Extensive properties, definition

Volume is an extensive property. Usually, we will be working with Vm, the molar volume. In solution, we will work with the partial molar volume V, which is the contribution per mole of component i in the mixture to the total volume. We will give the mathematical definition of partial molar quantities later when we describe how to measure them and use them. Volume is a property of the state of the system, and hence is a state function.1 That is... 

This relationship is expressed in extensive properties that depend on the extent of the system, as opposed to intensive properties that describe conditions at a point in the system. For example, extensive properties are made intensive by expressing them on a per unit mass basis, e.g. s = S/m density, p 1 /v, v V/m. For a pure system (one species), Equation (1.2) in intensive form allows a definition of thermodynamic temperature and pressure in terms of the intensive properties as... 

Comparison of the combined first/second law (4.28) with (4.30) leads to the more general and rigorous thermodynamic definitions for the intensive properties T, —P respectively conjugate to the extensive properties S, V ... 

We have previously emphasized (Section 2.10) the importance of considering only intensive properties Rt (rather than size-dependent extensive properties Xt) as the proper state descriptors of a thermodynamic system. In the present discussion of heterogeneous systems, this issue reappears in terms of the size dependence (if any) of individual phases on the overall state description. As stated in the caveat regarding the definition (7.7c), the formal thermodynamic state of the heterogeneous system is wholly / dependent of the quantity or size of each phase (so long as at least some nonvanishing quantity of each phase is present), so that the formal state descriptors of the multiphase system again consist of intensive properties only. We wish to see why this is so. 

Some authors state that the reaction rate is d /dt where t stands for time. But dfydt is proportional to the size of the reactor and, hence, is an extensive property like , and not an intensive property, as should be the reaction rate, according to the definition of the term. The derivative dt /dt is to be called the reactor productivity, but not the reaction rate. 

The SI units of heat capacity are J/K, as can be easily seen from the definition of C. Heat capacity is an extensive property of matter. That is, larger samples have greater heat capacities. 

The definition of subsystem available energy, A, which is an extensive property, is crucial to practical Second Law efficiency analysis. Before a process, device, or system can be analyzed, it is necessary to ascertain (or assume or approximate) the dead states of all relevant materials and equipment. 

The properties of random copolymers can be estimated by using weighted averages for all extensive properties, and the appropriate definitions for the intensive properties in terms of the extensive properties. Let lrq, m2,. .., mn denote the mole fractions (see Section l.D) of n different types of repeat units in a random copolymer. [The most common random copolymers have n=2. Terpolymers (n=3) are also often encountered.] The n mole fractions then add up to one, and the extensive properties of a random copolymer can be estimated by using mole fractions as weight factors ... 

Free-energy change, AG, is an extensive property i.e., it depends on the size of the system. If normalized, however, to a per-mole or per-atom basis, it becomes known as the chemical potential. The formal definition of the chemical potential of species, /, is... 

The value of any extensive property is obtained by summing the values of that property in every part of the system. Suppose that the system is subdivided into many small parts, as in Fig. 2.3. Then the total volume of the system is obtained by adding together the volumes of each small part. Similarly, the total number of moles (or total mass) in the system is obtained by summing the number of moles in (or mass of) each part. By definition, such properties are extensive. It should be clear that the value obtained is independent of the way in which the system is subdivided. 

The intrinsic properties are the amount of radioactivity and factors which give a measure of the risk for the worker (e.g. the radiotoxic properties of the particular nuclide as given by the ALI or DAC values). It is not possible to draw any definite conclusions about the hazard ft om a certain amount of a radiotoxic substance. The hazard risk may only be evaluated from its radiotoxicity value. For that purpose, it is also necessary to consider its chemical form and pathways to man, which are considered to be extensive properties. ALI values do take into account if the chemical form is "soluble" or "insoluble", but this is, of course, a rather crude subdivision. The DAC values consider the particle size and time of exposure to that particular air condition (e.g. in a factory). However, it does not consider the particular ways by which the substance is released to the environment (cf. 21.11.1). 

