State macroscopic

Relationships from thennodynamics provide other views of pressure as a macroscopic state variable. Pressure, temperature, volume and/or composition often are the controllable independent variables used to constrain equilibrium states of chemical or physical systems. For fluids that do not support shears, the pressure, P, at any point in the system is the same in all directions and, when gravity or other accelerations can be neglected, is constant tliroughout the system. That is, the equilibrium state of the system is subject to a hydrostatic pressure. The fiindamental differential equations of thennodynamics ... 

The term collectivism has sometimes been used to distinguish this AL philosophy from the more traditional top down and bottom up philosophies. Collectivism embodies the belief that in order to properly understand complex systems, such systems must be viewed as coherent wholes whose open-ended evolution is continuously fueled by nonlinear feedback between their macroscopic states and microscopic constituents. It is neither completely reductionist (which seeks only to decompose a system into its primitive components), nor completely synthesist (which seeks to synthesize the system out of its constituent parts but neglects the feedback between emerging levels). 

Self-organization seems to be counterintuitive, since the order that is generated challenges the paradigm of increasing disorder based on the second law of thermodynamics. In statistical thermodynamics, entropy is the number of possible microstates for a macroscopic state. Since, in an ordered state, the number of possible microstates is smaller than for a more disordered state, it follows that a self-organized system has a lower entropy. However, the two need not contradict each other it is possible to reduce the entropy in a part of a system while it increases in another. A few of the system s macroscopic degrees of freedom can become more ordered at the expense of microscopic disorder. This is valid even for isolated, closed systems. Eurthermore, in an open system, the entropy production can be transferred to the environment, so that here even the overall entropy in the entire system can be reduced. 

Almost all problems that require knowledge of free energies are naturally formulated or can be framed in terms of (1.15) or (1.16). Systems 0 and 1 may differ in several ways. For example, they may be characterized by different values of a macroscopic parameter, such as the temperature. Alternatively, they may be defined by two different Hamiltonians, 3%o and 3%, as is the case in studies of free energy changes upon point mutation of one or several amino acids in a protein. Finally, the definitions of 0 and 1 can be naturally extended to describe two different, well-defined macroscopic states of the same system. Then, Q0 is defined as ... 

One of the most powerful tools molecular simulation affords is that of measuring distribution functions and sampling probabilities. That is, we can easily measure the frequencies with which various macroscopic states of a system are visited at a given set of conditions - e.g., composition, temperature, density. We may, for example, be interested in the distribution of densities sampled by a liquid at fixed pressure or that of the end-to-end distance explored by a long polymer chain. Such investigations are concerned with fluctuations in the thermodynamic ensemble of interest, and are fundamentally connected with the underlying statistical-mechanical properties of a system. 

In thermodynamics the state of a system is specified in terms of macroscopic state variables such as volume, V, temperature, T, pressure,/ , and the number of moles of the chemical constituents i, tij. The laws of thermodynamics are founded on the concepts of internal energy (U), and entropy (S), which are functions of the state variables. Thermodynamic variables are categorized as intensive or extensive. Variables that are proportional to the size of the system (e.g. volume and internal energy) are called extensive variables, whereas variables that specify a property that is independent of the size of the system (e.g. temperature and pressure) are called intensive variables. 

Frequently, however, the stability and, more generally, the microstructure and the macroscopic states of dispersions are determined by kinetic and thermodynamic considerations. Thermodynamics dictates what the equilibrium state will be, but it is often the kinetics that determines if that equilibrium state will be reached and how fast. This becomes a consideration of special importance in practice since most processing operations involve dynamic variables such as flow, sedimentation, buoyancy, and the like. Although a detailed discussion of this is beyond our scope here, it is important that we consider at least one example so that we can place some of the topics we discuss in this chapter in proper context. 

IL-5 There exists a macroscopic state property U ( internal energy ) whose infinitesimal changes in processes involving only differential absorption of heat dq or performance of work dw on the system are given by dU = dq + dw. 

IL-6 There exists a macroscopic state property S ( entropy ) that achieves the character of a maximum with respect to variations that do not alter the energy of an isolated system at equilibrium, and whose differential changes at equilibrium are given by dS = dq/T. 

FUNDAMENTAL DEFINITIONS SYSTEM, PROPERTY, MACROSCOPIC, STATE... 

To a chemist the entropy of a system is a macroscopic state function, i.e., a function of the thermodynamic variables of the system. In statistical mechanics, entropy is a mesoscopic quantity, i.e., a functional of the probability distribution, viz., the functional given by (V.5.6) and (V.5.7). It is never a microscopic quantity, because on the microscopic level there is no irreversibility. ... 

There may be any number of them. From the fact that the master equation (unless decomposable or splitting) has a single stationary solution Ps one cannot conclude that the macroscopic equation cannot have more than one stationary macroscopic state, as will be seen in XIII. 1. 

The probability of finding a system in a given macroscopic state (i.e., given molecular weight) depends upon the multiplicity of that state, which is proportional... 

The system of our choice will usually prevail in a certain macroscopic state, which is not under the influence of external forces. In equilibrium, the state can be characterized by state properties such as pressure (P) and temperature (T), which are called "intensive properties." Equally, the state can be characterized by extensive properties such as volume (V), internal energy (U), enthalpy (H), entropy (S), Gibbs energy (G), and Helmholtz energy (A). 

Consider a set of 10 coins that forms the system. The most ordered macroscopic states are 10 heads up or 10 tails up. In either case, there is one possible configuration, and hence the entropy according to Eq. 7.4-13 is zero. The most disordered state consists of 5 heads up and 5 tails up, which allows for 252 different configurations. This would be the case if we reflip each coin sequentially for a long time. What we have done is mix the system. 

Entropy is interpreted as the number of microscopic arrangements included in the macroscopic definition of a system. The second law is then used to derive the distribution of molecules and systems over their states. This allows macroscopic state functions to be calculated from microscopic states by statistical methods. 

The probability of a macroscopic state is proportional to the number of distinguishable microscopic configurations that are consistent with the definition of that state. 

Indeed, since the macroscopic states of a protein are discrete, they are described by discrete surfaces in the phase space of considered variables (Pfeil and Privalov, 1976c). The small globular proteins, or individual cooperative domains, which have only two stable macroscopic states, the native (N) and denatured (D), are described by two surfaces in the phase space, corresponding to their extensive thermodynamic functions. The transition between these states is determined by the differences of... 

It should be emphasized that the native and denatured states of a protein depend on the environmental conditions, but these modify the protein states gradually and cannot be considered as phase transitions, i.e., as transitions between macroscopic states, but only as transitions between microscopic states, corresponding to the same macroscopic state (see, e.g., Griko et al., 1988a). 

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