Distinguishable state

A third definition of surface mobility is essentially a rheological one it represents the extension to films of the criteria we use for bulk phases and, of course, it is the basis for distinguishing states of films on liquid substrates. Thus as discussed in Chapter IV, solid films should be ordered and should show elastic and yield point behavior liquid films should be coherent and show viscous flow gaseous films should be in rapid equilibrium with all parts of the surface. 

The identity of several rows and columns in the matrix in Eq. (4.3) shows that the same information is contained in a more compact matrix. This matrix can be written with the four distinguishable states on the 2nnd lattice indexing the rows and columns, in the order in which they are listed in Table 4.1 [153] ... 

Here NA and Ag are the number of A and B atoms in the mixture. Each distinguishable arrangement of the atoms represents a unique state, and to obtain the degeneracy for the single energy state of an ideal solution with a specific composition we need to calculate the number of distinguishable states. 

Distinguishing States of Spec Types from Design Types... 

From the quantum mechanical standpoint the appearance of the factor 1/2 = 1/s for the diatomic case means the configurations generated by a rotation of 180° are identical, so the number of distinguishable states is only one-half the classical total. Thus the classical value of the partition function must be divided by the symmetry number which is 1 for a heteronuclear diatomic and 2 for a homonuclear diatomic molecule. 

The case of polyatomics, which is considered in more detail below, follows analogously. The point to keep in mind is that the symmetry restricts the number of distinguishable states. The symmetry number is the number of equivalent (indistinguishable) positions into which a molecule can be carried by rigid body rotation. For example s = 12 for CH4 since the molecule can be held by a CH bond and rotated into three equivalent positions, and there are four CH bonds. Similarly for benzene s = 12 since there are six indistinguishable positions for rotation about an axis perpendicular to the plane of the molecule (and through its center), and six more when the molecule is flipped over. 

The usual presentation of a spin-1/2 particle starts with two physically distinguishable states. These states are usually labeled by kets, such as +z) and —z> (in physics texts) or 1) and 0) (in quantum computing texts). The name and asymmetrical notation connote the right half of the complex scalar product (also known as a bracket) used in descriptions of quantum systems, (1). One posits that every quantum state can be written as a superposition of kets ... 

Molecular superconductors are of more immediate practical interest. Experimental evidence indicates superconductivity is incipient at 30 K, a temperature above the critical temperature for other known materials. Superconductors have a major potential use in providing two distinguishable states associated with tunneling of electron pairs through a... 

Symmetry is represented by the elements of a (mathematical) group and thus cannot change continuously. The a-0 phase transition therefore occurs at a distinct temperature. Let us now assume that we have identified an extensive thermodynamic variable which can distinguish states between the a and 0 phases. We call it an order parameter (/ ). For a quantitative description of order-disorder or continuous displacive processes, the order parameter is normalized (0< s 1). For example, if we regard the classic 0-0 brass transition, tj is defined as (2/Cu -1), where /Cu is the fraction of Cu atoms which occupy the (0,0,0) sites of the (Cu,Zn) bcc structure. 

Initial shock velocity is a very poor method of characterizing input shock. This becomes obvious upon examination of Fig 11. Let points A B in Fig 11 represent initiation thresholds for two different expls, both at a density p0. Points A B in this diagram are well separated in both P u. Thus no extraordinary exptl accuracy is required to distinguish states A B in terms of either P or u. The slopes of the Rayleigh lines (the lines connecting A or B to the origin), however, are almost equal. Since these slopes are equal to poU, extreme exptl accuracy is necessary to characterize thresholds in terms of U, the initial shock velocities in the test expls... 

With no further approximations we can now consider an ensemble of molecules, which represents the components of the mixture under consideration as swimming around individually in the infinitely extended virtual conductor. Because the electric field of the molecules is perfectly screened off by the conductor, there is no interaction of the molecules. We know the total energy of this ensemble by just summing the total energies of the individual molecules in this virtual, but distinguished state, which we consider as the north pole of our globe (Fig. 1.1). 

In general, more than one product asymptotic region can be reached from a single reactant asymptote, i.e., certain reactants may undergo several elementary processes, the so-called channels. Within a channel, processes leading from specific reactant states to specific product states can be distinguished (state-to-state elementary processes). 

Fig. 23. Potentiodynamic current-potential relations at Pt owing to the oxide film reconstruction effect. Distinguishable states of oxide, O, in anodic (A) and cathodic (C) sweeps are...

The relative numbers of distinguishable states in ice and in water at 273 K, calculated ... 

The number of distinguishable states in a molecular system is given by the thermodynamic probability W. This is related to entropy as... 

These expressions however are restricted to solids, as our expression for the number of microstates assumed that the molecules were in some way identifiable. Now this is true in solids where their positions on lattice sites can be specified but is not true in a gas where an exchange of molecules does not produce a distinguishable state. We must, in the case of perfect gases, divide the number of microstates by N to allow for this possibility of interchange This leads to an additional term in the expression for entropy... 

In chemical systems one may thus distinguish state variables and control parameters. Chemical reactions have been known from experience to be structurally stable (resistant to small changes in control parameters) under some conditions it is also known that the change in a reaction character (change in the stationary state or dynamics) may occur upon a continuous variation in control parameters in other words, the sensitive state may be created in a chemical system. The above remarks substantiate an attempt to apply catastrophe theory to a description of chemical reactions. 

This is an important phenomenon for any theory to account for, because ultra-violet absorptions of a given molecule are one of the criteria by which molecular structure is routinely tested. Because, in this application, we need to classify a particular absorption, it is important that we should be able to say something about excited states, rather than just the ground states, of molecules. The immediate difficulty which is encountered with the Hiickel method is that it does not distinguish states of different multiplicity (in this instance, the most important examples would be singlets and triplets). Let us suppose, purely for convenience, that we are dealing with a two-electron system (other electrons could also be considered if we wished). In this two-electron system we suppose that one electron is in an orbital, u, and the other is in a different orbital, v—i.e. 

Some problems and concerns typical in the teaching and learning of thermodynamics do not appear in the papers cited earlier. These are a concern with a distinction between system and surroundings a concern with distinguishing state quantities from process quantities and the use of a certain type of mathematics that is specifically suited to treat non-Mnear relations between three or more variables. We therefore conclude that there is consensus among physics education researchers that the teaching of an energy concept does not require a specifically thermodynamic framework. 

B. Yurke, D. Stoler, Generating quantum mechanical superpositions of macroscopically distinguishable states via amplitude dispersion, Phys. Rev Lett. 57 (1986) 13. 

The physical nature of an amorphous polymer is related to the extent of the molecular motion in the sample, which in turn is governed by the chain flexibility and the temperature of the system. Examination of the mechanical behavior shows that there are five distinguishable states in which a linear amorphous polymer can exist, and these are readily displayed if a parameter such as the elastic modulus is measured over a range of temperatures. 

Carrying out the mathematical analysis it is possible to obtain the dependences which allow to determine the values of the probabihties of system staying in the distinguished states. 

Between the two clearly distinguishable states of an emulsion lies flocculation, which refers to the mutual attachment of individual emulsion drops to form floes or loose assemblies of particles in which the identity of each is maintained (Fig. 11.2c), a condition that clearly differentiates it from the action of coalescence. Flocculation can be, in many cases, a reversible process, overcome by the input of much less energy than was required in the original emulsification process. 

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