Extensive quantities, definition

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. 

In amount of substance measurements, we almost never face the need to determine the sum of all components. We try to determine specific substances that form a part of the whole. Amount of substance, which is of course an extensive quantity, can be considered as having partial nature - this is supported by its definition, too (entities must be stated). Any prepared stand-... 

The result (11), AS = -nRB, is aparadox, a contradiction with our presumption of not influencing a thermodynamic state of A by diaphragms, and, leads to that result that the heat entropy S (of a system in equilibrium) is not an extensive quantity. But, by the definition of the differential dS, this is not true. 

The first term on the right hand side of this equation shows how much the observed quantity of state Z has increased by in the volume V, at time t. The second term indicates which part of the state quantity flows out with the material. The surface element d.4, is, as shown in Fig. 3.2, by definition in the outward direction so that positive widAi indicates a flow out of the volume. Clearly, according to (3.17), the change in the extensive quantity of state Z of a substance amount M with a volume V(t) that changes with time, is equal to the increase in the extensive quantity of state inside the volume V at time t and the fraction of the quantity of state that flows out of the volume with the material at the same time. 

Summarizing, we recall a common and simple definition of intensive and extensive variables. If two identical systems are gathered, the extensive term will double, whereas the intensive term will remain the same. Examples are the pairs (T, S), (-p, V), iix, n), (cp, q), i.e., (temperature, entropy), (pressure, volume), (chemical potential, mol number), (electrical potential, electrical charge). Often the extensive quantity may flow into or out of the system. 

To be sure what ensues within the system, one monitors carefully its state and its boundaries. All transport across the boundaries or creation which increases an extensive quantity within the system is counted as positive (+). All that is lost across the boundaries or within the system is counted as negative (-). This assignment of plus and minus is characteristic from the point of view of the scientists. They consider themselves as the spokespersons for the system. Engineers often have the opposite view since their responsibility is to make use of the products of a system. They will count quantities lost by the system as a gain. This double set of definitions causes confusion, but by now cannot be eliminated and requires care to avoid mistakes. 

The macroscopic dipole moment (Pe) of the material in the capacitor is by definition an extensive quantity. Hence it easily can be generalized to multi-component or multiphase systems, assuming all components or phases being homogenously distributed within the capacitor. Considering a sorption system consisting of three quasi-homogenous phases sorbent (s), sorbate (a), and sorptive gas (f), we get from (6.38) and (6.41) respectively... 

In the present section, we start our thermodynamic stability considerations by recalling and generalizing the definition of capacitances as introduced in Section 4.2. Quite generally, capacitances are the equilibrium parts of thermodynamic networks or, in other words, they are reversible storage elements for extensive quantities like energy U, volume V, and mole numbers Np 2 property of the reversibility of the storage process is expressed by Gibbs relation which in its version (3.65) for the volume densities s = S/V, u = U/V, c. = N. /V can be written as... 

It is noteworthy that a system, even when not in equilibrium, always has definite extensive variables as for the intensive variables, they are - as a rule - definite in a stable or metastable state only. Hence, the preference is for extensive quantities as independent variables. 

Adii/f is here an extensive quantity obtained for a given amount of 1 cm pure methanol added successively to a definite volume of solution in which the composition of poly(L-lysine), methanol and water is known. 

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. 

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... 

Facilitate pre-vulcanisation processing, increase softness, extensibility and flexibility of the vulcanised end-product. The rubber processing industry consumes large quantities of materials which have a plasticising function complex mixtures (paraffinic, naphthenic, aromatic) of mineral hydrocarbon additives, used with the large tonnage natural and synthetic hydrocarbon rubbers, are termed process oils. Because of the complexity of these products, precise chemical definition is usually not attempted. If the inclusion of an oil results in cost reduction it is functioning as an extender. The term plasticiser is commonly reserved for synthetic liquids used with the polar synthetic rubber. 

It is remarkable that in the same year, 1934, two independent approaches, those of Stoll et al. and of Kuhn, led to the definition of two quantities which are conceptually quite similar and can be practically identical in many actual cases. In either case the intramolecular reaction is compared to a corresponding intermolecular process. This is the dimerisation reaction of the bifunctional reactant in the definition of the cyclisation constant C in the case of the effective concentration Crff Winter must be determined with the aid of an inter-molecular model reaction, the choice of which is not always obvious and can possibly lead to conceptual as well as experimental difficulties. It is also worth noting that although these early workers established a firm basis for interpretation of physical as well as of preparative aspects of intramolecular reactions, no extensive use of quantities C and Qff appears to have been made in the chemical literature over more than three decades after their definition. This is in spite of the enormous development of studies in the field of... 

The possibility to have metastable hadronic stars, together with the feasible existence of two distinct families of compact stars, demands an extension of the concept of maximum mass of a neutron star with respect to the classical one introduced by Oppenheimer Volkoff (1939). Since metastable HS with a short mean-life time are very unlikely to be observed, the extended concept of maximum mass must be introduced in view of the comparison with the values of the mass of compact stars deduced from direct astrophysical observation. Having in mind this operational definition, we call limiting mass of a compact star, and denote it as Mum, the physical quantity defined in the following way ... 

Characteristics and implementation of the treatments depend on the expected results and on the properties of the material considered a variety of processes are employed. In ferrous alloys, in steels, a eutectoid transformation plays a prominent role, and aspects described by time-temperature-transformation diagrams and martensite formation are of relevant interest. See a short presentation of these points in 5.10.4.5. Titanium alloys are an example of the formation of structures in which two phases may be present in comparable quantities. A few remarks about a and (3 Ti alloys and the relevant heat treatments have been made in 5.6.4.1.1. More generally, for the various metals, the existence of different crystal forms, their transformation temperatures, and the extension of solid-solution ranges with other metals are preliminary points in the definition of convenient heat treatments and of their effects. In the evaluation and planning of the treatments, due consideration must be given to the heating and/or cooling rate and to the diffusion processes (in pure metals and in alloys). 

In general, neither the force nor the molecular extension can be controlled in the experiments so definitions in Eqs. (96), (98), and (99) result in approximations to the true mechanical work that satisfy Eqs. (40) and (41). The control parameter in single molecule experiments using optical tweezers is the distance between the center of the trap and the immobilized bead [88]. Both the position of the bead in the trap and the extension of the handles are fluctuating quantities. It has been observed [94—96] that in pulhng experiments the proper work that satisfies the FT includes some corrections to Eqs. (97) and (99) mainly due to the effect of the trapped bead. There are two considerations to take into account when analyzing experimental data. 

The quantities G, //, and S are called extensive thermodynamic functions because the magnitude of the quantity in each case depends on the amount of substance in the system. The change in Gibbs free energy under addition of unit concentration of component / at constant concentrations of the other components is called the partial Gibbs free energy of the /-component, i.e., the chemical potential of the /-component in the system. The chemical potential is an intensive thermodynamic quantity, like temperature and concentrations. The formal definition is... 

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. 

By analogy to other heat capacities, we identify (35/37 )z v as C) V/T, where Q r is the heat capacity at constant extension and volume, definitely a positive quantity. We then have... 

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