Equilibrium systems entropy

For equilibrium systems, diemiodynamic entropy is related to ensemble density distribution p as... 

To close this chapter we emphasize that Hie statistical mechanical definition of macroscopic parameters such as temperature and entropy are well designed to describe isentropic equilibrium systems, but are not immediately applicable to the discussion of transport processes where irreversible entropy increase is an essential feature. A macroscopic system through which heat is flowing does not possess a single tempera-... 

Next, an exploration of thermodynamics and equilibrium, based on a conceptual understanding of entropy and Gihbs free energy. This integrated presentation lays a common foundation for these concepts and provides a basis for understanding the origin and form of the equilibrium constant and the behavior of equilibrium systems. 

The generic case is a subsystem with phase function x(T) that can be exchanged with a reservoir that imposes a thermodynamic force Xr. (The circumflex denoting a function of phase space will usually be dropped, since the argument T distinguishes the function from the macrostate label x.) This case includes the standard equilibrium systems as well as nonequilibrium systems in steady flux. The probability of a state T is the exponential of the associated entropy, which is the total entropy. However, as usual it is assumed (it can be shown) [9] that the... 

What does entropy have to do with favourable chemical changes and equilibrium systems All favourable changes involve an increase in the total amount of entropy. Recall the endothermic reaction in Figure 7.4. 

Diffusion in general, not only in the case of thin films, is a thermodynamically irreversible self-driven process. It is best defined in simple terms, such as the tendency of two gases to mix when separated by a porous partition. It drives toward an equilibrium maximum-entropy state of a system. It does so by eliminating concentration gradients of, for example, impurity atoms or vacancies in a solid or between physically connected thin films. In the case of two gases separated by a porous partition, it leads eventually to perfect mixing of the two. 

An example of such a system is one in which the internal energy, U, and the volume, V, are constant. If these are the only two constraints on the system then, at thermodynamic equilibrium, the entropy, S, is at a maximum. On the other hand, if entropy and volume are constant for the isolated system then, at thermodynamic equilibrium, the internal energy is at a minimum. See also Closed System Open System... 

Gibbs criterion (I) In an isolated equilibrium system, the entropy function has the mathematical character of a maximum with respect to variations that do not alter the energy. 

In equilibrium thermodynamics, entropy maximization for a system with fixed internal energy determines equilibrium. Entropy increase plays a large role in irreversible thermodynamics. If each of the reference cells were an isolated system, the right-hand side of Eq. 2.4 could only increase in a kinetic process. However, because energy, heat, and mass may flow between cells during kinetic processes, they cannot be treated as isolated systems, and application of the second law must be generalized to the system of interacting cells. 

When dealing with general thermodynamic systems, the fact that entropy tends to a maximum in the trend toward equilibrium of a natural process generalizes the above mechanical consideration with respect to stability. An equilibrium state can be characterized as a stable equilibrium when the entropy is a maximum neutral equilibrium when displacement from one equilibrium state lo another does not involve changing entropy and unstable equilibrium when entropy is a minimum. Any slight disturbance from an unstable equilibrium state or a system will lead to transition to another state of equilibrium. 

Occasionally it is convenient to analyze an equilibrium system or a fixed compn system in terms of volume and entropy. In this case, the coefficients /3 and (defined below) become... 

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. 

It is a general observation that any system that is not in equilibrium will approach equilibrium when left to itself. Such changes of state that take place in an isolated system do so by irreversible processes. However, when a change of state occurs in an isolated system by an irreversible process, the entropy change is always positive (i.e., the entropy increases). Consequently, as the system approaches equilibrium, the entropy increases and will continue to do so until it obtains the largest value consistent with the energy of the system. Thus, if the system is already at equilibrium, the entropy of the system can only decrease or remain unchanged for any possible variation as discussed in Section 5.1. 

This fundamental equation for the entropy shows that S has the natural variables U, V, and n . The corresponding criterion of equilibrium is (dS) 0 at constant U, V, and n . Thus the entropy increases when a spontaneous change occurs at constant U, V, and ,. At equilibrium the entropy is at a maximum. When U, V, and , are constant, we can refer to the system as isolated. Equation 2.2-13 shows that partial derivatives of S yield 1/T, P/T, and pJT, which is the same information that is provided by partial derivatives of U, and so nothing is gained by using equation 2.2-13 rather than 2.2-8. Since equation 2.2-13 does not provide any new information, we will not discuss it further. 

