Entropy ideas

The first question we pose is of wider generality than is needed to consider the question of electron occupancies. In particular, we ask for the relevant probability distribution for those problems in which not only is the average value of the energy prescribed, but so too is the average number of particles, N). In this case, using the maximum entropy ideas introduced above, the entropy and its associated constraints may be written as... 

The use of business entropy idea allows evaluating the quality of business processes, in particular, for the obtaining of the values of their critical parameters. 

As we have seen, the third law of thermodynamics is closely tied to a statistical view of entropy. It is hard to discuss its implications from the exclusively macroscopic view of classical themiodynamics, but the problems become almost trivial when the molecular view of statistical themiodynamics is introduced. Guggenlieim (1949) has noted that the usefiihiess of a molecular view is not unique to the situation of substances at low temperatures, that there are other limiting situations where molecular ideas are helpfid in interpreting general experimental results ... 

Since entropy plays the determining role in the elasticity of an ideal elastomer, let us review a couple of ideas about this important thermodynamic variable ... 

For the evaporation process we mentioned above, the thermodynamic probability of the gas phase is given by the number of places a molecule can occupy in the vapor. This, in turn, is proportional to the volume of the gas (subscript g) 12- oc V In the last chapter we discussed the free volume in a liquid. The total free volume in a liquid is a measure of places for molecules to occupy in the liquid. The thermodynamic probability of a liquid (subscript 1) is thus V, oc V, frgg. Based on these ideas, the entropy of the evaporation process can be written as... 

The quantity in parentheses is always positive for a > 1, the case of elongation, making AS < 0 for stretching. Therefore AS is positive for the opposite process, showing that entropy alone is sufficient to explain the elastomer s snap. To get an idea of the magnitude of this entropy effect, consider the following example. 

In Chap. 8 we discuss the thermodynamics of polymer solutions, specifically with respect to phase separation and osmotic pressure. We shall devote considerable attention to statistical models to describe both the entropy and the enthalpy of mixtures. Of particular interest is the idea that the thermodynamic... 

The solvophobic model of Hquid-phase nonideaHty takes into account solute—solvent interactions on the molecular level. In this view, all dissolved molecules expose microsurface area to the surrounding solvent and are acted on by the so-called solvophobic forces (41). These forces, which involve both enthalpy and entropy effects, are described generally by a branch of solution thermodynamics known as solvophobic theory. This general solution interaction approach takes into account the effect of the solvent on partitioning by considering two hypothetical steps. Eirst, cavities in the solvent must be created to contain the partitioned species. Second, the partitioned species is placed in the cavities, where interactions can occur with the surrounding solvent. The idea of solvophobic forces has been used to estimate such diverse physical properties as absorbabiHty, Henry s constant, and aqueous solubiHty (41—44). A principal drawback is calculational complexity and difficulty of finding values for the model input parameters. 

Regular Solution Theory. The key assumption in regular-solution theory is that the excess entropy, is zero when mixing occurs at constant volume (3,18). This idea of a regular solution (26) leads to the equations ... 

A number of groups have criticized the ideas of Dauben and Noyce, especially the concept of PDC. Kamernitzsky and Akhrem, " in a thorough survey of the stereochemistry of addition reactions to carbonyl groups, accepted the existence of SAC but not of PDC. They point out that the reactions involve low energies of activation (10-13 kcal/mole) and suggest that differences in stereochemistry involve differences in entropies of activation. The effect favoring the equatorial alcohols is attributed to an electrostatic or polar factor (see also ref. 189) which may be determined by a difference in the electrostatic fields on the upper and lower sides of the carbonyl double bond, connected, for example, with the uncompensated dipole moments of the C—H bonds. The way this polar effect is supposed to influence the attack of the hydride is not made clear. 

A successful method to obtain dynamical information from computer simulations of quantum systems has recently been proposed by Gubernatis and coworkers [167-169]. It uses concepts from probability theory and Bayesian logic to solve the analytic continuation problem in order to obtain real-time dynamical information from imaginary-time computer simulation data. The method has become known under the name maximum entropy (MaxEnt), and has a wide range of applications in other fields apart from physics. Here we review some of the main ideas of this method and an application [175] to the model fluid described in the previous section. 

