Thermodynamic energy functions

Although the statistical approach to the derivation of thermodynamic functions is fairly general, we shall restrict ourselves to a) crystals with isolated defects that do not interact (which normally means that defect concentrations are sufficiently small) and b) crystals with more complex but still isolated defects (i.e., defect pairs, associates, clusters). We shall also restrict ourselves to systems at some given (P T), so that the appropriate thermodynamic energy function is the Gibbs energy, G, which is then constructed as... 

Fig. 2.2. Thermodynamic energy functions (a) Internal energy U, (b) Enthalpy H.

The chemical potential is defined as an intensive energy function to represent the energy level of a chemical substance in terms of the partial molar quantity of free enthalpy of the substance. For open systems permeable to heat, work, and chemical substances, the chemical potential can be used more conveniently to describe the state of the systems than the usual extensive energy functions. This chapter discusses the characteristics of the chemical potential of substances in relation with various thermodynamic energy functions. In a mixture of substances the chemical potential of an individual constituent can be expressed in its unitary part and mixing part. 

Among the four principal thermodynamic energy functions, U, H, F, and G, the free enthalpy G (Gibbs energy) associated with the intensive variables T and p is a homogeneous function of the first degree with respect to the extensive independent variable of the number of moles n. of the constituent substances present in the system considered, so that it can be expressed as the sum of the chemical potentials of all constituent substances at constant temperature and pressure. 

Starting from the definition 5.22 we now establish several important properties of thermodynamic potentials (partial molar quantities of thermodynamic energy functions) for an ideal system of mixture. Differentiating G-H-TS with respect to n, with Tand p constant, we have pt = ht- Tsl and furthermore [d(jWf IT) / dT pn = (1 IT) (dp, / dT) - (p, / T1) = - [(r s, + pt) / T2] = -h,l T2. From this equation we obtain Eq. 5.34 for the partial molar enthalpy hf of a constituent i in an ideal mixture ... 

Abstract Thermodynamic energy functions are related to six variables such as volume, pressure, temperature, entropy, chemical potential and amount of substance. They are rather cumbersome and perplexing to undergraduates who start to leam their relations. With a story and the two-dimensional Cartesian coordinate system, most of thermodynamic relations could be obtained in addition to the Maxwell relations for a reversible change in a closed system only in the presence of pressure-volume work and heat. 

Fig. 1 Thermodynamic energy functions with the four variables shown in the two-dimensional Cartesian coordinate system...

It would in general take considerable amount of time and pains for students to be familiar with any new concept. When the students knew the thermodynamic relations, it seemed much more effective to teach them more complex concepts. This simple scheme would facilitate teaching the thermodynamic energy functions and their pertinent relations to chemistry undergraduates or graduates. 

There is a need to find out the relationship between molecular pair potentials and thermodynamic energy functions, because all macroscopic bulk and surface properties arise from molecular interactions. For example, adsorption occurs when a molecule loses sufficient energy to the atoms in a surface by exciting them vibrationally, electrostatically or... 

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). 

Free energy calculations rely on the following thermodynamic perturbation theory [6-8]. Consider a system A described by the energy function = 17 + T. 17 = 17 (r ) is the potential energy, which depends on the coordinates = (Fi, r, , r ), and T is the kinetic energy, which (in a Cartesian coordinate system) depends on the velocities v. For concreteness, the system could be made up of a biomolecule in solution. We limit ourselves (mostly) to a classical mechanical description for simplicity and reasons of space. In the canonical thermodynamic ensemble (constant N, volume V, temperature T), the classical partition function Z is proportional to the configurational integral Q, which in a Cartesian coordinate system is... 

Figure 3 Mutation of a ligand Asp into Asn in solution and bound to a protein, (a) Thermodynamic cycle, (b) Dual topology description a hybrid ligand with two side chains. Blocks are used to define the hybrid energy function [Eq. (14)]. Only the ligand is shown the environment is either solvent or the solvated protein, (c) Single-topology description.

Actually the assumptions can be made even more general. The energy as a function of the reaction coordinate can always be decomposed into an intrinsic term, which is symmetric with respect to jc = 1 /2, and a thermodynamic contribution, which is antisymmetric. Denoting these two energy functions h2 and /zi, it can be shown that the Marcus equation can be derived from the square condition, /z2 = h. The intrinsic and thermodynamic parts do not have to be parabolas and linear functions, as in Figure 15.28 they can be any type of function. As long as the intrinsic part is the square of the thermodynamic part, the Marcus equation is recovered. The idea can be taken one step further. The /i2 function can always be expanded in a power series of even powers of hi, i.e. /z2 = C2h + C4/z. The exact values of the c-coefficients only influence the... 

Later on Cahn and Hilliard presented some thermodynamic estimates for the nucleation of liquid in vapour. Values of AO and the composition profiles c(r) of the embryos have been estimated using the mean-field and gradient expansion approximations for the free energy functional F c(7 ). A number of qualitative features in variation... 

A change in any thermodynamic state function is independent of the path used to accomplish that change. This feature of state functions tells us that the energy change in a chemical reaction is independent of the manner in which the reaction takes place. In the real world, chemical reactions often follow very complicated paths. Even a relatively simple overall reaction such as the combustion of CH4 and O2 can be very complicated at the... 

Why do some reactions go virtually to completion, whereas others reach equilibrium when hardly any of the starting materials have been consumed At the molecular level, bond energies and molecular organization are the determining factors. These features correlate with the thermodynamic state functions of enthalpy and entropy. As discussed In Chapter 14, free energy (G) is the state function that combines these properties. This section establishes the connection between thermodynamics and equilibrium. 

An important use of the free energy function is to obtain a simple criterion for the occurrence of spontaneous processes and for thermodynamic equilibrium. According to the second law of thermodynamics,... 

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