Equilibrium energy functions

Most of the molecules we shall be interested in are polyatomic. In polyatomic molecules, each atom is held in place by one or more chemical bonds. Each chemical bond may be modeled as a harmonic oscillator in a space defined by its potential energy as a function of the degree of stretching or compression of the bond along its axis (Fig. 4-3). The potential energy function V = kx j2 from Eq. (4-8), or W = ki/2) ri — riof in temis of internal coordinates, is a parabola open upward in the V vs. r plane, where r replaces x as the extension of the rth chemical bond. The force constant ki and the equilibrium bond distance riQ, unique to each chemical bond, are typical force field parameters. Because there are many bonds, the potential energy-bond axis space is a many-dimensional space. 

The equilibrium problem for the shell corresponds to minimization of the energy functional over the set of admissible displacements. To this end, introduce the convex sets... 

The free energy is the most important equilibrium thennodynamic function, but other quantities such as the enthalpy and entropy are also of great interest. Thermodynamic integration and permrbation fonnulas can be derived for them as well. For example, the derivative of the entropy can be written [24]... 

Estr is the energy function for stretching a bond between two atom types A and B. In its simplest form, it is written as a Taylor expansion around a namral , or equilibrium , bond length Rq- Tenninating the expansion at second order gives the expression... 

The critical hydrogen content for the ductility loss increased with increasing hydrogen solubility in the alloy. The fracture surfaces were not characteristic of those found under conditions of SCC. In terms of hydrogen and deuterium solubility in a similar series of bcc alloys, the equilibrium constants were determined at infinite dilution as a function of temperature The free energy function was expressed in terms of the bound-proton model. 

Using the fact that we have a well defined energy function (equation 10.9), we know from statistical mechanics that when the system has reached equilibrium, the probability that it is in some state S = Si, S, , Sm) is given by the Boltzman distribution ... 

The quantities n, V, and (3 /m) T are thus the first five (velocity) moments of the distribution function. In the above equation, k is the Boltzmann constant the definition of temperature relates the kinetic energy associated with the random motion of the particles to kT for each degree of freedom. If an equation of state is derived using this equilibrium distribution function, by determining the pressure in the gas (see Section 1.11), then this kinetic theory definition of the temperature is seen to be the absolute temperature that appears in the ideal gas law. 

The effect of temp on chemical equilibria is conventially determined via the free energy function AG°/RT and the ideal equilibrium constant K. Table 1 gives the free, energy function G°/RT for the important detonation products of CHNO expls. From these data A G°/(RT) can be obtained for different temps for the reactions of interest, and ideal equilibrium constants computed according to ... 

This contrasts with relation (5.16), which led to a non-physical conservation law for J. Eqs. (5.28) and Eq. (5.30) make it possible to calculate in the high-temperature limit the relaxation of both rotational energy and momentum, avoiding any difficulties peculiar to EFA. In the next section we will find their equilibrium correlation functions and determine corresponding correlation times. 

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,... 

The tools for calculating the equilibrium point of a chemical reaction arise from the definition of the chemical potential. If temperature and pressure are fixed, the equilibrium point of a reaction is the point at which the Gibbs free energy function G is at its minimum (Fig. 3.1). As with any convex-upward function, finding the minimum G is a matter of determining the point at which its derivative vanishes. 

The potential energy function U(R) that appears in the nuclear Schrodinger equation is the sum of the electronic energy and the nuclear repulsion. The simplest case is that of a diatomic molecule, which has one internal nuclear coordinate, the separation R of the two nuclei. A typical shape for U(R) is shown in Fig. 19.1. For small separations the nuclear repulsion, which goes like 1 /R, dominates, and liniR >o U(R) = oo. For large separations the molecule dissociates, and U(R) tends towards the sum of the energies of the two separated atoms. For a stable molecule in its electronic ground state U(R) has a minimum at a position Re, the equilibrium separation. 

Even for a diatomic molecule the nuclear Schrodinger equation is generally so complicated that it can only be solved numerically. However, often one is not interested in all the solutions but only in the ground state and a few of the lower excited states. In this case the harmonic approximation can be employed. For this purpose the potential energy function is expanded into a Taylor series about the equilibrium separation, and terms up to second order are kept. For a diatomic molecule this results in ... 

The need for entropy values is bypassed when the van t Hoff equation (d In K/dT) =AH/RT2 is used. This can be integrated, either assuming AH is temperature-independent, or by incorporating a specific heat-temperature variation. This is the so-called second law method which contrasts with the third law method. In the latter method, the standard enthalpy is obtained from each equilibrium constant using free-energy functions of all the species present, for example... 

Rose and Benjamin (see also Halley and Hautman ) utilized molecular dynamic simulations to compute the free energy function for an electron transfer reaction, Fe (aq) + e Fe (aq) at an electrodesolution interface. In this treatment, Fe (aq) in water is considered to be fixed next to a metal electrode. In this tight-binding approximation, the electron transfer is viewed as a transition between two states, Y yand Pf. In Pj, the electron is at the Fermi level of the metal and the water is in equilibrium with the Fe ion. In Pf, the electron is localized on the ion, and the water is in equilibrium with the Fe" ions. The initial state Hamiltonian H, is expressed as... 

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