Thermodynamic Derivation

An equivalent form of Eq. (2.8.12) or (2.8.14) in terms of equilibrium constants can be obtained as follows To establish the relationship between the two notations we 

Assuming that A is in equilibrium with an ideal-gas phase, we can define the equilibrium constant for the addition of one A to i by 

The average number of molecules adsorbed per site is therefore 

Note that i Kj may be interpreted as the equilibrium constant for the reaction 

This should be compared with Eq. (2.7.12). Of course, (2.9.12) is more general in the sense that it is not restricted by the assumption of ideality as used in (2.9.23). To establish 

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Hthiated 4-substituted-2-methylthia2oles (171) at -78 C (Scheme 80). Crossover experiments at—78 and 25°C using thiazoles bearing different substituents (R = Me, Ph) proved that at low temperature the lithioderivatives (172 and 173) do not exchange H/Li and that the product ratios (175/176) observed are the result of independent metala-tion of the 2-methyl and the C-5 positions in a kinetically controlled process (444). At elevated temperatures the thermodynamic acidities prevail and the resonance stabilized benzyl-type anion (Scheme 81) becomes more abundant, so that in fine the kinetic lithio derivative is 173, whereas the thermodynamic derivative is 172. 

The crystals are assumed to be circular disks. This geometry is consistent with previous thermodynamic derivations. It has the advantage of easy mathematical description. 

These two equations are the starting point for a number of thermodynamic derivations and calculations. 

Thermodynamic derivations and applications are closely associated with changes in properties of systems. It should not be too surprising, then, that the mathematics of differential and integral calculus are essential tools in the study of this subject. The following topics summarize the important concepts and mathematical operations that we will use. 

The total differential given by equation (A 1.1) is a useful starting point for many thermodynamic derivations. If we consider a process in which X and Y are changed such that Z remains constant, then dZ = 0, and equation (A 1.1) can be rearranged to yield... 

In our thermodynamic derivations, we will routinely make use of equations of the type represented by equation (A1.3) to replace AX7jdYz with (dX/dY)z. We may also represent this ratio of differentials as (dA/d Y)z in which the direct equality with the partial derivative (dX/0Y)Z is more immediately evident. That is... 

This volume also contains four appendices. The appendices give the mathematical foundation for the thermodynamic derivations (Appendix 1), describe the ITS-90 temperature scale (Appendix 2), describe equations of state for gases (Appendix 3), and summarize the relationships and data needed for calculating thermodynamic properties from statistical mechanics (Appendix 4). We believe that they will prove useful to students and practicing scientists alike. 

Redox potential (thermodynamic derivation). Suppose we take an electrochemical cell represented by Fig. 2.7. We shall now address the question of both the potential values and the equilibrium state that can be finally attained... 

The LFER that results when correlating partitioning in the octanol-water system and the humic substances-water system Implies that the thermodynamics of these two systems are related. Hence, much can be learned about humic substances-water partitioning by first considering partitioning In the simpler octanol-water system. The thermodynamic derivation that follows is based largely on the approach developed by Chlou and coworkers (18-20), Miller et al. (21), and of Karickhoff (J, 22). In the subsequent discussion, we will adopt the pure liquid as the standard state and, therefore, use the Lewls-Randall convention for activity coefficients, l.e., y = 1 if the mole fraction x 1. 

The Nernst equation defines the equilibrium potential of an electrode. A simplified thermodynamic derivation of this equation is given in the Sections 5.3 to 5.5. Here we will give the kinetic derivation of this equation. 

Good, R.J, (1952). A thermodynamic derivation of Wenzel s modification of Young s equation contact angles Together with a theory of hysteresis. J. Am. Chem. Soc. 74, 5041-5042. 

EXAMPLE 6.1 Laplace Equation for Spherical Surfaces A Thermodynamic Derivation. The Maxwell relations play an important role in thermodynamics. By including the term dA in the usual differential form for cfG, show that (dVfdA)pJ = (dyidp)AJ. Evaluate (dVidA)pJ assuming a spherical surface and, from this, derive the Laplace equation for this geometry. 

Laplace equation A thermodynamic derivation Determining surface tension from the Kelvin equation Heat of immersion from surface tension and contact angle Surface tension and the height of a meniscus at a wall Interfacial tensions from the Girifalco-Good-Fowkes equation... 

Regioselectivity in the LiTMP metalation of 2,4-dichloropyrimidine (109) was shown to be dependent on temperature and solvent (Scheme 33) (91JHC). In THF/Et20 at - 100°C, metalation followed by acetaldehyde quench gave the 5-substituted (kinetic) product 110, whereas in THF/ HMPA mixture at -70°C, treatment with the same electrophile afforded the corresponding 6-substituted (thermodynamic) derivative 111. Minimum neglect of differential overlap (MNDO) calculations support the observed results. 

The identities (1.11)-(1.14) are among the most commonly employed in thermodynamic derivations, because two degrees of freedom underlie the important special case of simple substances (pure, homogeneous), as will be subsequently described. 

Maxwell relations are a powerful tool for deriving thermodynamic relationships. Their use should be considered whenever it is desirable to replace thermodynamic derivatives involving S with equivalent derivatives involving variables P, V, T only. Sidebars 5.4-5.6 illustrate this derivation techniques for a number of standard thermodynamic identities. 

The Maxwell relations are powerful tools of thermodynamic derivation. With the help of these relations and derivation techniques analogous to those illustrated in Sidebars 5.3-5.6, the skilled student of thermodynamics can (in principle ) re-express practically any partial derivative in terms of a small number of base properties involving only PVT variables. Consider, for example, the eight most common variables... 

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