Entropy Subject

Alhassid, Y., and Levine, R. D. (1977), Entropy and Chemical Change IU The Maximal Entropy (Subject to Constraints) Procedure as a Dynamical Theory, J. Chem. Phys. 67, 4321. 

The distribution of fragment size is computed as one of the maximal entropy subject to two constraints (1) conservation of matter ... 

The reason we employ two rather distinct methods of inquiry is that neither, by itself, is free of open methodological issues. The method of molecular dynamics has been extensively applied, inter alia, to cluster impact. However, there are two problems. One is that the results are only as reliable as the potential energy function that is used as input. For a problem containing many open shell reactive atoms, one does not have well tested semiempirical approximations for the potential. We used the many body potential which we used for the reactive system in our earlier studies on rare gas clusters containing several N2/O2 molecules (see Sec. 3.4). The other limitation of the MD simulation is that it fails to incorporate the possibility of electronic excitation. This will be discussed fmther below. The second method that we used is, in many ways, complementary to MD. It does not require the potential as an input and it can readily allow for electronically excited as well as for charged products. It seeks to compute that distribution of products which is of maximal entropy subject to the constraints on the system (conservation of chemical elements, charge and... 

Keck, James C. Rate Controlled Constrained Equilibrium Method of Treating Reactions in Complex Systems Maximize the Entropy Subject to Constraints, proceedings of the conference on Maximum Entropy Formalism, M.I.T. Press, 1979. 

Experimental and computational results often do deviate from the distributions of maximal entropy subject only to a given total strength. Consider the following simple modification. The spectrum can be regarded as the set of expectation values of the state Pi... 

The entropy of P(y) is now given in terms of the deviance of P y) from yv/2-i The result of seeking the distribution of maximum entropy subject to a given total strength, , cf. (2.15), leads to... 

We now impose not only a given overall strength but also a given envelope. The average local intensity is thus also specified. This requires the introduction of the set of N constraints (3.13), each of which is assigned the Lagrange multiplier yf. The distribution of maximal entropy subject to the three constraints (1)—(3)is... 

The subsystems exchange energy, volume, and particles until they come to equilibrium. The equilibrium states XJ, XJj are those states which maximize the total entropy subject to the constraint Xt = + XB. The equilibrium entropy is therefore... 

In thermal equilibrium, the most probable distribution is the one that maximizes the entropy, subject to any constraints that may act on the system. These results are easily generalized to systems where the degrees of freedom are not necessarily those of position and momentum — e.g., two-state systems. For simplicity, in the following discussion, we consider position and momentum. Also for simplicity, we drop the index i and consider the vectors p and q to denote the positions and momenta of all of the particles e-g P = iPx,u Py,u Pz,u Px,2 Py,2 Pz,2-.), where 1,2... are the labels of the individual particles. 

A general prerequisite for the existence of a stable interface between two phases is that the free energy of formation of the interface be positive were it negative or zero, fluctuations would lead to complete dispersion of one phase in another. As implied, thermodynamics constitutes an important discipline within the general subject. It is one in which surface area joins the usual extensive quantities of mass and volume and in which surface tension and surface composition join the usual intensive quantities of pressure, temperature, and bulk composition. The thermodynamic functions of free energy, enthalpy and entropy can be defined for an interface as well as for a bulk portion of matter. Chapters II and ni are based on a rich history of thermodynamic studies of the liquid interface. The phase behavior of liquid films enters in Chapter IV, and the electrical potential and charge are added as thermodynamic variables in Chapter V. 

Dislocation theory as a portion of the subject of solid-state physics is somewhat beyond the scope of this book, but it is desirable to examine the subject briefly in terms of its implications in surface chemistry. Perhaps the most elementary type of defect is that of an extra or interstitial atom—Frenkel defect [110]—or a missing atom or vacancy—Schottky defect [111]. Such point defects play an important role in the treatment of diffusion and electrical conductivities in solids and the solubility of a salt in the host lattice of another or different valence type [112]. Point defects have a thermodynamic basis for their existence in terms of the energy and entropy of their formation, the situation is similar to the formation of isolated holes and erratic atoms on a surface. Dislocations, on the other hand, may be viewed as an organized concentration of point defects they are lattice defects and play an important role in the mechanism of the plastic deformation of solids. Lattice defects or dislocations are not thermodynamic in the sense of the point defects their formation is intimately connected with the mechanism of nucleation and crystal growth (see Section IX-4), and they constitute an important source of surface imperfection. 

Molecular moments of inertia are about 10 g/cm thus 7 values for benzene, N2, and NH3 are 18, 1.4, and 0.28, respectively, in those units. For the case of benzene gas, a = 6 and n = 3, and 5rot is about 21 cal K mol at 25°C. On adsorption, all of this entropy would be lost if the benzene were unable to rotate, and part of it if, say, rotation about only one axis were possible (as might be the situation if the benzene was subject only to the constraint of lying flat... 

Equation (A2.1.21) includes, as a special case, the statement dS > 0 for adiabatic processes (for which Dq = 0) and, a fortiori, the same statement about processes that may occur in an isolated system (Dq = T)w = 0). If the universe is an isolated system (an assumption that, however plausible, is not yet subject to experimental verification), the first and second laws lead to the famous statement of Clausius The energy of the universe is constant the entropy of the universe tends always toward a maximum. ... 