The crystal orbital approach (see ref. 94 for a review of the recent computational developments in this field) has dominated the electronic structure calculations on polymers for several years. However, the recently published reports on the finite-cluster calculations reveal that the latter methodology has several definite advantages over the traditional approach. Let P(N) be an extensive property of a finite cluster X-(-A-)j -Y, where N is the number of repeating units denoted by A, while X and Y stand for terminal groups. The corresponding intensive properties, p(N) = P(N)/N, are known only for integer values of N. However, provided the polymer in question is not metallic, P(v) can be approximated by a smooth function p(v) of v = 1/N, which in turn can be extrapolated to v = 0 yielding the property of the bulk polymer. 

These operational distinctions between extensive and intensive avoid ambiguities that can occur in other definitions. Some of those definitions merely say that extensive properties are proportional to the amount of material N in the system, while intensive properties are independent of N. Other definitions are more specific by identifying extensive properties to be those that are homogeneous of degree one in N, while intensive properties are of degree zero (see Appendix A). 

Any extensive property can be made intensive by dividing it by the total amount of material in the system however, not all extensive properties are proportional to the amount of material. For example, the interfacial area between the system and its boundary satisfies our definition of an extensive property, but this area changes not only when we change the amount of material but also when we merely change the shape of the system. Further, although some intensive properties can be made extensive by multiplying by the amount of material, temperature and pressure carmot be made extensive. 

Since the definition (4.2.1) is a linear combination of thermodynamic properties, all relations among extensive properties, such as those in Chapter 3, can be expressed in terms of residual properties. Examples of such relations include the four forms of the fundamental equation and the Maxwell relations. Moreover, using the expressions developed in 4.1.4 for ideal-gas mixtures, the following intensive forms for residual properties are obtained ... 

Since the enthalpy is an extensive property, it could be expected that the enthalpy of a humid gas is the sum of the partial enthalpies of the constituents and a term to take into account the heat of mixing and other effects. The humid enthalpy Iq is defined as the enthalpy of a unit mass of dry gas and its associated moisture. With this definition of enthalpy. 

With this definition, the differential of an extensive property is written in the more compact form,... 

Another useful formulation of the extensive property Z is based on the definition of apparent molar quantities Oz of the solute, which yields... 

The additivity property of entropy and internal energy of subsystems demands that both S and U of the subsystems be first-order homogeneous functions of extensive properties which define the subsystem. From the definition of the first-order homogeneous property for U, one... 

Sometimes a more restricted definition of an extensive property is used The property must be not only additive, but also proportional to the mass or the amount when intensive properties remain constant. According to this definition, mass, volume, amount, and energy are extensive, but surface area is not. 

Clearly, the next step is the handling of a molecule as a real object with a spatial extension in 3D space. Quite often this is also a mandatory step, because in most cases the 3D structure of a molecule is closely related to a large variety of physical, chemical, and biological properties. In addition, the fundamental importance of an unambiguous definition of stereochemistry becomes obvious, if the 3D structure of a molecule needs to be derived from its chemical graph. The moleofles of stereoisomeric compounds differ in their spatial features and often exhibit quite different properties. Therefore, stereochemical information should always be taken into ac-count if chiral atom centers are present in a chemical structure. 

Within the scope of the original definition, a very wide variety of ionomers can be obtained by the introduction of acidic groups at molar concentrations below 10% into the important addition polymer families, followed by partial neutralization with metal cations or amines. Extensive studies have been reported, and useful reviews of the polymers have appeared (3—8). Despite the broad scope of the field and the unusual property combinations obtainable, commercial exploitation has been confined mainly to the original family based on ethylene copolymers. The reasons for this situation have been discussed (9). Within certain industries, such as flexible packaging, the word ionomer is understood to mean a copolymer of ethylene with methacrylic or acryhc acid, partly neutralized with sodium or zinc. 

From the definition of a partial molar quantity and some thermodynamic substitutions involving exact differentials, it is possible to derive the simple, yet powerful, Duhem data testing relation (2,3,18). Stated in words, the Duhem equation is a mole-fraction-weighted summation of the partial derivatives of a set of partial molar quantities, with respect to the composition of one of the components (2,3). For example, in an / -component system, there are n partial molar quantities, Af, representing any extensive molar property. At a specified temperature and pressure, only n — 1) of these properties are independent. Many experiments, however, measure quantities for every chemical in a multicomponent system. It is this redundance in reported data that makes thermodynamic consistency tests possible. 

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