Entropy classic definition

The classical definition of entropy based on the second law of thermodynamics has given the total differential of entropy in the form of dQrev / / . With a reversible heat transfer into a closed system receiving a differential amount of heat dQrev, the system changes its entropy by the differential amount of dS as shown in Eq. 3.8 ... 

The Gibbs free energy is a measure of the probability that a reaction occurs. It is composed of the enthalpy, H, and the entropy, S° (Eq. 5). The enthalpy can be described as the thermodynamic potential, which ensues H = U + p V. where U is the internal energy, p is the pressure, and V is the volume. The entropy, according to classical definitions, is a measure of molecular order of a thermodynamic system and the irreversibility of a process, respectively. 

Two other information indices derived from the indices of neighborhood symmetry were proposed modifying the classical definition of Shaimon s entropy [King, 1989]. 

The classical definition of entropy goes through the Clausius inequality ... 

The reason why the classical definition of cooperativity is not sufficient to explain multivalency effects in interactions between polyvalent carbohydrates with multiple carbohydrate-binding sites can be found when changes of entropy are considered. 

For a classical system, the definition of the statistical entropy in Eq. (26.1-1) cannot be used because we cannot count states in the same way as with a quantum system. The classical definition of the statistical entropy is analogous to that in Problem 27.6. 

There is thus assumed to be a one-to-one correspondence between the most probable distribution and the thermodynamic state. The equilibrium ensemble corresponding to any given thermodynamic state is then used to compute averages over the ensemble of other (not necessarily thermodynamic) properties of the systems represented in the ensemble. The first step in developing this theory is thus a suitable definition of the probability of a distribution in a collection of systems. In classical statistics we are familiar with the fact that the logarithm of the probability of a distribution w[n is — J(n) w n) In w n, and that the classical expression for entropy in the ensemble is20... 

With this definition, the classical entropy per system equals the ensemble average of the expectation value of 8 in occupation number representation. 

It has been seen thus far that the first law, when applied to thermodynamic processes, identifies the existence of a property called the internal energy. It may in other words be stated that analysis of the first law leads to the definition of a derived property known as internal energy. Similarly, the second law, when applied to such processes, leads to the definition of a new property, known as the entropy. Here again it may in other words be said that analysis of the second law leads to the definition of another derived property, the entropy. If the first law is said to be the law of internal energy, then the second law may be called the law of entropy. The three Es, namely energy, equilibrium and entropy, are centrally important in the study of thermodynamics. It is sometimes stated that classical thermodynamics is dominated by the second law. 

In this tribute and memorial to Per-Olov Lowdin we discuss and review the extension of Quantum Mechanics to so-called open dissipative systems via complex deformation techniques of both Hamiltonian and Liouvillian dynamics. The review also covers briefly the emergence of time scales, the definition of the quasibosonic pair entropy as well as the precise quantization relation between the temperature and the phenomenological relaxation time. The issue of microscopic selforganization is approached through the formation of certain units identified as classical Jordan blocks appearing naturally in the generalised dynamical picture. 

Chapter 5 gives a microscopic-world explanation of the second law, and uses Boltzmann s definition of entropy to derive some elementary statistical mechanics relationships. These are used to develop the kinetic theory of gases and derive formulas for thermodynamic functions based on microscopic partition functions. These formulas are apphed to ideal gases, simple polymer mechanics, and the classical approximation to rotations and vibrations of molecules. 

The kinetic theory leads to the definitions of the temperature, pressure, internal energy, heat flow density, diffusion flows, entropy flow, and entropy source in terms of definite integrals of the distribution function with respect to the molecular velocities. The classical phenomenological expressions for the entropy flow and entropy source (the product of flows and forces) follow from the approximate solution of the Boltzmann kinetic equation. This corresponds to the linear nonequilibrium thermodynamics approach of irreversible processes, and to Onsager s symmetry relations with the assumption of local equilibrium. 

