Intensive 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. 

Both of these partial derivatives are divided by V to make them intensive quantities. The SI units of a are K-1 and those of k are Pa 1. A negative sign is used in the definition of k, because volumes always decrease as pressure increases, and we would prefer to tabulate positive quantities. 

We have made mention earlier (Frame 5, section 5.4) albeit very briefly, of the definition of the chemical potential, //, of a substance i. For a two component system having components labelled as 1 and 2, this intensive quantity (Frame 1, section 1.3) is defined as the rate of change of Gibbs energy per mole of substance present ... 

Now since this is a one-component system the Gibbs free energy F, should have the property Fs = riigi, and we use this as our definition of F Integrating Eq. (61) keeping the intensive quantities T, P, 0, and in constant,... 

For more than a hundred years, quantities that appear in this role have been called intensive factors, intensive quantities, or simply intensive. Unfortunately, this descripticMi does not agree completely with the definition in Sect. 1.6. In order to avoid misunderstandings, we have no choice but to look for a new name. The German physicist and physician Hermann von Helmholtz came up with one that would be helpful to us. Using Joseph Louis De Lagrange s concept of forces in the field of mechanics, he generalized it. We refer to this and call the quantities force-Uke. ... 

These rates have dimensions of [g T" h" ] or [kJ T h ] and are intensive quantities that may be substituted directly into the mass balance equation. On the other hand, these absolute rates may take on any value and are therefore not characteristic of the system. Comparable variables, which are biologically representative, are the so-called specific rates of production or utilization, which refer to the catalytically active mass (to a first approximation, the concentration of biomass, x, is taken as the dry cell weight). With this, one has the definitions of the specific rates for bioprocesses of Equs. 2.5a-f where, in each case, the specific rate has the dimension [h ]. For growth, the specific rate is p... 

We note here that for nanoscale systems in which the contribution to their energy (or other quantity) from the surface atoms is comparable with that from the bulk volume atoms the non-additivity exists forever. The surface energy is increased in times when one increases the number of particles in n times. That is why the question about the definition of temperature for nanoscale system is very intriguing not only from the point of view of its physical measurements but also from the estimations for the minimal length scale on which this intensive quantity exists (Hartmann et al., 2004). 

The statistical ensemble framework of equilibrium statistical mechanics gives us the tools to analyze experimental data and to make theoretical predictions. The concept of entropy, its maximization, and the ensuing definition of intensive quantities such as pressure and temperature reduces the complexity of a statistical system of 10 particles to manageable proportions. In principle, the problem of predicting the collective behavior of equilibrium systems starting from microscopic interactions is solved. In practice, exact calculations are rarities. Computational tools such as the Monte Carlo Metropolis method can, however, fill in this void a priori knowledge of the probability distribution of microstates is at the heart of the Metropolis algorithm. 

Section BT1.2 provides a brief summary of experimental methods and instmmentation, including definitions of some of the standard measured spectroscopic quantities. Section BT1.3 reviews some of the theory of spectroscopic transitions, especially the relationships between transition moments calculated from wavefiinctions and integrated absorption intensities or radiative rate constants. Because units can be so confusing, numerical factors with their units are included in some of the equations to make them easier to use. Vibrational effects, die Franck-Condon principle and selection mles are also discussed briefly. In the final section, BT1.4. a few applications are mentioned to particular aspects of electronic spectroscopy. 

Several additional terms related to the absorption of x-radiation require definition energy of a x-ray photon is properly represented in joules but more conveniently reported in eV fluence is the sum of the energy in a unit area intensity or flux is the fluence per unit time and the exposure is a measure of the number of ions produced in a mass of gas. The unit of exposure in medicine is the Rn ntgen, R, defined as the quantity of radiation required to produce 2.58 x C/kg of air. The absorbed dose for a tissue is a measure of energy dissipated per unit mass. The measure of absorbed dose most... 

How then, can one recover some quantity that scales with the local charge on the metal atoms if their valence electrons are inherently delocalized Beyond the asymmetric lineshape of the metal 2p3/2 peak, there is also a distinct satellite structure seen in the spectra for CoP and elemental Co. From reflection electron energy loss spectroscopy (REELS), we have determined that this satellite structure originates from plasmon loss events (instead of a two-core-hole final state effect as previously thought [67,68]) in which exiting photoelectrons lose some of their energy to valence electrons of atoms near the surface of the solid [58]. The intensity of these satellite peaks (relative to the main peak) is weaker in CoP than in elemental Co. This implies that the Co atoms have fewer valence electrons in CoP than in elemental Co, that is, they are definitely cationic, notwithstanding the lack of a BE shift. For the other compounds in the MP (M = Cr, Mn, Fe) series, the satellite structure is probably too weak to be observed, but solid solutions Coi -xMxl> and CoAs i yPv do show this feature (vide infra) [60,61]. 

On continuous illumination (i.e. when the incident light intensity is constant), the measured anisotropy is called steady-state anisotropy r. Using the general definition of an averaged quantity, with the total normalized fluorescence intensity as the probability law, we obtain... 

The conditions for eqnilibrinm have not changed, and application of the phase rnle is conducted as in the previous section. The difference now is that composition can be counted as an intensive variable. Composition is accounted for through direct introduction into the thermodynamic quantities of enthalpy and entropy. The free energy of a mixtnre of two pure elements, A and B, is still given by the definition... 

The pressure intensity is related to the piezometric pressure p, through the definition of the latter quantity ... 

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. 

Remark. When U is a field strength one may regard UU as an instantaneous intensity I (ignoring for the moment that it should be divided by T). Its moments are the same as those of UU. However, the cumulants are not the same because their definition depends on whether UU is regarded as a product of two quantities or as a single quantity /. For example,... 

The reflectivity within the halo is not constant one observes a maximum close to the radioactive source with a decrease outwards. The tracing of reflectivity curves with a photocell, recorder, and moving stage has shown that the decrease curves are smooth but sometimes stepwise (Figure la, b). The width of the halos is difficult to estimate by a subjective method, and our instrumental measurements are still too few to allow, definite interpretation. However, the width generally varies between 20 and 50 microns. Preliminary observations suggest that the intensity of the halos is proportional to the quantity of uranium (Figure 4c). 

Remark 6.1. The definition of the energy recovery number follows the same principle as that of the recycle number Rc (Definition 3.1). Both numbers characterize the intensity of recycling/recovery of a process inventory (see, e.g., Farschman et al. 1998) - that is, energy and mass, respectively. From the perspective of inventory recycling, the two numbers Ere and Rc are, in effect, particular cases of the same dimensionless quantity. 

Note also that the form of the pulse does not appear in the expression [Eq. (2.78)] for the cross section. This is because resolving the energy, embodied in the orthogonality expression [Eq. (2.47)], extracts a single frequency component of e(to), whose contribution is canceled in the division by the incident light intensity. Therefore, as shown in Appendix 2A, we can use any convenient pulse shape to compute energy-resolved quantities. This is not the case if we want to follow the real-time dynamics of the system, where the pulse shape is intimately linked with the observables. Indeed this link prevents a pulse-free definition of concepts such as the lifetime of a state. This issue is addressed in Appendix 2A. 

Fortunately most problems involving aerosol optics do not require the use of absolute quantities but instead make use of such unitless radiometric indicators as absorbance, reflectance, and the like. Table 16.3 lists several of the more common of these indicators. Figure 16.1 illustrates some of the more common definitions of light intensity. 

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