If the system is at equilibrium, the entropy change (of the universe) for this process must be zero ... 

A possible way of the description of reversible polymerization is based on the assumption of maximum entropy in equilibrium systems. Then, different structures could be taken into account based on the analysis of the configurational entropy [8]. However, the problem of the evaluation of the configurational entropy in the general case is very complicated, and this complexity replaces the initial one of the direct evaluation of the weight distribution. 

A system reaches the thermodynamic equilibrium state when it is left for a long time with no external disturbances. At equilibrium the internal properties are fully determined by the external properties. This makes it easy to describe such systems for example, if the temperature is not uniform within the system, heat is exchanged with the immediate surrounding until the system reaches a thermal equilibrium, at which the total internal eneigy U and entropy S are completely specified by the temperature, volume, and number of moles. Therefore, at equilibrium there are no thermodynamic forces operating within the system (Figure 2.1). Equilibrium systems are stable. For small deviations, the system can spontaneously return to the state of equilibrium. Equilibrium correlations result from short-range intermolecular interactions. Existence of the extremum principles is a characteristic property of equilibrium thermodynamics. 

The change of total entropy is dS d,.S + d S. The term dJS is the entropy exchange through the boundary, which can be positive, zero, or negative, while the term dtS is the rate of entropy production, which is always positive for irreversible processes and zero for reversible ones. The rate of entropy production is djS/dt = %JkXk. A near equilibrium system is stable to fluctuations if the change of entropy production is negative A S < 0. For isolated systems,... 

The vividness of our world does not rely on processes that are characterized by linear force-flux relations, rather they rely on the nonlinearity of chemical processes. Let us recapitulate some results for proximity to equilibrium (see also Section VI.2.H.) In equilibrium the entropy production (n) is zero. Out of equilibrium, II = T<5 S/I8f > 0 according to the second law of thermodynamics. In a perturbed system the entropy production decreases while we reestablish equilibrium (II < 0), (Fig. 72). For the cases of interest, the entropy production can be written as a product of fluxes and corresponding forces (see Eq. 108). If some of the external forces are kept constant, equilibrium cannot be achieved, only a steady state occurs. In the linear regime this steady state corresponds to a minimum of entropy production (but nonzero). Again this steady state is stable, since any perturbation corresponds to a higher II-value (<5TI > 0) and n < 0.183 The linear concentration profile in a steady state of a diffusion experiment (described in previous sections) may serve as an example. With... 

Thus, monitoring the behavior of the entropy production rate (or the energy dissipation rate) in near equilibrium systems allows the identifica tion of the system transition to its stationary state. In fact, as the system evolves from some initial state to the stationary state, the possible value ofP = TdjS/dt decreases monotonicaUy but remains positive and gradually approaches the minimal and constant positive value related to the final sta tionary state. 

The Onsager reciprocal relations are not satisfied in open strongly non equilibrium systems. As a result, the assumption on minimization of the entropy production rate is not substantiated. Therefore, the universal criterion of the system that is evolution far from equilibrium should be a generalization of the principle of the minimized entropy production rate in specific terms of nonlinear thermodynamics. 

Finally, we need to remind you about free energy. In many of our discussions we will be trying to establish whether or not a process will occur spontaneously— will this polymer dissolve in that solvent , for example. Conceptually, this is easily done, because once a system reaches equilibrium, its entropy is a maximum, so all we need to do is calculate if the entropy change for that process is positive. This is not so easily done for real systems, however, because they are not isolated from their surroundings. Dis-... 

Figure 2.7. Gibbs function, system enthalpy, and system entropy variations with the extent of reaction for the dissolution of gaseous CO2 in water C02(g) = C02(a

The previously accepted value (2, 3, 4) -280.2 0.4 kcal mol", was based on the pressure-teraperature-composition studies of Huttig and Kurre (5) for the WOj-H20 system. We suspect that the data do not correspond to equilibrium (see entropy section) for the reaction H2WO (cr) = WOg(cr) + H20(g). 

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