Entropy owes its existence to the second law, from wliicli it arises in much the same way as internal energy does from the first law. Equation (5.11) is the ultimate source of all equations that relate the entropy to measurable quantities. It does not represent a definition of entropy there is none in the context of classical themiodynamics. What it provides is tlie means for calculating changes in tills property. Its essential nature is summarized by the following axiom ... 

Standard definitions from classical thermodynamics are used to calculate the enthalpy H, entropy S, and Gibbs free energy G, as functions of the temperature, from the observed Cp(T) ... 

This section summarizes the classical, equilibrium, statistical mechanics of many-particle systems, where the particles are described by their positions, q, and momenta, p. The section begins with a review of the definition of entropy and a derivation of the Boltzmann distribution and discusses the effects of fluctuations about the most probable state of a system. Some worked examples are presented to illustrate the thermodynamics of the nearly ideal gas and the Gaussian probability distribution for fluctuations. 

We stress that we use only the classical partial thermodynamic quantities, calculable (say) by (4.269), (4.270), but there are also other possible definitions, e.g. partial entropies by — (different by (4.272)), cf. [17, 18]. These are, however, not so useful as those classical. 

A simple model for the determination of conformational entropy (ASl) is based on the assumption that accessible conformational space has the same potential energy. From the classical (nonquantum mechanical) definition of the partition function, the conformational entropy is given by the following equation ... 

Classical thennodynamics deals with the interconversion of energy in all its forms including mechanical, thermal and electrical. Helmholtz [1], Gibbs [2,3] and others defined state functions such as enthalpy, heat content and entropy to handle these relationships. State functions describe closed energy states/systems in which the energy conversions occur in equilibrium, reversible paths so that energy is conserved. These notions are more fully described below. State functions were described in Appendix 2A however, statistical thermodynamics derived state functions from statistical arguments based on molecular parameters rather than from basic definitions as summarized below. 

In contrast to other textbooks on thermodynamics, we assume that the readers are familiar with the fundamentals of classical thermodynamics, that means the definitions of quantities like pressure, temperature, internal energy, enthalpy, entropy, and the three laws of thermodynamics, which are very well explained in other textbooks. We therefore restricted ourselves to only a brief introduction and devoted more space to the description of the real behavior of the pure compounds and their mixtures. The ideal gas law is mainly used as a reference state for application examples, the real behavior of gases and liquids is calculated with modern g models, equations of state, and group contribution methods. 

In normal classical statistical mechanics, it is assumed that all states which are fixed by the same external constraints, e.g., total volume V, average energy < ), average particle number N), are equally probable. All possible states of the system are generated and are assigned weight unity if they are consistent with these constraints and, zero otherwise. Thus in the case of an iV-particle system with classical Hamiltonian //j, the microcanonical ensemble entropy S E) is obtained from the total number of states ( ) via the definition... 

With this new definition, classical thermodynamic manipulations can be ap-phed to yield a variety of useful expressions defining properties such as the entropy of interface formation per imit area... 

Our aim is not to introduce the reader into the field of thermodynamic theory, its historical development and problems. The latter are numerous, starting from the general definition of entropy itself. We shall simply assume that the thermodynamic state functions are those found in tables, or computed using the classical thermodynamic relations. For example we have the relations... 

A rigorous interpretation is provided by the discipline of statistical mechanics, which derives a precise expression for entropy based on the behavior of macroscopic amounts of microscopic particles. Suppose we focus our attention on a particular macroscopic equilibrium state. Over a period of time, while the system is in this equilibrium state, the system at each instant is in a microstate, or stationary quantum state, with a definite energy. The microstate is one that is accessible to the system—that is, one whose wave function is compatible with the system s volume and with any other conditions and constraints imposed on the system. The system, while in the equilibrium state, continually jumps from one accessible microstate to another, and the macroscopic state functions described by classical thermodynamics are time averages of these microstates. 

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