Mechanically definition

In Chapter III, surface free energy and surface stress were treated as equivalent, and both were discussed in terms of the energy to form unit additional surface. It is now desirable to consider an independent, more mechanical definition of surface stress. If a surface is cut by a plane normal to it, then, in order that the atoms on either side of the cut remain in equilibrium, it will be necessary to apply some external force to them. The total such force per unit length is the surface stress, and half the sum of the two surface stresses along mutually perpendicular cuts is equal to the surface tension. (Similarly, one-third of the sum of the three principal stresses in the body of a liquid is equal to its hydrostatic pressure.) In the case of a liquid or isotropic solid the two surface stresses are equal, but for a nonisotropic solid or crystal, this will not be true. In such a case the partial surface stresses or stretching tensions may be denoted as Ti and T2-... 

We begin by remembering the mechanical definition of work and apply that definition to the stretching process of Fig. 3.1. Using the notation of Fig. 3.1, we can write the increment of elastic work we associated with an increment in elongation dL as... 

To close this chapter we emphasize that Hie statistical mechanical definition of macroscopic parameters such as temperature and entropy are well designed to describe isentropic equilibrium systems, but are not immediately applicable to the discussion of transport processes where irreversible entropy increase is an essential feature. A macroscopic system through which heat is flowing does not possess a single tempera-... 

Reaction mechanism, definition, 4, 12 Reaction order apparent, 7 defined, 5... 

The energy associated with position of a body or entity with respect to another body or entity or with respect to relative positions of parts within that same body. For an entity or body at position r associated with a force F, the potential energy, Ep or U, is —/F-dr. The strictly classical mechanical definition of potential energy typically leads one to consider a system in terms of stored energy and position, and in that case, one deals with the magnitude of force and distance between interacting... 

Although the definition (3.35) allows practical progress, it rests on other concepts of temperature and thermal capacity that border on circular reasoning. Accordingly, we shall first attempt to formulate an alternative mechanical definition of heat that is of no practical significance, but satisfies the thermodynamicist s penchant for logical order. As in the case of temperature (Section 2.3), we attempt to characterize heat in terms of mechanical variables only (e.g., P, V), which are well defined in a pre-thermodynamic context. 

It may not be initially apparent why this definition coincides with (3.35), but the right-hand side of (3.36) clearly depends only on PV measurements, and thus meets the criterion of a purely mechanical definition of heat. [We shall see in the following section that adiabatic is identical to the internal energy change, so the equivalence of (3.35) and... 

The most interesting example of a quantum mechanical object is the photon itself. By using the relativistic and quantum mechanical definition of the photon energy, we can obtain a quantitative formulation of the concepts just described. The relativistic form of the total energy of a particle with rest mass m and momentum p is ... 

The expression for J is derived via the general quantum mechanical definition (32), introducing the perturbation expansion for the current density and the a-state molecular wave-function (depending on n-electron space-spin coordinates ), yb—Zd e-... 

The quantum mechanical definition of a distribution function in the classical phase-space is an old theme in theoretical physics. Most frequently used is the so-called Wigner distribution function (Wigner 1932 Hillery, O Connell, Scully, and Wigner 1984). Let us consider a onedimensional system with coordinate R and corresponding classical momentum P. The Wigner distribution function is defined as... 

BASIC CONCEPTS AND MIXING MECHANISMS Definitions of Mixtures and Mixing Mechanisms... 

Despite the success in parameterizing acid/base and many other properties for a range of different compounds, it is obvious that the simple electrostatic model used by Taft and extended by others [27, 28] have fundamental weaknesses - both with regards to the domain of validity and at a more fundamental level. The model is more intuitive than physical, in the sense that the inductive effect, polarisation effect, resonance effect, mesomeric effect and steric effect have no proper quantum mechanical definition, and can therefore not be derived directly from the system s wave function [29,30]. 

