Equilibrium time definition

The elastic and viscoelastic properties of materials are less familiar in chemistry than many other physical properties hence it is necessary to spend a fair amount of time describing the experiments and the observed response of the polymer. There are a large number of possible modes of deformation that might be considered We shall consider only elongation and shear. For each of these we consider the stress associated with a unit strain and the strain associated with a unit stress the former is called the modulus, the latter the compliance. Experiments can be time independent (equilibrium), time dependent (transient), or periodic (dynamic). Just to define and describe these basic combinations takes us into a fair amount of detail and affords some possibilities for confusion. Pay close attention to the definitions of terms and symbols. 

Thermo-reversible has the following meaning if an amorphous polymer is heated to above Tg, it readily reaches thermodynamic equilibrium by definition the sample has then "forgotten" its history, any previous ageing it may have undergone below Tg having been erased. In other words it is completely rejuvenated. Ageing therefore is a thermo-reversible process to which one and the same sample can be subjected an arbitrary number of times. It has just to be retreated each time to the same temperature above Tg. 

The phase space trajectory r (Z), p (Z) is uniquely determined by the initial conditions r (Z = 0) = r p (Z = 0) = p. There are therefore no probabilistic issues in the time evolution from Z = 0 to Z. The only uncertainty stems from the fact that our knowledge of the initial condition is probabilistic in nature. The phase space definition of the equilibrium time correlation function is therefore. 

At equilibrium, by definition, the concentration distribution no longer changes with time, and the Lamm equation is equal to zero. Then, an exponential solution may be found, of the form... 

By analogy with Eq. (3.1), we seek a description for the relationship between stress and strain. The former is the shearing force per unit area, which we symbolize as as in Chap. 2. For shear strain we use the symbol y it is the rate of change of 7 that is involved in the definition of viscosity in Eq. (2.2). As in the analysis of tensile deformation, we write the strain AL/L, but this time AL is in the direction of the force, while L is at right angles to it. These quantities are shown in Fig. 3.6. It is convenient to describe the sample deformation in terms of the angle 6, also shown in Fig. 3.6. For distortion which is independent of time we continue to consider only the equilibrium behavior-stress and strain are proportional with proportionality constant G ... 

Electrochemical cells may be used in either active or passive modes, depending on whether or not a signal, typically a current or voltage, must be actively appHed to the cell in order to evoke an analytically usehil response. Electroanalytical techniques have also been divided into two broad categories, static and dynamic, depending on whether or not current dows in the external circuit (1). In the static case, the system is assumed to be at equilibrium. The term dynamic indicates that the system has been disturbed and is not at equilibrium when the measurement is made. These definitions are often inappropriate because active measurements can be made that hardly disturb the system and passive measurements can be made on systems that are far from equilibrium. The terms static and dynamic also imply some sort of artificial time constraints on the measurement. Active and passive are terms that nonelectrochemists seem to understand more readily than static and dynamic. 

In order to eharaeterize the dewetting kineties more quantitatively, the time dependenee of the average thiekness of the film and the deerease of adsorbed fraetion Fads(0 with time (Fig. 34) are monitored. The standard interpretation of the behavior of sueh quantities is in terms of power laws, ads(0 with some phenomenologieal exponents. From Fig. 34(a), where sueh power-law behavior is indeed observed, one finds that the exponent a is about 2/3 or 3/4 for small e and then deereases smoothly to a value very elose to zero at the eritieal value e k. —. 2 where the equilibrium adsorbed fraetion F s(l l) starts to be definitely nonzero. If, instead, one analyzes the time dependenee of — F ds(l l) observes a eollapse... 

The growth of crystals—or more generally the solidification of a sohd from a fluid phase—is definitely not an equilibrium problem. Why, therefore, should we discuss here equihbrium thermodynamics, instead of treating directly, for example the coagulation of two atoms and then simply following the growth of the cluster by adding more particles with time ... 

By definition, the equilibrium distribution /o is one which does not depend on time. For simplicity assume further that the system is uniform in space so that /o is not a function of position, and set all external forces F = 0. The LHS of equation 9.32... 

The work of Higbie laid the basis of the penetration theory in which it is assumed that the eddies in the fluid bring an element of fluid to the interface where it is exposed to the second phase for a definite interval of time, after which the surface element is mixed with the bulk again. Thus, fluid whose initial composition corresponds with that of the bulk fluid remote from the interface is suddenly exposed to the second phase. It is assumed that equilibrium is immediately attained by the surface layers, that a process... 

The response time (relaxation time, adjustment time) of a reservoir is a time scale that characterizes the adjustment to equilibrium after a sudden change in the system. A precise definition is not easy to give except in special circumstances like in the following example. 

Figure 5. A schematic representation of superposed steady-state reservoirs of constant volumes Vi (fractional crystallization is omitted in this schema). At steady-state, Vi/xi=V2/x2=..., where x is the residence time. This is analogous to the law of radioactive equilibrium between nuclides 1 and 2 Ni/Ti=N2/T2=...A further interest of this simple model is to show that residence times by definition depend on the volume of the reservoirs.

That the terminal acceleration should most likely vanish is true almost by definition of the steady state the system returns to equilibrium with a constant velocity that is proportional to the initial displacement, and hence the acceleration must be zero. It is stressed that this result only holds in the intermediate regime, for x not too large. Hence and in particular, this constant velocity (linear decrease in displacement with time) is not inconsistent with the exponential return to equilibrium that is conventionally predicted by the Langevin equation, since the present analysis cannot be extrapolated directly beyond the small time regime where the exponential can be approximated by a linear function. 

For conservative systems with time-independent Hamiltonian the density operator may be defined as a function of one or more quantum-mechanical operators A, i.e. g= tp( A). This definition implies that for statistical equilibrium of an ensemble of conservative systems, the density operator depends only on constants of the motion. The most important case is g= [Pg.463]

This technique was employed to study the binding dynamics of Pyronine Y (31) and B (32) with /)-CD/ s The theoretical background for this particular system has been discussed with the description of the technique above. Separate analysis of the individual correlation curves obtained was difficult since the diffusion time for the complex could not be determined directly because, even at the highest concentration of CD employed, about 20% of the guest molecules were still free in solution. The curves were therefore analyzed using global analysis to obtain the dissociation rate constant for the 1 1 complex (Table 12). The association rate constant was then calculated from the definition of the equilibrium constant. 

It will be obvious from the description of Lewis s and Donnan and Barker s experiments that equilibrium is assumed to establish itself during the time of contact between the mercury or air surface and the liquid in fact this point was checked by increasing the time and showing that the result was not affected, i.e., that no further quantity of the solute was removed from solution. Experiments to decide this question had, however, been made at an earlier date by Wilhelm Ostwald. The strict definition of an equilibrium requires that it should be independent of the mass of the phases in contact thus, a soluble substance and its concentrated solution are in equilibrium at a given temperature and pressure, and this obviously remains unaffected by altering the quantity of either solid substance or solution. Ostwald placed a quantity of charcoal in a given volume of dilute hydrochloric acid and determined the decrease in concentration after a short time. If, then, a part of either the charcoal or the dilute solution was... 

On the other hand, in a sufficient number of cases a definite equilibrium is undoubtedly reached in a short time, and if we confine ourselves to these, it becomes possible to approach the second question we have put, that referring to the connection between concentration and amount adsorbed. Among the investigators who have treated this problem both mathematically and experimentally Freundlich deserves to be mentioned particularly. 

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