Calculation of AS

AS j can be computed as the difference between the entropy of an ideal gas and the entropy of the crystal at a given temperature and pressure. If the intra- and intermolecular contributions to the entropy of the crystal are considered to be decoupled, such that the change in intramolecular vibrational entropy for transfer from crystal to gas can be taken to be zero, then the sublimation entropy of rigid molecules can be approximated by AS where and are the 

MOLECULAR SIMULATION METHODS TO COMPUTE INTRINSIC AQUEOUS 

A critical understanding of the availability and accuracy of experimental data in the published literature is an important step in developing or testing computational methods. This is especially true since there is currently a lack of accurate and well-documented experimental data for polyfunctional organic molecules [53,133], 

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In equation (1.17), S is entropy, k is a constant known as the Boltzmann constant, and W is the thermodynamic probability. In Chapter 10 we will see how to calculate W. For now, it is sufficient to know that it is equal to the number of arrangements or microstates that a molecule can be in for a particular macrostate. Macrostates with many microstates are those of high probability. Hence, the name thermodynamic probability for W. But macrostates with many microstates are states of high disorder. Thus, on a molecular basis, W, and hence 5, is a measure of the disorder in the system. We will wait for the second law of thermodynamics to make quantitative calculations of AS, the change in S, at which time we will verify the relationship between entropy and disorder. For example, we will show that... 

Calculation of AS for the Reversible Isothermal Expansion of an Ideal Gas Integration of equation (2.38) gives... 

The calculation of AS from equations (2.69) to (2,74), along with equations (2.78) or (2.79), all demonstrate that an increase in entropy causes an increase in disorder. For example ... 

Calculation of As was carried out for hole transfer in finite DNA duplexes in water, in which one strand includes a G3(T) G3 sequence, n = 0-6, where G and T are, respectively, guanine and thymine nucleotide bases (in the complementary strand, of course, G and T are paired respectively, with cytosine and adenine bases) [23], The D and A sites (the solute) were taken as the middle unit of each G3 triad, or as alternative models for the n = 0 case, using one or both of the inner members of the G3 units). These structures lead to D/A sites in contact or separated by intervening bases ranging in number from 1 to 8. Using a base stacking separation of 3.4 A yields rDA = (m+ 1) (3.4 A), m = 0-8 (an estimate closely supported by detailed molecular force field calculations). 

We now proceed to illustrate the practical calculation of AS in a number of specific cases. 

The standard state chosen for the calculation of AS controls its magnitude and even its sign. The standard state is established when the concentration scale is selected. For most solution kinetic work the molar concentration scale is used, so AS values reported by different workers are usually comparable. Nevertheless, an important chemical question is implied Because the sign of AS may depend upon the concentration scale used for the evaluation of the rate constant, which concentration scale should be used when AS is to serve as a mechanistic criterion The same question appears in studies of equilibria. The answer (if there is a single answer) is not known, though some analyses of the problem have been made. Further discussion of this issue is given in Section 6.1. 

A rather simple problem is the calculation of AS between two states at the same pressure. To perform this calculation we devise a reversible constant-pressure path between the two states. If no phase change occurs, the heat along this path the heat is... 

The last equation demonstrates that the starting point for the solution of the problem is the calculation of ci(double layer (this makes low-frequency dielectric dispersion [LFDD] measurements a most valuable electrokinetic technique). Probably, the first theoretical treatment is the one due to Schwarz [61], who considered only surface diffusion of counterions (it is the so-called surface diffusion model). In fact, the model is inconsistent with any explanation of dielectric dispersion based on double-layer polarization. The generalization of the theory of diffuse atmosphere polarization to the case of alternating external fields and its application to the explanation of LFDD were first achieved by Dukhin and Shilov [20]. A full numerical approach to the LFDD in suspensions is due to DeLacey and White [60], and comparison with this numerical model allowed to show that the thin double-layer approximations [20,62,63] worked reasonably well in a wider than expected range of values of both and ku [64]. Figure 3.12 is an example of the calculation of As. From this it will be clear that (i) at low frequencies As can be very high and (ii) the relaxation of the dielectric constant takes place in the few-kHz frequency range, in accordance with Equations (3.56) and (3.57). 

Naturally, this strongly affects the values of AH calculated by expression (O 53.2). In the case of polyatomic molecules, when one has to take into account the entropy variation due to variation of internal degrees of freedom, rotation of the molecule, and probably chemical bond of molecules with the surface, a theoretical calculation of AS becomes much more complicated. Thus, the uncertainty in evaluation of AS results in approximate values of AH only. Several attempts have been made to calculate empirically AS and AH for various compounds, e.g., chlorides and oxides (B. Eichler 1976), and to determine experimentally these two main parameters of the adsorption interaction, but their values obtained for a majority of adsorbate-adsorbent pairs are still estimates. Therefore, it is desirable to cite these values together, no matter how they were obtained. 

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