Equation (5-43) has the practical advantage over Eq. (5-40) that the partition functions in (5-40) are difficult or impossible to evaluate, whereas the presence of the equilibrium constant in (5-43) permits us to introduce the well-developed ideas of thermodynamics into the kinetic problem. We define the quantities AG, A//, and A5 as, respectively, the standard free energy of activation, enthalpy of activation, and entropy of activation from thermodynamics we now can write... 

It would appear that the tradeoffs between these two requirements are optimized at the phase transition. Langton also cites a very similar relationship found by Crutchfield [crutch90] between a measure of machine complexity and the (per-symbol) entropy for the logistic map. The fact that the complexity/entropy relationship is so similar between two different classes of dynamical systems in turn suggests that what we are observing may be of fundamental importance complexity generically increases with randomness up until a phase transition is reached, beyond which further increases in randomness decrease complexity. We will have many occasions to return to this basic idea. 

Returning now to silver chloride, let us apply these ideas to its saturated aqueous solution at 25°. From the value given in Table 42, we see that in solid AgCl the entropy per ion pair is almost exactly 1 milli-electron-volt per degree, which is equivalent to 23.0 cal/deg/mole. It makes no difference whether we express the entropies per ion pair in electron-volts per degree or in the equivalent calories per degree per mole. In the electrochemical literature the calorie per degree per mole is used and is called one entropy unit. (This is abbreviated e.u.) ... 

What Are the Key Ideas Tlic direction of natural change coi responds 10 the increasing disorder of energy and matter. Disorder is measured by the thermodynamic quantity called entropy. A related quantity—the Gibbs free energy—provides a link between thermodynamics and the description of chemical equilibrium. 

In the language of thermodynamics, this simple idea is expressed as the entropy,... 

Some changes are accompanied by a change in volume. Because a larger volume provides a greater range of locations for the molecules, we can expect the positional disorder of a gas and therefore its entropy to increase as the volume it occupies is increased. Once again, we can use Eq. 1 to rum this intuitive idea into a quantitative expression of the entropy change for the isothermal expansion of an ideal gas. 

The expressions in Eq. 1 and Eq. 6 are two different definitions of entropy. The first was established by considerations of the behavior of bulk matter and the second by statistical analysis of molecular behavior. To verify that the two definitions are essentially the same we need to show that the entropy changes predicted by Eq. 6 are the same as those deduced from Eq. 1. To do so, we will show that the Boltzmann formula predicts the correct form of the volume dependence of the entropy of an ideal gas (Eq. 3a). More detailed calculations show that the two definitions are consistent with each other in every respect. In the process of developing these ideas, we shall also deepen our understanding of what we mean by disorder. ... 

What Are the Key Ideas Equilibrium between two phases is reached when the rates of conversion between the two phases are the same in each direction. The rates are equal when the molar Gibbs free energy of the substance is the same in each phase and therefore there is no tendency to change in either direction. The same concepts apply to the dissolving of a solute. The presence of a solute alters the entropy of a solvent and consequently affects its thermodynamic properties. 

Qualitative and quantitative relations between enthalpy and entropy were observed several times in the 1920 s, and their importance was rightly recognized by some authors. However, some ideas from this early work seem to have been overlooked later, perhaps because they were connected with obsolete theories or because they were developed independently in the fields of organic chemistry, catalysis, and pure physical chemistry. For this reason, a brief historical survey seems appropriate. 

The third law of thermodynamics establishes a starting point for entropies. At 0 K, any pure perfect crystal is completely constrained and has S = 0 J / K. At any higher temperature, the substance has a positive entropy that depends on the conditions. The molar entropies of many pure substances have been measured at standard thermodynamic conditions, P ° = 1 bar. The same thermodynamic tables that list standard enthalpies of formation usually also list standard molar entropies, designated S °, fbr T — 298 K. Table 14-2 lists representative values of S to give you an idea of the magnitudes of absolute entropies. Appendix D contains a more extensive list. 

The antibody-catalyzed Diels-Alder reaction developed by Schultz utilized a Diel-Alderase enzyme-like catalyst evolved from an antibody-combining site (Eq. 12.13). The idea is that the generation of antibodies to a structure that mimics the transition state for the Diels-Alder reaction should result in an antibody-combining site that lowers the entropy of activation by binding both the diene and dienophile in a reactive conformation. 

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