Problems associated with the quantum-mechanical definition of molecular shape do not diminish the importance of molecular conformation as a chemically meaningful concept. To find the balanced perspective it is necessary to know that the same wave function that describes an isolated molecule, also describes the chemically equivalent molecule, closely confined. The distinction arises from different sets of boundary conditions. The spherically symmetrical solutions of the free molecule are no longer physically acceptable solutions for the confined molecule. 

A few simple calculations would show that, unless both the energy levels are present, the statistical mechanical definition of temperature has no meaning. Also, the following relations must hold, or Eq. (5) will be violated ... 

In the past few years, development of new theories have led to completely new ways of determining free energy changes. Traditionally, the difference in the free energy of two equilibrium state is (AFi 2) and the free energy change of a process can be obtained directly from the statistical mechanical definition of the free energy, F, in terms of the partition function. For the canonical ensemble F = —k T In J = —ksTln Z, where ka is Boltzmann s constant, //(F) is the phase... 

A similar correlation diagram can be drawn to interpret the exclusive observation of BaOH chemiluminescence in the reaction of Ba(6s6p P) with water [172]. Here again, the reaction of excited barium can be explained by H-atom migration and formation of the intermediate HBaOH. This mechanism definitely carries some generality since it has been used also by Oberlander and Parson to account for re-... 

This interesting field was initiated by Bader [158]. Topological analysis provides the means for a concise description of multivariate functions. For functions that describe physical observables, the number and location of critical points, where the gradient vanishes, and their mutual relationship are often directly related to the properties of the system under study. The application of topological analysis to the one electron density is even more productive, furnishing rigorous quantum-mechanical definitions of and bonds in molecules. Cioslowski has extended this analysis to the study of the electron-electron interactions, based on the analysis of the intracule and extracula densities [159,160]. 

Mechanical Definitions of dwell and contact times disregard material properties and concentrate on press and punch geometry (Fig. 17). Contact time can be defined as the time the punch is in contact with the compression wheel. Dwell time is defined as the time the flat portion of punch head is in contact with the compression wheel (time at maximum punch displacement, or time when the punch does not move in vertical direction). In dwell time calculations, the length of the punch head flat and horizontal component of punch speed (as determined by RPM and pitch circle diameter) are used. In case of a round head tooling, the dwell time, as defined here, is zero. But it should be kept in mind that mechanical definition is given here as a convention, a yardstick, or a common measure, to compare press speeds for different presses, and its absolute value is meaningless. A proposed convention to quantify linear speed of a press is to use an... 

Fig. 17 Dwell time and contact time—mechanical definitions.

Finally we recall [3.6.15] which is a mechanical definition for an infinitesimally thin monolayer. The interfacial stress tensor is a more general quantify than y because it also contains the shear components. When shear stresses are absent reduces to... 

TTie definition of a bound atom—an atom in a molecule— must be such that it enables one to define all of its average properties. For reasons of physical continuity, the definition of these properties must reduce to the quantum mechanical definitions of the corresponding properties for an isolated atom. The atomic values for a given property should, when summed over aU the atoms in a molecule, yield the molecular average for that property The atomic properties must be additive in the above sense to account for the observation that, in certain series of molecules, the atoms and their properties are transferable between molecules, leading to what are known as additivity schemes. An additivity scheme requires both that the property be additive over the atoms in a molecule and that the atoms be essentially transferable between molecules. 

If the topological property which defines an atom is also one of physical significance, then it should be possible to obtain from quantum mechanics an equivalent mechanical definition. As demonstrated in Chapters 5 and 8, this can be accomplished through a generalization of the quantum action principle to obtain a statement of this principle which applies equally to the total system or to an atom within the system. The result is a single variational principle which defines the observables, their equations of motion, and their average values for the total system or for an atom within the system. 

In the LCAO MO description, the H2 molecnle in its ground state has a pair of electrons in a bonding MO, and thus a single bond (that is, its bond order is 1). Later in this chapter, as we describe more complex diatomic molecules in the LCAO approximation, bond orders greater than 1 are discussed. This quantum mechanical definition of bond order generalizes the concept first developed in the Lewis theory of chemical bonding—a shared pair of electrons corresponds to a single bond, two shared pairs to a double bond, and so forth. 

---