Extensive variable definition

This equation is valid for all species Ah a fact that is a consequence of the law of definite proportions. The molar extent of reaction is a time-dependent extensive variable that is measured in moles. It is a useful measure of the progress of the reaction because it is not tied to any particular species A. Changes in the mole numbers of two species j and k can be related to one another by eliminating between two expressions that may be derived from equation 1.1.4. 

In the present work, the general mathematical scheme of construction of the equilibrium statistical mechanics on the basis of an arbitrary definition of statistical entropy for two types of thermodynamic potential, the first and the second thermodynamic potentials, was proposed. As an example, we investigated the Tsallis and Boltzmann-Gibbs statistical entropies in the canonical and microcanonical ensembles. On the example of a nonrelativistic ideal gas, it was proven that the statistical mechanics based on the Tsallis entropy satisfies the requirements of the equilibrium thermodynamics only in the thermodynamic limit when the entropic index z is an extensive variable of state of the system. In this case the thermodynamic quantities of the Tsallis statistics belong to one of the classes of homogeneous functions of the first or zero orders. 

The molar extent of reaction introduced previously has the disadvantage of an extensive variable, since depends on the amount of reactants. It is often more convenient to use conversion as an intensive measure of a chemical transformation. The use of a reference reactant is compulsory. If Nis the initial molar amount of the reactant A, and the amount after reaction, the conversion is by definition ... 

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. 

For convenience, thermodynamic systems are usually assumed closed, isolated from the surroundings. The laws that govern such systems are written in terms of two types of variables intensive (or intrinsic) that do not depend on the mass and extensive that do. By definition, extensive variables are additive, that is, their value for the whole system is the sum of their values for the individual parts. For example, volume, entropy, and total energy of a system are extensive variables, but the specific volume (or its reciprocity - the density), molar volume, or molar free energy of mixing are intensive. It is advisable to use, whenever possible, intensive variables. 

It is understood that under the small strain state the variable of u is e and d Ua is composed of the increments of the extensive variable e and the intensive variable (see Appendix D for the definition of the extensive and intensive variables). 

Where entropy S and volume V are the extensive variables and temperature T and pressure p are the conjugated intensive variables (the signs indicated follow from the definition of the positive direction of the energy fiow, namely, one directed into the system). Thus, the thermodynamic potential of this system is E=E(S, V) if this function is known, the system can be regarded as completely described. But the variables S and V cannot be controlled in a simple, direct manner. Thus, it would be preferable to select the pressure p and the temperature T as variables because they can easily be set experimentally. Which quantity is then the proper thermodynamic potential ... 

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. 

This definition of the variable of mixing is appUed to aity extensive variable. Let us consider the following examples. 

We can use this definition for different extensive variables and we will do this for Gibbs energy, entropy, enthalpy and heat capacity. 

Knowing that F, the Helmholtz free energy, is U - TS from Eq. (3) of Fig. 5.11 lets one easily derive Eq. (2). Since the extensive variables volume and length are dlfhcult to keep constant during a measurement, one commonly changes to the intensive variables pressure and force. The equations to be derived are simple only if one uses the modified definition of enthalpy, H, as proposed in Eq. (8b) of Fig. 1.2. 

Quantities such as the volume V, the mass m, the number of moles <, the thermodynamic potentials. .. are called extensive variables since their values depend on the extent of the S5retem. On the other hand, variables such as the temperature, the pressure, the mole fraction Xi (= m/n) are intensive variables since they have definite values at each point in the system and do not depend on the total extent of the system. The ratio of two extensive variables is an intensive variable. To each extensive variable Y there correspond intensive variables defined by the partial derivative at constant pressure and temperature... 

From the definition of temperature, pressure and chemical potential as partial derivatives it follows that these variables may be written as functions of the extensive parameters ... 

Inheritance, derivation, or extension mean that the definition of one class is based on that of one or more others. The extended class has by default the variables and operations of the class(es) it extends augmented by some of its own. The extension can also override an inherited operation definition by having one of its own of the same name. 

Since X is a random variable, this extension of (6.29) to / x is not necessarily obvious. However, by working backwards from (6.177), it is possible to show that this definition is the only choice which permits f% to remain uniform. 

A fundamental idea in multivariate data analysis is to regard the distance between objects in the variable space as a measure of the similarity of the objects. Distance and similarity are inverse a large distance means a low similarity. Two objects are considered to belong to the same category or to have similar properties if their distance is small. The distance between objects depends on the selected distance definition, the used variables, and on the scaling of the variables. Distance measurements in high-dimensional space are extensions of distance measures in two dimensions (Table 2.3). 

In practice, however, this extension is not as straightforward as in DMC. In multivariable DMC, there is a definite design procedure to follow. In multi-variable IMC, there are steps in the design procedure that are not quantitative but involve some art. The problem is in the selection of the invertible part of the process transfer function matrix. Since there are many possible choices, the design procedure becomes cloudy. 

Phase III studies represent the confirmatory phase of drug development, which takes several years and usually involves several thousand patients at multiple trial centers. Large patient numbers are required in these trials to provide convincing documentation of clinical efficacy and safety, a more complete adverse event profile and covariates and estimates of variability in dose response relationship due to individual differences in pharmacokinetics and pharmacodynamics. They are aimed at definitively determining a drug s effectiveness and side-effect profile. Most of these studies are double-blind and placebo-controlled, sometimes with the option of open-label long-term extensions. 

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