Minimum, in free energy

By setting the derivative of the right-hand side of the equation with respect to L to zero, the distance for the minimum in free energy, L, is found to be... 

ASllllxmg and AGreaction as functions of the extent of reaction are plotted in Figure 1.8. The minimum value of-T ASlllmng is positioned at about 8 = 0.57. The position of the minimum depends on the ratio of the number of moles of reactants and products. The greater the number of moles of reactants, the closer the position is toward the product side. However, the minimum in free energy change is located at about 8 = 0.81 (i.e dAG/dS = 0). The equilibrium constant is then calculated ... 

Unique dominant minimum in free energy configuration space... 

This is a common feature in carbocation chemistry. Various rearrangements will run, usually in equilibrium with each other, between structures of similar energy, until one structure is reached which represents a minimum in free energy. Eventually all of the material will fall into this... 

Warneck, P. (2000) Chemistry of the natural atmosphere. Academic Press, New York, 969 pp Warneck, P. (2003) In-cloud chemistry opens pathway to the formation of oxalic acid in the marine atmosphere. Atmospheric Environment 37, 2423-2427 Wasai, K., G. Kaptay, K. Mukai and N. Shinozaki (2007) Modified classical homogeneous nucleation theory and a new minimum in free energy change 1. A new minimum and Kelvin equation. Fluid Phase Equilibria 254, 67-74 Waterman, L. S. (1983) Comments on The Montsouris series of carbon dioxide concentration measurements, 1877-1910 by Stanhill . Climate Change 5, 413-415... 

Most surprisingly, Helfand and Tagami (1972) found evidence for a significant rarefication of matter at the interface of incompatible blends. The density profile for the polystyrene/poly(methyl methacrylate) interface is shown in Figure 13.10. They found that the shape of the density profiles depends on both energetic and entropic factors. For the case of macromolecules near the interface, a minimum in free energy must be struck... 

Adsorption is an entropically driven process by which molecules diffuse preferentially from a bulk phase to an interface. Due to the affinity that a surfactant molecule encounters towards both polar and non-polar phases, thermodynamic stability (i.e. a minimum in free energy or maximum in entropy of the system) occurs when these surfactants are adsorbed at a polar/non-polar (e.g. oil/water or air/water) interface. The difference between solute concentration in the bulk and that at the interface is the surface excess concentration. The latter... 

Using those three molecular parameters, all of which can be measured or calculated with some degree of accuracy, the geometric approach allows one to predict the shape and size of aggregates that will produce a minimum in free energy for a given surfactant structure. 

Figure 12.12 I Chemical reactions always proceed toward a minimum in free energy. The left-hand plot illustrates what this means for a reaction with AG° < 0. In this case, the equilibrium constant wiU be greater than one, and equilibrium wiU favor products. A mixture of reactants will react to form products, whereas a mixture heavy in products will react backward to form reactants. The right-hand figure shows a reaction with AG° > 0. Here the equilibrium constant will be less than one, and the situation wiU be reversed. In either case, equilibrium will be the point at which free energy is minimized.

Spontaneous reactions are associated with a decrease in the free energy, AG < 0. Chemical equilibrium therefore corresponds to the minimum in free energy,... 

An energy diagram illustrating that AG is dependent on concentration. A minimum in free energy is achieved at particular concentrations (equilibrium). 

The phase behavior of polymer blends and solutions is, like any other mixtures, governed by enthalpic interactions between the different units and entropic effects, as described in thermcxlynamics [1]. The stable phase is determined from the minimum in free energy. 

While a well-defined equilibrium situation allows a clear characterization of fleq. the displacement of the contact line (which evidently implies a velocity of displacement) requires a deviation from this value, resulting in a dynamic contact angle (see O Fig. 5.4). In fact, whenever on ideal surfaces the contact angle deviates from 0eq (which represents the absolute minimum in free energy of the system) the contact line will show the tendency to move in order to try to reestablish equilibrium. In general, the stronger deviates from ( eq> the faster will be the movement of the contact line (see Fig. 5.5). Thus, for fldyn 0eq> instead of a perfect balance of interfacial tensions at equilibrium, a net force F (per unit length of the contact hne) remains ... 

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