state function. Its value depends on the temperature, pressure, and composition of the mixture. 

We obtain an important relation between the mixture volume and the partial molar volumes by imagining the following process. Suppose we continuously pour pure water and pure methanol at constant but not necessarily equal volume rates into a stirred, thermostat-ted container to form a mixture of increasing volume and constant composition, as shown schematically in Fig. 9.2. If this mixture remains at constant T and p as it is formed, none of its intensive properties change during the process, and the partial molar volumes Fa and Fb remain constant. Under these conditions, we can integrate Eq. 9.2.8 to obtain the additivity rule for volume ... 

The osmotic pressure 77 is an intensive property of a solution and was defined in Sec. 12.2.2. In a dilute solution of low 77, the approximation used to derive Eq. 12.2.11 (that the partial molar volume Fa of the solvent is constant in the pressure range from p to p+ 11) becomes valid, and we can write... 

Partial molar properties are intensive properties. For a system at constant temperature and pressure, the total value of extensive property z is given by... 

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... 

This is a rare example of a partial molar derivative of an intensive property z,. However, we see that it is the logical result of representing an extensive property V by the number of mols and a set of intensive properties. This causes no difficulties, and the resulting equation is widely used. 

This expression represents the Gibbs-Duhem equation and indicates that the intensive properties of the mixture temperature, pressure and partial molar properties, cannot vary independently. Restricted to constant T and P, Eq. 11.6.3 becomes ... 

Eq. 12.7.1 represents the Phase Rule and, as its development indicates, it deals with the intensive properties of the system (T, P, composition, and partial molar properties). 

Equation (6.17) indicates that the extensive total solution property K is equal to the sum of the partial molar properties of its constituent species, each adjusted in proportion to the quantity of that species present. Similarly, Equation (6.18) shows that the intensive solution property k is simply the weighted average of the partial molar properties of each of the species present. The partial molar property, Kj, can then be thought of as species is contribution to the total solution property, K. 

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... 

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