General Properties of Entropy

Entropy, as formulated here and in the previous chapter, encompasses all aspects of matter transformations changes in energy, volume and composition. Thus, every system in Nature, be it a gas, an aqueous solution or a living cell, is associated with a certain entropy. We shall obtain explicit expressions for entropies of various systems in the following chapters and study how entropy production is related to irreversible processes. At this stage, however, we shall note some general properties of entropy as a function of state. 

Entropy is a function of the total energy U, volume V and mole numbers Nk-. 

For a function of many variables, we have the general relation 

Furthermore, from the general relation dU = TdS — pdV+ Y ij dNk, it follows that 

If the change in mole number Nk is only due to a chemical reaction, the entropy can also be expressed as a function of U, V and (Example 4.1). Then one can 

---

Vesicles can be considered as minimal mimics of enzymes in terms of desolvation effects and entropy factors. If the general property of the vesicle membrane in promot-ing/accelerating chemical reactions is closely related to the one described above with micelles, it usually occurs with improved efficiency because of the higher structural organization of the bilayers. This was shown, for instance, in the study of the acceleration of the Kemp elimination in the presence of cationic micelles or cationic vesicles (Figure 17). ... 

Secondly, in the theory of irreversible processes, variation principles may be expected to help establish a general statistical method for a system which is not far from equilibrium, just as the extremal property of entropy is quite important for establishing the statistical mechanics of matter in equilibrium. The distribution functions are determined so as to make thermod5mamic probability, the logarithm of which is the entropy, be a maximum under the imposed constraints. However, such methods for determining the statistical distribution of the s retem are confined to the case of a system in thermodynamic equilibrium. To deal with a system out of equilibrium, we must use a different device for each case, in contrast to the method of statistical thermodynamics, which is based on the general relation between the Helmholtz free energy and the partition function of the system. 

These fascinating bicontinuous or sponge phases have attracted considerable theoretical interest. Percolation theory [112] is an important component of such models as it can be used to describe conductivity and other physical properties of microemulsions. Topological analysis [113] and geometric models [114] are useful, as are thermodynamic analyses [115-118] balancing curvature elasticity and entropy. Similar elastic modulus considerations enter into models of the properties and stability of droplet phases [119-121] and phase behavior of microemulsions in general [97, 122]. 

P rtl IMol r Properties. The properties of individual components in a mixture or solution play an important role in solution thermodynamics. These properties, which represent molar derivatives of such extensive quantities as Gibbs free energy and entropy, are called partial molar properties. For example, in a Hquid mixture of ethanol and water, the partial molar volume of ethanol and the partial molar volume of water have values that are, in general, quite different from the volumes of pure ethanol and pure water at the same temperature and pressure (21). If the mixture is an ideal solution, the partial molar volume of a component in solution is the same as the molar volume of the pure material at the same temperature and pressure. 

The second general principle is based on the properties of the entropy function, and is contained in the aphorism of Clausius... 

Equation (2.66) indicates that the entropy for a multipart system is the sum of the entropies of its constituent parts, a result that is almost intuitively obvious. While it has been derived from a calculation involving only reversible processes, entropy is a state function, so that the property of additivity must be completely general, and it must apply to irreversible processes as well. 

The binary systems we have discussed so far have mainly included phases that are solid or liquid solutions of the two components or end members constituting the binary system. Intermediate phases, which generally have a chemical composition corresponding to stoichiometric combinations of the end members of the system, are evidently formed in a large number of real systems. Intermediate phases are in most cases formed due to an enthalpic stabilization with respect to the end members. Here the chemical and physical properties of the components are different, and the new intermediate phases are formed due to the more optimal conditions for bonding found for some specific ratios of the components. The stability of a ternary compound like BaCC>3 from the binary ones (BaO and CC>2(g)) may for example be interpreted in terms of factors related to electron transfer between the two binary oxides see Chapter 7. Entropy-stabilized intermediate phases are also frequently reported, although they are far less common than enthalpy-stabilized phases. Entropy-stabilized phases are only stable above a certain temperature,... 

Interestingly, the standard entropies (and in turn heat capacities) of both phases were found to be rather similar [69,70]. Considering the difference in standard entropy between F2(gas) and the mixture 02(gas) + H2(gas) taken in their standard states (which can be extracted from general thermodynamic tables), the difference between the entropy terms of the Gibbs function relative to HA and FA, around room temperature, is about 6.5 times lower than the difference between enthalpy terms (close to 125 kJ/mol as estimated from Tacker and Stormer [69]). This indicates that FA higher stability is mostly due to the lower enthalpy of formation of FA (more exothermic than for HA), and that it is not greatly affected by entropic factors. Jemal et al. [71] have studied some of the thermodynamic properties of FA and HA with varying cationic substitutions, and these authors linked the lower enthalpy of formation of FA compared to HA to the decrease in lattice volume in FA. 

---