Derived thermodynamic functions

Carell and Olin (58) were the first to derive thermodynamic functions relating to beryllium hydrolysis. They determined the enthalpy and entropy of formation of the species Be2(OH)3+ and Be3(OH)3+. Subsequently, Mesmer and Baes determined the enthalpies for these two species from the temperature variation of the respective equilibrium constants. They also determined a value for the species Be5(OH) + (66). Ishiguro and Ohtaki measured the enthalpies of formation of Be2(OH)3+ and Be3(OH)3+ calorimetrically in solution in water and water/dioxan mixtures (99). The agreement between the values is satisfactory considering the fact that they were obtained with different chemical models and ionic media. 

Steele, W.V., Chirico, R.D., Nguyen, A., Knipmeyer, S.E. (1995) Vapor pressure, high-temperature heat capacities, critical properties, derived thermodynamic functions, and barriers to methyl-group rotation, for the six dimethylpyridines. J. Chem. Thermodyn. 27, 311-334. 

Extended virial equations of many terms and constants have been developed for the highly accurate representation of experimental data. Some are developed specifically for standard tables of density and derived thermodynamic functions such as entropy and enthalpy. Bender [31] extended the virial equation to a 20-constant equation to represent argon, oxygen, methane, hydrogen, ethene. 

The IKBI method requires, as said, highly accurate data, because it employs derivative thermodynamic functions of the composition, see below. Following are the working expressions for the Kirkwood-Buff integrals ... 

MOLECULAR VIBRATIONS OF QUINONES. VI. A VIBRATIONAL ASSIGNMENT FOR RHO-BENZOQUINONE AND SIX ISOTROPIC DERIVATIVES, THERMODYNAMIC FUNCTIONS OF RHO-BENZOQUINONE. 

For perfect, crystalline solids, the entropy at 0 K is zero, Sc° = 0. Thus, absolute entropies can be calculated directly. Information to the available data is given in Table 1. For crystalline, linear macromolecnles, the derived thermodynamic functions are reported as follows... 

The procedure would then require calculation of (2m+2) partial derivatives per iteration, requiring 2m+2 evaluations of the thermodynamic functions per iteration. Since the computation effort is essentially proportional to the number of evaluations, this form of iteration is excessively expensive, even if it converges rapidly. Fortunately, simpler forms exist that are almost always much more efficient in application. 

It has long been known that the adsorption of a gas on a solid surface is always accompanied by the evolution of heat. Various attempts have been made to arrive at a satisfactory thermodynamic analysis of heat of adsorption data, and within the past few years broad agreement has been achieved in setting up a general system of adsorption thermodynamics. Here we are not concerned with the derivation of the various thermodynamic functions but only with the more relevant definitions and the principles involved in the thermodynamic analysis of adsorption data. For more detailed treatments, appropriate texts should be consulted. " ... 

Whereas this two-parameter equation states the same conclusion as the van der Waals equation, this derivation extends the theory beyond just PVT behavior. Because the partition function, can also be used to derive aH the thermodynamic functions, the functional form, E, can be changed to describe this data as weH. Corresponding states equations are typicaHy written with respect to temperature and pressure because of the ambiguities of measuring volume at the critical point. 

The protonation equilibria for nine hydroxamic acids in solutions have been studied pH-potentiometrically via a modified Irving and Rossotti technique. The dissociation constants (p/fa values) of hydroxamic acids and the thermodynamic functions (AG°, AH°, AS°, and 5) for the successive and overall protonation processes of hydroxamic acids have been derived at different temperatures in water and in three different mixtures of water and dioxane (the mole fractions of dioxane were 0.083, 0.174, and 0.33). Titrations were also carried out in water ionic strengths of (0.15, 0.20, and 0.25) mol dm NaNOg, and the resulting dissociation constants are reported. A detailed thermodynamic analysis of the effects of organic solvent (dioxane), temperature, and ionic strength on the protonation processes of hydroxamic acids is presented and discussed to determine the factors which control these processes. 

Macroscopic observables, such as pressme P or heat capacity at constant volume C v, may be calculated as derivatives of thermodynamic functions. 

The difference can also be seen directly from eq. (16.1). Properties which depend on derivatives of Q are independent of the actual value of Q, only its variation with an external variable (T or V) matters. Thermodynamical functions such as A, S and G, however, depend on the actual value of Q, i.e. the whole volume of phase space. 

