Entropy predictions about

It is possible to derive an expression equivalent to Eq. (4.67) starting from entropy rather than free volume concepts. We have emphasized the latter approach, since it is easier to visualize and hence to use for qualitative predictions about Tg. 

We need a quantitative definition of entropy to measure and make precise predictions about disorder. Provided that the temperature is constant, it turns out that a We generalize the definition in the change in the entropy of a system can be calculated from the following expression next section to changes in which the... 

The models proposed lead to a number of predictions about the variations of the enthalpy and entropy changes as ligands of different character are consecutively coordinated to a metal ion. To see whether these changes really occur as expected, experimental data have been collected for hard ligand atoms (F, O Table 1) as well as for more or less soft ones (Cl, Br, I S, Se P C Table 2). 

One then observes that the overall index of Scheme 1.4, Af = 1.58 = H[p°], predicting about 3/2 jr-bond multiplicity in allyl, can be reconstructed by adding to this additive-ionicity measure, the sum of the bonding (positive) entropy-covalency Si of the first MO and the antibonding (negative) contribution (—S2) due to the second MO ... 

The Kauzmann temperature plays an important role in the most widely applied phenomenological theories, namely the configurational entropy [100] and the free-volume theories [101,102]. In the entropy theory, the excess entropy ASex obtained from thermodynamic studies is related to the temperature dependence of the structural relaxation time xa. A similar relation is derived in the free-volume theory, connecting xa with the excess free volume AVex. In both cases, the excess quantity becomes zero at a distinguished temperature where, as a consequence, xa(T) diverges. Although consistent data analyses are sometimes possible, the predictive power of these phenomenological theories is limited. In particular, no predictions about the evolution of relaxation spectra are made. Essentially, they are theories for the temperature dependence of x.-jT) and r (T). 

How can we use the fact that any spontaneous process is irreversible to make predictions about the spontaneity of an unfamiliar process Understanding spontaneity requires us to examine the thermodynamic quantity called entropy, which was first mentioned in Section 13.1. In general, entropy is associated either with the extent of randomness in a system or with the extent to which energy is distributed among the various motions of the molecules of the system. In this section we consider how we can relate entropy changes to heat transfer and temperature. Our analysis will bring us to a profound statement about spontaneity that we call the second law of thermodynamics. 

We can usually make qualitative predictions about entropy changes by focusing on these factors. For example, when water vaporizes, the molecules spread out into a larger volume. Because they occupy a larger volume, there is an increase in their freedom of motion, giving rise to a greater number of possible microstates, and hence an increase in entropy. 

If you observe s) < 3.5, then the maximuni-entropy principle predicts an exponentially decreasing distribution (see Figure 6.3(b)), with more I s than 2 s, more 2 s than 3 s, etc. If you observe (f) > 3.5, then maximmn entropy predicts an exponentially increasing distribution (see Figure 6.3(c)) more 6 s than 5 s, more 5 s than 4 s, etc. For any given value of (f), the exponential or flat distribution gives the most impartial distribution consistent with that score. The flat distribution is predicted either if the average score is 3.5 or if you have no information at all about the score. 

As we said earlier, entropy often is described as a measure of randomness or disorder. Although this can be a useful description, it should be used with caution and not taken too literally. It is generally preferable to view the change in entropy of a system in terms of the change in the number of microstates of the system. Nevertheless, we can use the concept of disorder to make some qualitative predictions about the entropy changes that accompany certain processes. 

Chapter 17 introduced some of the basic concepts that led to the development of a statistical approach to energy and entropy. This is statistical thermodynamics. By the end of the chapter, equations were applied to monatomic gases, and thermodynamic state functions—mostly entropy—were calculated whose values were very close to experimental values. Also, in some of the exercises you were asked to derive some simple expressions that were also derived from phenomenological thermodynamics. For example, we know from early chapters in this book that the equation AS = i In (V2/V1) is applicable for an isothermal change in volume of an ideal gas. We can also get this expression using the Sackur-Tetrode statistical thermodynamic expression for S. These correspondences are just two examples where phenomenological and statistical thermodynamics are consistent with each other. That is, they ultimately make the same predictions about the state functions of a system, and how they change with a process. 

Thermodynamics rests largely on the consohdation of many observations of nature into two fundamental postulates or laws. Chapter 2 addressed the first law— the energy of the universe is conserved. We camiot prove this statement, but based on over a hundred years of observation, we believe it to be true. In order to use this law quantitatively— that is, to make numerical predictions about a system—we cast it in terms of a thermodynamic property internal energy, u. Likewise, the second law summarizes another set of observations about nature. We will see that to quantify the second law, we need to use a different thermodynamic property entropy, s. Like internal energy, entropy is a conceptual property that allows us to quantify a law of nature and solve engineering problems. This chapter examines the observations on which the second law is based explores how the property s quantifies these observations illustrates ways we can use the second law to make numerical predictions about closed systems, open systems, and thermodynamic cycles and discusses the molecular basis of entropy. 

If one would ask a chemist not burdened with any knowledge about the peculiar thermodynamics that characterise hydrophobic hydration, what would happen upon transfer of a nonpolar molecule from the gas phase to water, he or she would probably predict that this process is entropy driven and enthalpically highly unfavourable. This opinion, he or she wo ild support with the suggestion that in order to create room for the nonpolar solute in the aqueous solution, hydrogen bonds between water molecules would have to be sacrificed. 

The discovery of a transition which we identify with this has been reported by Simon, Mendelssohn, and Ruhemann,16 who measured the heat capacity of hydrogen with nA = 1/2 down to 3°K. They found that the heat capacity, after following the Debye curve down to about 11°K, rose at lower temperatures, having the value 0.4 cal/deg., 25 times that of the Debye function, at 3°K. The observed entropy of transition down to 3°K, at which the transition is not completed, was found to be about 0.5 E.U. That predicted by Eq. (15) for the transition is 2.47 E.U. 

In 1950, Pomeranchuck (1913-1966) predicted that for 3He on the melting curve below about 0.3 K, the entropy of liquid is smaller than that of solid. It was only after 15 years that Anufriev [2], after Pomeranchuck s suggestion, succeeded in reducing the temperature from 50 to 18mK in an experiment based on this 3He property. Four years later, the Pomeranchuck method produced a temperature of 2mK [3],... 

In many atomization processes, physical phenomena involved have not yet been understood to such an extent that mean droplet size could be expressed with equations derived directly from first principles, although some attempts have been made to predict droplet size and velocity distributions in sprays through maximum entropy principle.I252 432] Therefore, the correlations proposed by numerous studies on droplet size distributions are mainly empirical in nature. However, the empirical correlations prove to be a practical way to determine droplet sizes from process parameters and relevant physical properties of liquid and gas involved. In addition, these previous studies have provided insightful information about the effects of process parameters and material properties on droplet sizes. 

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