Basic Thermodynamic Functions

In order to calculate the equOibrium composition of a system consisting of one or more phases in equilibrium with an aqueous solution of electrolytes, a review of the basic thermodynamic functions and the conditions of equilibrium is important, This is particularly true inasmuch as the study of aqueous solutions requires consideration of chemical and/or ionic reactions in the aqueous phase as well as a thermodynamic framework which is, for the most part, quite different from those definitions associated with nonelectrolytes. Therefore, in this section we will review the definition of the basic thermodynamic functions, the partial molar quantities, chemical potentials, conditions of equilibrium, activities, activity coefficients, standard states, and composition scales encountered in describing aqueous solutions. 

The thermodynamic properties of a system at equilibrium consists of two types of properties, intensive and extensive properties. The roost common intensive properties encountered are the temperature, T, and pressure, P, which are 

The thermodynamic properties most often encountered in describing phase equilibria of a system are functions of the state of the system. This is important since the calculation of these thermodynamic properties depends only on the existing state of the system and not the route by which this state has been reached. The following energy and energy related properties are extensive properties if they refer to the system as a whole  

The above thermodynamic properties are intensive properties when their values are expressed on a per mole basis. Other useful relationships which we will encounter are obtained through differentiation of these basic thermodynamic functions and include  

These relationships express the first and second laws of thermodynamics for a closed system. Furthermore, from the last expression above we can also obtain 

Chemical bonding is not considered, i.e. molecules are taken as they are and do not react. 

The analysis is limited to compounds made of the usual elements of organic chemistry, with the exclusion of organometaUics, rocks, silicates, and metals. 

Molecules are considered only up to a size of some 1000 da, with the exclusion of polymers and biological macromolecules. 

Only pure substances or binary mixtures are considered. 

Within the above-mentioned restrichons, a phase is a piece of matter containing one or two chemical species in a distinguishable state of aggregation. Admittedly, this oversimplifies the real world, where even one of the purest liquids, drinking water, is a solution of hundreds of components, and most chemical systems consist of many different chemicals in different and perhaps variable states of aggregation. 

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Adopting an atoms in molecules viewpoint, the atoms can be regarded as "open subsystems" and it thus becomes natural to use the grand potential 2 [83] as the basic thermodynamic function to describe the system ... 

As a means to predict the properties of drug delivery systems, it is useful to briefly review some basic thermodynamic functions. The rate or speed of a reaction is given by... 

The basic thermodynamic functions are internal energy U, enthalpy H, entropy S, and Gibbs free energy G. These are extensive properties of a thermodynamic system and they are first order homogenous functions of the components of the system. Pressure and temperature are intensive properties of the system and they are zero-order homogenous functions of the components of the system. Electrochemical potentials are the driving force in an electrochemical system. The electrochemical potential comprises chemical potential and electrostatic potential in the following relation. 

We could introduce Hess s generalization into thermodynamics as another empirical law, which is similar to the first law. However, a firm theoretical framework depends on a minimum of empirical postulates. Thermodynamics is so powerful a method precisely because it leads to so many predictions from only two or three basic assumptions. Hess s law need not be among these postulates, because it can be derived directly from the first law of thermodynamics perhaps most conveniently by using a new thermodynamic function, enthalpy. 

Carnot efficiency is one of the cornerstones of thermodynamics. This concept was derived by Carnot from the impossibility of a perpetuum mobile of the second kind [ 1]. It was used by Clausius to define the most basic state function of thermodynamics, namely the entropy [2]. The Carnot cycle deals with the extraction, during one full cycle, of an amount of work W from an amount of heat Q, flowing from a hot reservoir (temperature Ti) into a cold reservoir (temperature T2 < T ). The efficiency r] for doing so obeys the following inequality ... 