The thermodynamic functions of primary interest in chemistry are Cp.m, Sm, and Gm-Ho.m- The translational, rotational, and vibrational contributions are summarized in Table 10.4.u We will not attempt to derive all the equations in this table but will do enough to show how it is done. 

Another distinction that we make among the thermodynamic functions is to describe p, V, T, U, and 5 as the fundamental properties of thermodynamics. The other quantities, H, A, and G are derived properties, in that they are defined in terms of the fundamental properties, with... 

The process we have followed Is Identical with the one we used previously for the uranium/oxygen (U/0) system (1-2) and Is summarized by the procedure that Is shown In Figure 1. Thermodynamic functions for the gas-phase molecules were obtained previously (3) from experimental spectroscopic data and estimates of molecular parameters. The functions for the condensed phase have been calculated from an assessment of the available data, Including the heat capacity as a function of temperature (4). The oxygen potential Is found from extension Into the liquid phase of a model that was derived for the solid phase. Thus, we have all the Information needed to apply the procedure outlined In Figure 1. 

Thermodynamic Functions of the Gases. To apply Eqs. (1-10), the free energies of formation, Ag , for all gaseous species as a function of temperature are required. Tabulated data were fit by a least-squares procedure to derive an analytical equation for AG° of each vapor species. For the plutonium oxide vapor species, the data calculated from spectroscopic data (3 ) were used for 0(g) and 02(g), the JANAF data (.5) were used and for Pu(g), data from the compilation of Oetting et al. (6) were used. The coefficients of the equations for AG° of the gaseous species are included in Table I. 

Surface composition. The principle of surface segregation in ideal systems is easy to understand and to derive thermodynamically the equilibrium relations (surface concentration Xg as a function of the bulk concentration Xb at various temperatures) is also very easy (4,8). Even easier is a kinetic description which can also comprise some of the effects of the non-ideality (9). We consider an equilibrium between the surface(s) and the bulk(b) in the exchange like ... 

Other thermodynamic functions may be derived from the partition function Q, or from the expression for the osmotic pressure. The chemical potential of the solvent in the solution (not to be confused with the excess chemical potential (mi —within a region of uniform segment expectancy, or density) is given, of course, by ... 

The calculation of entropy is required for compression and expansion calculations. Isentropic compression and expansion is often used as a reference for real compression and expansion processes. The calculation of entropy might also be required in order to calculate other derived thermodynamic properties. Like enthalpy, entropy can also be calculated from a departure function ... 

Here Hm is the sum (in the stoichiometric ratio of the compound in question) of A298-15tf° of the elements in their defined standard state, a, b, c and dn are coefficients and n integers. This form of expression is useful for storing thermodynamic information in databases. A number of such expressions are often required for a given phase to cover the whole temperature range of interest. From eq. (2.41) all other thermodynamic functions can be derived, e.g. 

Here again G = G(T) - G(0). Other thermodynamic functions such as enthalpy, entropy and volume may be derived by the thermodynamic relationships discussed in Chapter 1. 

In the case of reciprocal systems, the modelling of the solution can be simplified to some degree. The partial molar Gibbs energy of mixing of a neutral component, for example AC, is obtained by differentiation with respect to the number of AC neutral entities. In general, the partial derivative of any thermodynamic function Y for a component AaCc is given by... 

Errors also arise from the long extrapolations of AG or Alf . to 298 K using calculated thermodynamic functions. Although values derived by this technique are never as precise as those found calorimetrically, they are often the only ones available for many species. Representative examples of the technique are given in Table VI. 

Once the cluster expansion of the partition function has been made the remaining thermodynamic functions can be obtained as cluster expansions by taking suitable derivatives. Of particular interest are the expressions for the equilibrium concentrations of intrinsic point defects for the various types of lattice disorder. Since the partition function is a function of Nx, N2, V, and T, it is convenient for the derivation of these expressions to introduce defect chemical potentials for each of the species in the set (Nj + N2) defined, by analogy with ordinary Gibbs chemical potentials (cf. Section I), by the 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. 

Whether obtained from an actual experimentally feasible process or from a thought process, As i Gg, which is obtained from Eq. (2.9) by re-arrangement, pertains to the solvation of the solute and expresses the totality of the solute-solvent interactions. It is a thermodynamic function of state, and so are its derivatives with respect to the temperature (the standard molar entropy of solvation) or pressure. This means that it is immaterial how the process is carried out, and only the initial state (the ideal gaseous solute B and the pure liquid solvent) and the final state (the dilute solution of B in the liquid) must be specified. 

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