This probability distribution can be found by extremizing the generalization of the entropy Eq. (1) subject to the constraints... 

The Boltzmann distribution is fundamental to statistical mechanics. The Boltzmann distribution is derived by maximising the entropy of the system (in accordance with the second law of thermodynamics) subject to the constraints on the system. Let us consider a system containing N particles (atoms or molecules) such that the energy levels of the... 

Water-soluble globular proteins usually have an interior composed almost entirely of non polar, hydrophobic amino acids such as phenylalanine, tryptophan, valine and leucine witl polar and charged amino acids such as lysine and arginine located on the surface of thi molecule. This packing of hydrophobic residues is a consequence of the hydrophobic effeci which is the most important factor that contributes to protein stability. The molecula basis for the hydrophobic effect continues to be the subject of some debate but is general considered to be entropic in origin. Moreover, it is the entropy change of the solvent that i... 

The subject of entropy is introduced here to illustrate treatment of experimental data sets as distinct from continuous theoretical functions like Eq. (1-33). Thermodynamics and physical chemistry texts develop the equation... 

The equations we have written until now in this section impose no restrictions on the species they describe or on the origin of the interaction energy. Volume and entropy effects associated with reaction (8.A) will be less if x is not too large. Aside from this consideration, any of the intermolecular forces listed above could be responsible for the specific value of x- The relationships for ASj in the last section are based on a specific model and are subject to whatever limitations that imposes. There is nothing in the formalism for AH that we have developed until now that is obviously inapplicable to certain specific systems. In the next section we shall introduce another approximation... 

The conformational characteristics of PVF are the subject of several studies (53,65). The rotational isomeric state (RIS) model has been used to calculate mean square end-to-end distance, dipole moments, and conformational entropies. C-nmr chemical shifts are in agreement with these predictions (66). The stiffness parameter (5) has been calculated (67) using the relationship between chain stiffness and cross-sectional area (68). In comparison to polyethylene, PVF has greater chain stiffness which decreases melting entropy, ie, (AS ) = 8.58 J/(molK) [2.05 cal/(molK)] versus... 

The ring-opening polymerization of is controUed by entropy, because thermodynamically all bonds in the monomer and polymer are approximately the same (21). The molar cycHzation equihbrium constants of dimethyl siloxane rings have been predicted by the Jacobson-Stockmayer theory (85). The ring—chain equihbrium for siloxane polymers has been studied in detail and is the subject of several reviews (82,83,86—89). The equihbrium constant of the formation of each cycHc is approximately equal to the equihbrium concentration of this cycHc, [(SiR O) Thus the total... 

The actual amount and stmcture of this "bound" water has been the subject of debate (83), but the key factor is that in water, PVP and related polymers are water stmcture organi2ers, which is a lower entropy situation (84). Therefore, it is not unexpected that water would play a significant role in the homopolymeri2ation of VP, because the polymer and its reactive terminus are more rigidly constrained in this solvent and termination k is reduced... 

Chemistry can be divided (somewhat arbitrarily) into the study of structures, equilibria, and rates. Chemical structure is ultimately described by the methods of quantum mechanics equilibrium phenomena are studied by statistical mechanics and thermodynamics and the study of rates constitutes the subject of kinetics. Kinetics can be subdivided into physical kinetics, dealing with physical phenomena such as diffusion and viscosity, and chemical kinetics, which deals with the rates of chemical reactions (including both covalent and noncovalent bond changes). Students of thermodynamics learn that quantities such as changes in enthalpy and entropy depend only upon the initial and hnal states of a system consequently thermodynamics cannot yield any information about intervening states of the system. It is precisely these intermediate states that constitute the subject matter of chemical kinetics. A thorough study of any chemical reaction must therefore include structural, equilibrium, and kinetic investigations. 

For all common substances the temperature coefficient of t is negative that is to say, the temperature coefficient of 1/e is positive hence (11) represents a loss of entropy on subjecting the homogeneous dielectric to the field. 

Entropy is often described as a measure of disorder or randomness. While useful, these terms are subjective and should be used cautiously. It is better to think about entropic changes in terms of the change in the number of microstates of the system. Microstates are different ways in which molecules can be distributed. An increase in the number of possible microstates (i.e., disorder) results in an increase of entropy. Entropy treats tine randomness factor quantitatively. Rudolf Clausius gave it the symbol S for no particular reason. In general, the more random the state, the larger the number of its possible microstates, the more probable the state, thus the greater its entropy. 

In Chapter 1 we described the fundamental thermodynamic properties internal energy U and entropy S. They are the subjects of the First and Second Laws of Thermodynamics. These laws not only provide the mathematical relationships we need to calculate changes in U, S, H,A, and G, but also allow us to predict spontaneity and the point of equilibrium in a chemical process. The mathematical relationships provided by the laws are numerous, and we want to move ahead now to develop these equations.1... 

In the previous chapter, we saw that entropy is the subject of the Second Law of Thermodynamics, and that the Second Law enabled us to calculate changes in entropy AS. Another important generalization concerning entropy is known as the Third Law of Thermodynamics. It states that ... 

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