Equilibrium displacement from

Note that a displacement from the initial equilibrium, either by compression or by stretching, produces positive work on the system. A sittiation analogous to the stretching of a sprmg is the stretching of a chain polymer. 

In this section we concentrate on the electronic and vibrational parts of the wavefimctions. It is convenient to treat the nuclear configuration in temis of nomial coordinates describing the displacements from the equilibrium position. We call these nuclear nomial coordinates Q- and use the symbol Q without a subscript to designate the whole set. Similarly, the symbol v. designates the coordinates of the th electron and v the whole set of electronic coordinates. We also use subscripts 1 and ii to designate the lower and upper electronic states of a transition, and subscripts a and b to number the vibrational states in the respective electronic states. The total wavefiinction f can be written... 

In many cases the variation is not very strong for reasonable displacements from equilibrium, and it is sufficient to use only the zero-order temi in die expansion. If diis is inserted hito equation (B 1.1.6) we get... 

Equation (Bl.8.6) assumes that all unit cells really are identical and that the atoms are fixed hi their equilibrium positions. In real crystals at finite temperatures, however, atoms oscillate about their mean positions and also may be displaced from their average positions because of, for example, chemical inlioniogeneity. The effect of this is, to a first approximation, to modify the atomic scattering factor by a convolution of p(r) with a trivariate Gaussian density function, resulting in the multiplication ofy ([Pg.1366]

The lattice atoms in the simulation are assumed to vibrate independently of one another. The displacements from the equilibrium positions of the lattice atoms are taken as a Gaussian distribution, such as... 

Xj 1. The sample relaxes fast with the displacement from equilibrium synclnonous with the sound wave. 

Figure B2.5.4. Periodic displacement from equilibrium through a sound wave. The frill curve represents the temporal behaviour of pressure, temperature, and concentrations in die case of a very fast relaxation. The other lines illustrate various situations, with 03Xj according to table B2.5.1. 03 is the angular frequency of the sound wave and x is the chemical relaxation time. Adapted from [110].

When we say that reactions 1 and 2 go we actually mean that the equilibrium between eactants and products is displaced from the reactants towards the products. We represent this strictly by the equation (for reaction 2)... 

This term is associated with deformation of a bond from its standard equilibrium length. For small displacements from equilibrium, a harmonic function is often used ... 

If we now pass from one equilibrium state to another (an equilibrium displacement") at constant temperature, then (cf. Equation (3.6)... 

Since depends on nuclear coordinates, because of the term, so do and but, in the Bom-Oppenheimer approximation proposed in 1927, it is assumed that vibrating nuclei move so slowly compared with electrons that J/ and involve the nuclear coordinates as parameters only. The result for a diatomic molecule is that a curve (such as that in Figure 1.13, p. 24) of potential energy against intemuclear distance r (or the displacement from equilibrium) can be drawn for a particular electronic state in which and are constant. 

When a system is displaced from an equilibrium state, it undergoes a.process, a. change of state, which continues until its properties attain new equilibrium values. During such a process the system may be... 

Vibrational energy, which is associated with the alternate extension and compression of die chemical bonds. For small displacements from the low-temperature equilibrium distance, the vibrational properties are those of simple harmonic motion, but at higher levels of vibrational energy, an anharmonic effect appears which plays an important role in the way in which atoms separate from tire molecule. The vibrational energy of a molecule is described in tire quantum theory by the equation... 

As was shown above (Section 2.2, Eq. 5), Young s equation (Eq. 4) may be derived by considering the small displacement from equilibrium of a sessile drop on a plane surface. If the same derivation is applied to the situation where the solid surface has a roughness factor (Eq. 19) of r, it is readily seen that Eq. 5 becomes [28]... 

The value of the integration constant is determined by the magnitude of the displacement from the equilibrium position at zero time. King also gives a solution for Scheme IV, and Pladziewicz et al. show how these equations can be used with a measured instrumental signal to estimate the rate constants by means of nonlinear regression. 

This device of A, the displacement from equilibrium, is used in the study of very fast reversible reactions by relaxation kinetics. We will see, in Chapter 4, that if A is very small, all reactions follow first-order kinetics, thus simplifying the interpretation of the kinetics. This approach might be extended to slow reversible reactions. 

Study of reversible reactions close to equilibrium. This possibility was discussed in eonnection with Scheme II and is further treated in Chapter 4. It turns out that if the displacement from equilibrium is small, the kinetics approach first-order behavior. 

If now the resistance in the external circuit is decreased slightly the reaction will proceed at a finite rate, and the electrodes constituting the cell will become mutually polarised and displaced from their equilibrium values, i.e. the polarised potential of the anode (Zn /Zn) will become more positive, whilst that of the cathode (2H /H2> will become more negative (Fig. 1.23). 

To a good approximation, only atoms within the dotted circles in Figs. 20.30a and b are displaced from their equilibrium position in a real, three-dimensional crystal the diameter d of these circles would be very much less than the length / of the dislocation, i.e. the length, perpendicular to the page, of the extra half plane of atoms ab in Fig. 20.30a, or of the line cd in Fig. 20.306. Dislocations strictly, therefore, are cylindrical defects of diameter d and length / however, since I d they are referred to as line defects. 

Overpotential (overvoltage, polarisation) the displacement from the equilibrium (or steady-state) electrode potential required to cause a reaction to proceed at a given rate. 

If the experimental conditions are such that the equilibrium is completely displaced from left to right the cation C+ is completely fixed on the cation exchanger. If the solution contains several cations (C+, D+, and E + ) the exchanger may show different affinities for them, thus making separations possible. A typical example is the displacement of sodium ions in a sulphonate resin by calcium ions ... 

If the atom is displaced from its equilibrium position through the distance Sr, the force of restitution is ... 

Equations (5.63) and (5.64) are actually more general than is apparent from the derivation. Consider a closed system at a given temperature and pressure with /t , moles of the components 1.2,3,... distributed among the phases A,B, C,... For the flow of mass between the phases due to an infinitesimal reversible (equilibrium) displacement we can write... 

The isotopic difference between the mean squares of the displacements in equation (7) can be calculated if the carbon-hydrogen oscillator is treated as a diatomic molecule. It is easily shown that for constant potential the mean square of the displacement from the equilibrium position of the harmonic oscillator will be inversely proportional to the square root of the reduced mass, /x, and hence... 

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