The well-known thermodynamic rule says that the two substances of different nature are miscible if the process brings about a gain in the value of the Gibbs function, AG, also called Gibbs energy or free enthalpy—that is, if AG > 0. The Gibbs function is connected with further basic thermodynamic quantities enthalpy and entropy, by the relation... 

From the experimental temperature dependence of A2 (and the corresponding inferred temperature dependence of juE the other basic excess thermodynamic functions can be determined using general thermodynamic relationships. This then provides a complete thermodynamic characterization of the system as a whole. Thus, for the determination of the excess molar enthalpy of the system at constant pressure, the following equation can be used (Prigogine and Defay, 1954) ... 

Only closed systems have so far been considered. However, mass can be varied and is an important variable for all thermodynamic functions. The introduction of mass as an independent variable into the basic differential expressions for the thermodynamic functions yields the equations that Gibbs called fundamental . It is on these equations that much of the development of the applications of thermodynamics to chemical systems is based. 

We have developed the basic equations for the thermodynamic functions of the defined surface in the preceding paragraphs, but have not discussed the determination of the position of the boundary. Actually, the position is somewhat arbitrary, and as a result we must also discuss the dependence of the properties of the surface on the position. The position can be fixed by assigning the value of zero to one of Equations (13.25)—(13.27) that is, by making one of the nf equal to zero. For a one-component system there is only one such equation. For multicomponent systems we have to choose one of the components for which nf is made zero. The value of nf for the other components then would not be zero in general. The most appropriate choice for dilute solutions would be the solvent. The position of the surface for a one-component system is illustrated in Figure 13.2, where the line c is determined by making the areas of the two shaded portions equal. 

We take up this topic not only because of its intrinsic interest, but because it is pedagogically valuable to note the various descriptions that arise from the multitude of available choices for the basic thermodynamic potential functions. The system under consideration consists of a thin layer of atoms held on the surface of a solid or liquid exposed to a gas phase. The solid or liquid is termed the adsorbent. whereas the material held on the surface is called the adsorbate the process by which the thin surface layer is formed from the transfer of gas molecules to the surface phase is called adsorption. 

It is interesting to note that no such linear correlations have been found between the other thermodynamic functions or AS or the corresponding A// and AS. This is undoubtedly a reflection of the more erratic behavior of these functions (Table XV.4) and the way in which changes in them tend to cancel each other in yielding AF / However, it makes very suspect many of the so-called basic interpretations of the correlation constants which have been given. 

In addition the reader may find tables with selection rules for the Resonance Raman and Hyper Raman Effect in the book of Weidlein et al. (1982). Special discussions about the basics of the application of group theory to molecular vibrations are given in the books of Herzberg (1945), Michl and Thulstrup (1986), Colthup et al. (1990) and Ferraro and Nakamoto (1994). Herzberg (1945) and Brandmiiller and Moser (1962) describe the calculation of thermodynamical functions (see also textbooks of physical chemistry). For the calculation of the rotational contribution of the partition function a symmetry number has to be taken into account. The following tables give this number in Q-... 

Another advantage of the simulation is its abihty to make direct tests on the range of validity of basic thermodynamical theorems such as the fluctuation-dissipation theorem. In the second paper of the series by Evans, he considers these points for the simplest type of torque mentioned above, —X F. Consider the return to equilibrium of a dynamical variable A after taking off at r = 0 the constant torque appUed prior to this instant in time. If the torque is removed instantaneously, the first fluctuation-dissipation theorem implies that the normalized fall transient will decay with the same dependence as the autocorrelation function (A(t)A(O))- Al /(A 0)) — Therefore,... 

Reliable values of thermodynamic functions of H bonds are derived from the equilibrium constant, K, and its variation with temperature. The experimental techniques vary only in their approach to finding the concentration or pressure values needed to determine K, The basic relations are... 

Description of thermochemical properties of chemical compounds, including that of polymers can be done using a few thermodynamic functions. One basic function is Gibbs free enthalpy that is expressed as follows [1] ... 

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