Isolated equilibrium state

In summary, Clausius states that entropy strives toward a maximum in isolated processes tending toward equilibrium, while Gibbs states that entropy is at a maximum in isolated equilibrium states. 

Wlien H has reached its minimum value this is the well known Maxwell-Boltzmaim distribution for a gas in themial equilibrium with a unifomi motion u. So, argues Boltzmaim, solutions of his equation for an isolated system approach an equilibrium state, just as real gases seem to do. Up to a negative factor (-/fg, in fact), differences in H are the same as differences in the themiodynamic entropy between initial and final equilibrium states. Boltzmaim thought that his //-tiieorem gave a foundation of the increase in entropy as a result of the collision integral, whose derivation was based on the Stosszahlansatz. 

The second law reqmres that the entropy of an isolated system either increase or, in the limit, where the system has reached an equilibrium state, remain constant. For a closed (but not isolated) system it requires that any entropy decrease in either the system or its surroundings be more than compensated by an entropy increase in the other part or that in the Emit, where the process is reversible, the total entropy of the system plus its surroundings be constant. 

It is clear that the strong form of the QCT is impossible to obtain from either the isolated or open evolution equations for the density matrix or Wigner function. For a generic dynamical system, a localized initial distribution tends to distribute itself over phase space and either continue to evolve in complicated ways (isolated system) or asymptote to an equilibrium state (open system) - whether classically or quantum mechanically. In the case of conditioned evolution, however, the distribution can be localized due to the information gained from the measurement. In order to quantify how this happ ens, let us first apply a cumulant expansion to the (fine-grained) conditioned classical evolution (5), resulting in the equations for the centroids (x = (t), P= (P ,... 

The preparation period consists of the creation of a non-equilibrium state and, possibly, of the frequency labeling in 2D experiments. Usually, the preparation period should be designed in such a way that in the created non-equilibrium state, the population differences or coherences under consideration deviate as much as possible from the equilibrium values. During the relaxation period, the coherences or populations evolve towards an equilibrium (or a steady-state) condition. The behavior of the spin system during this period can be manipulated in order to isolate one specific type of process. The detection period can contain also the mixing period of the 2D experiments. The purpose of the detection period is to create a signal which truthfully reflects the state of the spin system at the end of the relaxation period. As always in NMR, sensitivity is a matter of prime concern. 

The second law of thermodynamics states that an isolated system in equilibrium has maximum entropy. This is the basis for a variational principle often used in determining the equilibrium state of a system. When the system contains several elements which are allowed to exchange mass with each other, the variational principle yields the condition that all elements must have equal chemical potential once equilibrium is established. 

It was the principal genius of J. W. Gibbs (Sidebar 5.1) to recognize how the Clausius statement could be recast in a form that made reference only to the analytical properties of individual equilibrium states. The essence of the Clausius statement is that an isolated system, in evolving toward a state of thermodynamic equilibrium, undergoes a steady increase in the value of the entropy function. Gibbs recognized that, as a consequence of this increase, the entropy function in the eventual equilibrium state must have the character of a mathematical maximum. As a consequence, this extremal character of the entropy function makes possible an analytical characterization of the second law, expressible entirely in terms of state properties of the individual equilibrium state, without reference to cycles, processes, perpetual motion machines, and the like. 

Figures 4.44 and 4.45, best viewed in color, show a benign complication of the problem caused by the Lewis numbers. If, however, we reduce the Lewis number LeA further to 0.07, the system trajectories indicate periodic explosions of the underlying system throughout all time, and the trajectories do not converge to the steady state at all, even with what we thought to be proper feedback. The trajectory that these curves settle at is called a periodic attractor of the system in contradistinction to the earlier encountered point attractor of Figures 4.43 or 4.44, for example. A point attractor, or more accurately a fixed-point attractor, is a more commonly encountered steady state in chemical and biological engineering systems. It could be called a stationary nonequilibrium state to distinguish it from the stationary equilibrium states associated with closed or isolated batch processes.

We know from experience that any isolated system left to itself will change toward some final state that we call a state of equilibrium. We further know that this direction cannot be reversed without the use of some other system external to the original system. From all experience this characteristic of systems progressing toward an equilibrium state seems to be universal, and we call the process of such a change an irreversible process. In order to characterize an irreversible process further, we use one specific example and then discuss the general case. In doing so we always use a cyclic process. 

If the system is isolated and already at equilibrium, then any variation in the state of the system cannot increase the entropy if the energy is kept constant, and cannot decrease the energy if the entropy is kept constant. In other words, for an isolated system at equilibrium, the entropy must have the largest possible value consistent with the energy of the system, and the energy must have the smallest possible value consistent with the entropy of the system. There may be many states of the system that are equilibrium states, but these conditions must be applicable to each such equilibrium state. 

Equation (5.12) effectively corresponds to the dynamics of the individual process units that are part of the recycle loop. The description of the fast dynamics (5.12) involves only the large flow rates u1 of the recycle-loop streams, and does not involve the small feed/product flow rates us or the purge flow rate up. As shown in Chapter 3, it is easy to verify that the large flow rates u1 of the internal streams do not affect the total holdup of any of the components 1,..., C — 1 (which is influenced only by the small flow rates us), or the total holdup of I (which is influenced exclusively by the inflow Fjo, the transfer rate Af in the separator, and the purge stream up). By way of consequence, the differential equations in (5.12) are not independent. Equivalently, the quasi-steady-state condition 0 = G (x)u corresponding to the dynamical system (5.12) does not specify a set of isolated equilibrium points, but, rather, a low-dimensional equilibrium manifold. 

Another example is the thermal and photochemical cis-trans isomerization of Cp2Fe2(CO)2( -CO)( -Sip-TolH).25 In this case, both cis(H) and trans isomers can be isolated at full purity by flash chromatography. Interconversion between these isomers occurs both thermally and photochemi-cally in cyclohexane-d12, and the composition in the thermal equilibrium state (cis(H) trans = 2 98 at 25°C) is extremely different from that in the photostationary state (cis(H) trans = 70 30). Kinetics of the thermal isomerization in decalin afforded the activation parameters shown in Eq. (58). The large negative activation entropies imply that this reaction also... 

All isolated systems move, rapidly or slowly, by one path or another, toward equilibrium. In fact essentially all motion stems from the universal drift to eventual equilibrium. Therefore, if we wish to obtain a certain displacement of a component through some medium, we must generally establish equilibrium conditions that favor the desired displacement. Clearly, a knowledge of the equilibrium state is indispensable to the study of the displacements leading to separation. 

If a physical system is isolated, its state changes irreversibly to a time-invariant state in which no physical or chemical change occurs, and a state of equilibrium is reached in a finite time. Some conditions of equilibrium are (i) for a system thermally insulated with an infinitesimal change at constant volume dS 0, dV = 0, dU = 0, (ii) for a system thermally insulated with an infinitesimal change at constant pressure dS = 0,dP = 0, dH = 0, (iii) for a system thermally insulated with an infinitesimal change at constant volume and temperature dA = 0, dV = 0, dT = 0, and (iv) for a system thermally insulated with an infinitesimal change at constant pressure and temperature dG = 0,dT= 0, dP = 0. 

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