SUBJECTS free energy

In Eq. (16), yci stands for the number of segments of a chain in conformation c located in layer i. The two equations express the obvious conditions that each lattice layer must be occupied and that the total number of chains is constant. The Lagrange multiplier method is used to calculate the minimum free energy subject to the above constraints. By introducing the multipliers at, for each of the constraints given by Eq. (16), and (i for the constraint expressed by Eq. (17), one can write... 

The Gibbs reactor solves the full reaction (and optionally phase) equilibrium of all species in the component list by minimization of the Gibbs free energy, subject to the constraint of the feed mass balance. A Gibbs reactor can be specified with restrictions such as a temperature approach to equilibrium or a fixed conversion of one species. 

The other extreme of behavior involves the "phantom chain" approximation. Here, it is assumed that the individual chains and crosslink points may pass through one another as if they had no material existence that is, they may act like phantom chains. In this approximation, the mean position of crosslink points in the deformed network is consistent with the affine transformation, but fluctuations of the crosslink points are allowed about their mean positions and these fluctuations are not affected by the state of strain in the network. Under these conditions, the distribution function characterizing the position of crosslink points in the deformed network cannot be simply related to the corresponding distribution function in the undeformed network via an affine transformation. In this approximation, the crosslink points are able to readjust, moving through one another, to attain the state of lowest free energy subject to the deformed dimensions of the network. 

Let s state the problem in non-mathematical language first. We need to find the unique concentrations of species in a system that minimize the total free energy subject to the following constraints or conditions. 

The total interfacial free energy per unit area, consists of the sum of /o and the free energy per unit area that comes from the liquid-vapor interface. In equilibrium, one minimizes the total free energy subject to the conservation constraint — i.e., one works at fixed chemical potential. As explained in the discussion of the gas-fiquid interface in Chapter 2, the appropriate bulk free energy to minimize to find the interfacial profile is the grand potential per unit area, gs, which is written ... 

In the spirit of the local random mixing approximation, we neglect fluctuations and determine the counterion charge density through a functional minimization of the free energy, subject to the constraint of charge conservation. We thus minimize the grand potential, G ... 

In order to derive the equilibrium states, the RCCE makes use of minimizing the Gibbs free energy subject to conservation of enthalpy, elements and mass for each constraint i ... 

The Gibbs surface free energy will usually be different for different facets of a crystal. Such variations, often referred to as surface free energy anisotropies, are key to determining the equilibrium crystal shape (as well as many other properties) of materials because at equilibrium, a crystal seeks to minimize its total surface free energy subject to the constraint of constant volume. 

A general prerequisite for the existence of a stable interface between two phases is that the free energy of formation of the interface be positive were it negative or zero, fluctuations would lead to complete dispersion of one phase in another. As implied, thermodynamics constitutes an important discipline within the general subject. It is one in which surface area joins the usual extensive quantities of mass and volume and in which surface tension and surface composition join the usual intensive quantities of pressure, temperature, and bulk composition. The thermodynamic functions of free energy, enthalpy and entropy can be defined for an interface as well as for a bulk portion of matter. Chapters II and ni are based on a rich history of thermodynamic studies of the liquid interface. The phase behavior of liquid films enters in Chapter IV, and the electrical potential and charge are added as thermodynamic variables in Chapter V. 

A second source of standard free energies comes from the measurement of the electromotive force of a galvanic cell. Electrochemistry is the subject of other articles (A2.4 and B1.28). so only the basics of a reversible chemical cell will be presented here. For example, consider the cell conventionally written as... 

As the simulation proceeds, the values of A fluctuate, subject to the constraint in Equatic 11.38). The free energy difference between two molecules i and j can be determined t identifying the probability that each molecule occupies the state A, = 1 or A = 1, respe lively. Thus ... 

The distribution coefficient is an equilibrium constant and, therefore, is subject to the usual thermodynamic treatment of equilibrium systems. By expressing the distribution coefficient in terms of the standard free energy of solute exchange between the phases, the nature of the distribution can be understood and the influence of temperature on the coefficient revealed. However, the distribution of a solute between two phases can also be considered at the molecular level. It is clear that if a solute is distributed more extensively in one phase than the other, then the interactive forces that occur between the solute molecules and the molecules of that phase will be greater than the complementary forces between the solute molecules and those of the other phase. Thus, distribution can be considered to be as a result of differential molecular forces and the magnitude and nature of those intermolecular forces will determine the magnitude of the respective distribution coefficients. Both these explanations of solute distribution will be considered in this chapter, but the classical thermodynamic explanation of distribution will be treated first. 

The standard free energy can be divided up in two ways to explain the mechanism of retention. First, the portions of free energy can be allotted to specific types of molecular interaction that can occur between the solute molecules and the two phases. This approach will be considered later after the subject of molecular interactions has been discussed. The second requires that the molecule is divided into different parts and each part allotted a portion of the standard free energy. With this approach, the contributions made by different parts of the solvent molecule to retention can often be explained. This concept was suggested by Martin [4] many years ago, and can be used to relate molecular structure to solute retention. Initially, it is necessary to choose a molecular group that would be fairly ubiquitous and that could be used as the first building block to develop the correlation. The methylene group (CH2) is the... 

The interface free energy per unit area fi,u is taken to be that of a planar interface between coexisting phases. Considering a solution v /(z) that minimizes Eq. (5) subject to the boundary conditions vj/(z - oo) = - v /coex, v /(z + oo) = + vj/ oex one finds the excess free energy of a planar interface ... 

The Alexander model and its descendants impose strong restrictions on the allowed chain configurations within the tethered assembly. The equilibrium state thus found is subject to constraints and may not attain the true minimum free energy of the constraint-free system. In particular, the Alexander model constrains the segment density to be uniform and all the chain ends to be at the same distance from the grafting surface. Related treatments of curved systems retain only the second... 

With the valence bond structures of the exercise, we can try to estimate the effect of the enzyme just in terms of the change in the activation-free energy, correlating A A g with the change in the electrostatic energy of if/2 and i/r3 upon transfer from water to the enzyme-active site. To do this we must first analyze the energetics of the reaction in solution and this is the subject of the next exercise. 

The Gibbs free energy is named for Josiah Willard Gibbs (Fig. 7.23), the nineteenth-century American physicist who was responsible for turning thermodynamics from an abstract theory into a subject of great usefulness. 

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