Equilibrium number

When there are ti interstitial defects distributed on N sites each requiring E l energy of formation, the equilibrium number of these at temperature T is given by... 

Since in most practical circumstances at temperatures where vapour transport is used and at around one atmosphere pressure, die atomic species play a minor role in the distribution of atoms, it is simpler to cast the distribution equations in terms of the elemental molecular species, H2, O2 and S2, tire base molecules, and the derived molecules H2O, H2S, SO2 and SO3, and eliminate any consideration of the atomic species. In this case, let X, be tire initial mole fraction of each atomic species in the original total of atoms, aird the variables Xi represent the equilibrium number of each molecular species in the final number of molecules, N/. Introducing tire equilibrium constants for the formation of each molecule from tire elemental atomic species, with a total pressure of one aurros, we can write... 

The state variables define a point on the diagram the "constitution point". If this point is given, then the equilibrium number of phases can be read off. So, too, can their composition and the quantity of each phase - but that comes later. So the diagram tells you the entire constitution of any given alloy, at equilibrium. Refer back to the definition of eonstitution (p. 311) and check that this is so. 

The lamellar thickening proceeds through many metastable states, each metastable state corresponding to a particular number of folds per chain, as illustrated in Fig. 8. In the original simulations of [22], Kg was monitored. Rg is actually very close to the lamellar thickness due to the asymmetric shape of the lamella. The number of folds indicated in Fig. 8 were identified by inspection of the coordinates of the united atoms. This quantization of the number of folds has been observed in experiments [50], as already mentioned. The process by which a state with p folds changes into a state with p - 1 folds is highly cooperative. The precursor lives in a quiescent state for a substantial time and suddenly it converts into the next state. By a succession of such processes, crystals thicken. If the simulation is run for a reasonably long time, the lamella settles down to the equilibrium number of folds per chain. 

Use the formula Eq. (2.44) for chemical potential to show that the equilibrium number abundance of a nucleus i with mass number A, = Z, + /V, partition function u, and binding energy B, with respect to free protons and neutrons is given by... 

The frequency vx is interpreted as an attempt frequency that describes the number of times per second that the atom tries to escape. The rest of the expression represents the probability of escape at any given attempt. The term a /ay specifies the ratio of the window sizes at A and X through which the atom has to pass. The final term is a temperature-dependent Boltzmann factor that specifies the ratio of equilibrium number densities at A and X. 

Figure 2.1 Change in Gibbs energy, AG, of a crystal as a function of the number of point defects present (a) Variation of AG with number of point defects (schematic) at equilibrium, neq defects are present in the crystal (b) calculated variation of AG for hy = 0.6 eV, kT = 0.1, N = 1000 the equilibrium number of point defects is 2.5 per 1000.

A plot of this equation (Fig. 2.1 b) closely resembles Figure 2.1a. The minimum in the curve gives the equilibrium number of vacancies present and confirms that vacancies exist in all crystals at temperatures above 0 K. For this reason these defects cannot be removed by thermal treatment but are always present in a crystal. Such defects are thus intrinsic defects. At equilibrium, AGy will be equal to zero and the minimum in the AGy versus tiy curve is given by... 

The papers of Wagner and Schottky contained the first statistical treatment of defect-containing crystals. The point defects were assumed to form an ideal solution in the sense that they are supposed not to interact with each other. The equilibrium number of intrinsic point defects was found by minimizing the Gibbs free energy with respect to the numbers of defects at constant pressure, temperature, and chemical composition. The equilibrium between the crystal of a binary compound and its components was recognized to be a statistical one instead of being uniquely fixed. 

In Eq. (1.36), Nj is the equilibrium number of point defects, N is the total number of atomic sites per volume or mole, Ej is the activation energy for formation of the defect, is Boltzmann s constant (1.38 x 10 J/atom K), and T is absolute temperature. Equation (1.36) is an Arrhenius-type expression of which we will see a great deal in subsequent chapters. Many of these Arrhenius expressions can be derived from the Gibbs free energy, AG. 

Calculate the energy of vacancy formation in aluminum, given that the equilibrium number of vacancies at 500°C is 7.57 x 10 m. State your assumptions. 

In order to construct the expression for the equilibrium number of nuclei in a unit volume (the dimension of b(x)dx is cm-3, the dimension of b(x), when x is defined as the radius, is cm-2), we must multiply the exponent exp(- /fcT), where is determined by (17), by a quantity of dimension cm-2. Exact evaluation of a pre-exponential factor is presently an unsolved problem of statistical mechanics. Erom dimensional considerations we may propose d 2 or x 2, where d is the linear size of a molecule of liquid and x is the radius of a bubble. In the present problem of evaluating the critical (i.e., minimum) value of the equilibrium concentration, we are dealing with a region where the factor in the exponent is large and exact evaluation of the pre-exponential factor is not actually necessary. 

Non-steady solutions of the equation for the probability of formation of a nucleus. Above we sought a steady solution of the fundamental equation (4), which gave us the number of virile (exceeding the critical size) nuclei which form in a unit volume per unit time under constant conditions. As is obvious from the form of solution (15), in a steady state the number of formations of subcritical size does not differ from the equilibrium number ... 

Fig. 2. Temperature dependence of the storage modulus G, loss modulus G", relaxation time t and ratio (n2/nj) of the equilibrium numbers of conformers for a single relaxation time model...

The enthalpies associated with several reactions are listed in Table 5.1 The equilibrium number of defect pairs is given by... 

The equilibrium number of vacancies depends exponentially on temperature ... 

This expression describes the number of vibrons Nq in a nonequilibrium state, Nq = fg (uiq) is the equilibrium number of vibrons. In the linear approximation the polarization operator is independent of Nq and — 2lmIIR(u q) describes additional dissipation. Note that in equilibrium Nq = Nq because lmll<(ujq) = 2 mnR(u)q)fg(wq). See also detailed discussion of vibron emission and absorption rates in Refs. [113-116]. 

To find c, we construct a table giving the initial, reacting, and equilibrium number of moles, mole fractions, and partial pressures. 

A neutrino lighter than 1 MeV decouples while relativistic. If it is so light to be still relativistic today (rnv 0.1 meV), its relic density is pp = 77t27 / I20. If it became non-relativistic after decoupling, its relic density is determined by its equilibrium number density as pv = mI,3C(3)T,3/27r2. Here Tv = (3/ll)1//3T7, where T7 = 2.725 0.002K is the cosmic microwave background temperature. (We use natural units, c- h, I.)... 

According to Eq. (23), the equilibrium number of chains in conformation c depends on the number gc of arrangements of a chain in that conformation, the conformational energy Ec, the surface parameter rjs, as well as the weight factors due to the chain-solvent interactions (/>,) and to the nearest-neighbor bond correlations In the above for-... 

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