State Functions from Fundamental Forms

There is no need to have the complete fundamental form as such in an analytical equation, to get some information on the state functions. We may postulate some properties of the fundamental forms. 

For example, we assume that the molar energy and the molar enthalpy are functions of the temperature alone. When we expand the energy O as a general function of temperature and volume U = U(T,V) 

In order to get the equation of the ideal gas, we must further postulate that g(T) -f(T) = R, since d(py) = (Cp - C )d7 . However, this assumption is not sufficient to obtain the adiabatic equation properly, we must more strictly assume that both Cp and Cp are constants. Further, we have to adjust the constant of integration properly. 

In this way, we can obtain equations of states using certain assumptions on the fundamental form. 

Before we leave this topic, we mention that we could not use also the free enthalpy and the free energy 

---

In the broadest sense, thermodynamics is concerned with mathematical relationships that describe equiUbrium conditions as well as transformations of energy from one form to another. Many chemical properties and parameters of engineering significance have origins in the mathematical expressions of the first and second laws and accompanying definitions. Particularly important are those fundamental equations which connect thermodynamic state functions to real-world, measurable properties such as pressure, volume, temperature, and heat capacity (1 3) (see also Thermodynamic properties). 

The construction of exchange correlation potentials and energies becomes a task for which not much guidance can be obtained from fundamental theory. The form of dependence on the electron density is generally not known and can only to a limited extent be obtained from theoretical considerations. The best one can do is to assume some functional dependence on the density with parameters to satisfy some consistency criteria and to fit calculated results to some model systems for which applications of proper quantum mechanical theory can be used as comparisons. At best this results in some form of ad-hoc semi-empirical method, which may be used with success for simulations of molecular ground state properties, but is certainly not universal. 

The efficiency of any photophysical or photochemical process is a function of both the properties of the reaction environment and the character of the excited state species. The fundamental quantity which is used to describe the efficiency of any photo process is the quantum yield (0) it is useful in both quantifying the process and in elucidating the reaction mechanism. Quantum yield has the general definition of the number of events occurring divided by the number of photons absorbed. Therefore, for a chemical process 0 is defined as the number of moles of reactant consumed or product formed divided by the number of einsteins (an einstein is equal to 6.02 X 10 photons) absorbed. Since the absorption of light by a molecule is a one-quantum process, then the sum of the quantum yields for all primary processes occurring must be one. Where secondary reactions are involved, however, the overall quantum yield can exceed unity and for chain reactions reach values in the thousands. When values of 0 are known or can be measured for a specific photochemical reaction the rate can be determined from ... 

Two patterns occur in this chapter, and we draw your attention to them here. One is the degree to which elements in thermodynamics are isomorphic to elements in the calculus. For example, the state functions of thermodynamics are, in the calculus, merely those quantities that form exact differentials. Several such isomorphisms are cited in Table 3.3, suggesting that much of fundamental thermod)mamics is merely an application of the calculus. One striking consequence is that although the first and second laws, formulated in Chapter 2, did not explicitly contain anything about mixtures, we were, nevertheless, able to show formally how properties of mixtures may differ from those of pure substances. 

This energy conservation law, based on experience, was tacitly assumed in the preceding sections otherwise, the fundamental form would be inconceivable and the physical quantity known as energy would not comply with the mathematical requirements. From the first law, it follows that energy is a state function. The total energy exchanged does not depend on the path the process performs rather, it depends only on the initial and the final states. 

This equation forms the fundamental connection between thermodynamics and statistical mechanics in the canonical ensemble, from which it follows that calculating A is equivalent to estimating the value of Q. In general, evaluating Q is a very difficult undertaking. In both experiments and calculations, however, we are interested in free energy differences, AA, between two systems or states of a system, say 0 and 1, described by the partition functions Qo and (), respectively - the arguments N, V., T have been dropped to simplify the notation ... 

An important step in developing the mean-field concept was done by Landau [8, 10]. Without discussing the relation between such fundamental quantities as disorder-order transitions and symmetry lowering, we just want to note here that his theory is based on thermodynamics and the derivation of the temperature dependence of the order parameter via the thermodynamic potential minimization (e.g., the free energy A(r),T)) which is a function of the order parameter. It is assumed that the function A(rj,T) is analytical in the parameter 77 and thus near the phase transition point could be expanded into the series in 77 usually it is a polynomial expansion with temperature-dependent coefficients. Despite the fact that such a thermodynamical approach differs from the original molecular field theory, they are quite similar conceptually. In particular, the r.h.s. of the equation of state for the pressure of gases or liquids and the external field in ferromagnetics, respectively, have the same polynomial form. 

However, a question arises - could similar approach be applied to chemical reactions At the first stage the general principles of the system s description in terms of the fundamental kinetic equation should be formulated, which incorporates not only macroscopic variables - particle densities, but also their fluctuational characteristics - the correlation functions. A simplified treatment of the fluctuation spectrum, done at the second stage and restricted to the joint correlation functions, leads to the closed set of non-linear integro-differential equations for the order parameter n and the set of joint functions x(r, t). To a full extent such an approach has been realized for the first time by the authors of this book starting from [28], Following an analogy with the gas-liquid systems, we would like to stress that treatment of chemical reactions do not copy that for the condensed state in statistics. The basic equations of these two theories differ considerably in their form and particular techniques used for simplified treatment of the fluctuation spectrum as a rule could not be transferred from one theory to another. 

---